CHAPTER V

Universal Enveloping Algebras

In this chapter we shall define the concept of the universal enveloping (associative) algebra of a Lie algebra . The principal function of is to reduce the theory of representations of to that of representations of the associative algebra . An important property of is that is isomorphic to a subalgebra of _L. In this way one can obtain a faithful representation of every Lie algebra. For finite-dimensional Lie algebras we shall obtain in the next chapter a sharpening of this result, namely, that every such algebra has a faithful finite-dimensional representation. In this chapter we obtain the basic properties of . Some of these will be used to prove an important formula due to Campbell and Hausdorff on the product of exponentials in an associative algebra. We shall give also the Cartan-Eilenberg definition of the cohomology groups of a Lie algebra.

The discussion here will not be confined to finite dimensional algebras nor to algebras of characteristic zero. In fact, a part of the chapter will be devoted to some notions which are peculiar to the characteristic p ≠ 0 case. The main concept here is that of a restricted Lie algebra of characteristic p, which arises in considering subspaces of an associative algebra which are closed relative to the mapping a → a^p as well as the Lie product [ab] = ab − ba. Restricted Lie algebras have restricted representations, restricted derivations, etc., and we can define a “restricted” universal enveloping algebra called the u-algebra. We discuss these notions briefly and consider the theory of abelian restricted Lie algebras.

1. Definition and basic properties

The central notion of this chapter—the universal enveloping algebra of a Lie algebra—is a basic tool for the study of representations and more generally for the study of homomorphisms of a Lie algebra into a Lie algebra _L where is associative with an identity element. Any such homomorphism can be “extended” to a homomorphism of the (associative) universal enveloping algebra into . Throughout this chapter we shall be concerned with Lie algebras and with associative algebras containing identities. We recall our conventions (of Chapter 1) on terminology: “algebra” will mean associative algebra containing an identity element 1, “subalgebra” means subalgebra in the usual sense containing 1, and “homomorphism” for algebras means homomorphism in the usual sense mapping 1 into 1.

DEFINITION 1. Let be a Lie algebra (arbitrary dimensionality and characteristic.) A pair (, i) where is an algebra and i is a homomorphism of into _L is called a universal enveloping algebra of if the following holds: If is any algebra and θ is a homomorphism of into _L, then there exists a unique homomorphism θ′ of into such that θ = iθ′. Diagramatically, we are given

where i and θ are homomorphisms of and we can complete this diagram to the commutative diagram

where θ′ is a homomorphism of .

A number of important properties of (, i) are built into this definition, or are easy consequences of basic facts on representations. We state these in the following

THEOREM 1

1. Let (, i), (, j) be universal enveloping algebras for . Then there exists a unique isomorphism j′ of onto such that j = ij′.

2. is generated by the image i.

3. Let ₁, ₂ be Lie algebras with (_i, i₁, (₂, i₂) respective universal enveloping algebras and let α be a homomorphism of ₁ into ₂. Then there exists a unique homomorphism α′ of ₁ into ₂ such that αi₂ = i₁α′ that is, we have a commutative diagram:

4. Let be an ideal in and let be the ideal in generated by _i. If l ∈ then j: l + → li + is a homomorphism of / into _L where = / and (, j) is a universal enveloping algebra for /.

5. has a unique anti-automorphism π such that iπ = –i. Moreover, π² = 1.

6. There is a unique homomorphism δ′ called the diagonal mapping of into ⊗ such that aiδ′ ai⊗ 1 + 1 ⊗ ai, a ∈ .

7. If D is a derivation in a unique derivation D′ in such that Di = iD′.

Proof:

1. If we use the defining property of (, i) and the homomorphism θ = j of into _L we obtain a unique homomorphism j′ of into such that j = ij′. Similarly, we have a homomorphism i′ of into such that i = ji′. Hence; = ji′j′ and i = ij′i′. On the other hand, we have j = jl_v where 1_v is the identity mapping in . If we apply the uniqueness part of the defining property of (, j) to θ = j we see that i′j′ = 1_v. Similarly, j′i′ = 1_u the identity in . It follows that j′ is an isomorphism of onto .

2. Let be the subalgebra generated by i. The mapping i can be considered as a mapping of into _L. Hence there is a unique homomorphism i′ of into such that i = ii′. Since i = i1_u and i′ can be considered as a mapping of into , the uniqueness condition gives i′ = l_u. Hence = 1_u = i′ . Hence = .

3. If α is a homomorphism of _i into ₂, αi₂ is a homomorphism of _i into _2L. Hence there exists a unique homomorphism α′ of ₁ into ₂ such that i₁α′ = αi₂.

4. We note first that the mapping l → li + of into = / is a homomorphism of into _L. Since i is mapped into 0 by this homomorphism. Hence we have an induced homomorphism l + → li + of / into . This is the mapping j. Now let θ be a homomorphism of / into _L, an algebra. Then η: l → (l + )θ is a homomorphism of into _L. Hence there exists a homomorphism η′ of into such that iη′ = η. If b ∈ , bη = 0 so that bi is in the kernel of the homomorphism η′. It follows that is in the kernel of η′ and consequently we have the induced homomorphism into . Now and . Hence θ = jθ′ as required. It remains to prove that θ′ is unique. This will follow by showing that (/)j; generates . Now, by 2., is generated by i, which implies that is generated by the cosets li + . Since it follows that is generated by the set of elements (l + )j, that is, by (/)j. This completes the proof of 4.

5. If μ′ is an anti-homomorphism of an algebra into an algebra , then one verifies directly that — μ′ is a homomorphism of _L into _L. Let be an algebra anti-isomorphic to under a mapping u → uμ′ of . (Such algebras exist trivially). The mapping θ = –iμ′ is a homomorphism of into _L. Hence there exists a homomorphism θ′ of into such that θ = -iμ′ = iθ′. Let π = θ′ (μ′)^-1. Then π is an anti-homomorphism of into itself such that iπ = –i. Hence iπ² = i and since i generates and π² is a homomorphism, π² = 1. Hence π is an anti-automorphism in . The uniqueness of π is also immediate from the fact that i generates .

6. The reasoning used to define the product of representations and modules (§1.6) shows that is a homomorphism of into (⊗)_L. Hence there exists a unique homomorphism δ′ of into ⊗ such that .

7. Let D be a derivation in . We form the algebra ₂ of 2 × 2 matrices with entries in the universal enveloping algebra and we consider the mapping

of into ₂. This is a linear mapping and

Hence θ is a homomorphism of into _2L. It follows that there is a homomorphism θ′ of into ₂ such that θ = iθ′. Since

and the ai generate , we have for any x ∈ ,

where y is uniquely determined by x. We write y = xD′ and a calculation like (2) shows that D′ is a derivation in . Then (3) shows that aiD′ = aDi. Hence iD′ = Di as required. The uniqueness of D′ follows from the fact that i generates , and a derivation is determined by its effect on a set of generators.

We now give a construction of a universal enveloping algebra. Let denote the tensor algebra based on the vector space . By definition,

where times. The vector space operations in are as usual and multiplication in is indicated ⊗ and is characterized by

Let be the ideal in which is generated by all the elements of the form

and let = /. Let i denote the restriction to = ₁ of the canonical homomorphism of onto . We have

Hence i is a homomorphism of into _L. We shall now prove

THEOREM 2. (, i) is a universal enveloping algebra for .

Proof. We recall first the basic property of the tensor algebra, that any linear mapping θ of into an algebra can be extended to a homomorphism of into . Thus let {u_j\j ∈ J} be a basis for , then it is well-known and easy to prove that the distinct “monomials” u_j1 ⊗ u_j2 ⊗ … u_jn of degree n form a basis for _n. Here u_j1 ⊗ u_j2 ⊗ … ⊗ u_jn = u_k1 ⊗ u_k2 ⊗ … ⊗ u_kn if and only if j_r = k_r, r = 1, 2, …, n. The element 1 and the different monomials of degrees 1,2, … form a basis for . It is easy to check that the linear mapping θ′′ of into such that 1θ′′ = 1, (u_j1 ⊗ u_j2 ⊗ … ⊗ u_jn)θ′′ = (u_j1 θ)(u_j2 θ) … (u_jnθ) is a homomorphism of into such that . Now let θ be a homomorphism of into _L and let θ′′ be its extension to a homomorphism of into . If a, b ∈ ,

Hence the generators (7) of belong to the kernel of θ′′. We therefore have an induced homomorphism θ′ of into such that aiθ′ = (a + )θ′ = aθ′′ = aθ. Thus θ = iθ′ as required. The tensor algebra is generated by and this implies that is generated by i. Since two homomorphisms which coincide on generators are necessarily identical there is only one homomorphism θ′ such that iθ′ = θ.

2. The Poincaré-Birkhoff-Witt theorem

We have noted that if {u_i | j ∈ J}, where J is set, is a basis for , then the monomials u_jl ⊗ u_j2 ⊗ … ⊗ u_jn of degree n form a basis for _n., n 1. We suppose now that the set J of indices is ordered and we proceed to use this ordering to introduce a partial order in the set of monomials of any given degree n 1. We define the index of a monomial u_j1 ⊗ u_j2 ⊗ … ⊗ u_jn as follows.

For i, k, i < k, set

and define the index

Note that ind = 0 if and only if j₁ j₂ … j_n. Monomials having this property will be called standard. We now suppose j_k > j_k+1 and we wish to compare

where the second monomial is obtained by interchanging uj_k, uj_k+1. Let denote the η’s for the second monomial. Then we have if if and Hence

We apply these remarks to the study of the algebra = / for which we prove first the following

LEMMA 1. Every element of is congruent mod to a Φ-linear combination of 1 and standard monomials.

Proof: It suffices to prove the statement for monomials. We order these by degree and for a given degree by the index. To prove the assertion for a monomial u_j1 ⊗ u_j2 ⊗ … ⊗ u_jn it suffices to assume it for monomials of lower degree and for those of the same degree n which are of lower index than the given monomial. Assume the monomial is not standard and suppose j_k > j_k+1. We have

The first term on the right-hand side is of lower index than the given monomial while the second is a linear combination of monomials of lower degree. The result follows from the induction hypothesis.

We wish to show that the cosets of 1 and the standard monomials are linearly independent and so form a basis for . For this purpose we introduce the vector space _n with the basis u_i1u_i2 … u_in, i₁ i₂ … i_n, i_i ∈ J, and the vector space . The required independence will follow easily from the following

LEMMA 2. There exists a linear mapping σ of into such that

Proof : Set 1σ = 1 and let _{n, i} be the subspace of _n spanned by the monomials of degree n and index j. Suppose a linear mapping σ has already been defined for satisfying (9) and (10) for the monomials in this space. We extend σ linearly to by requiring that (u_i1 ⊗ u_i2 ⊗ … ⊗ u_in)σ = u_i1u_i2 … u_in for the standard monomials of degree n. Next assume σ has already been defined for satisfying (9) and (10) for the monomials belonging to this space and let u_j1 ⊗ … ⊗ u_jn be of index i 1. Suppose j_k > j_k+1. Then we set

This makes sense since the two terms on the right are in . We show first that (11) is independent of the choice of the pair (j_k j_k+1), j_k > j_k+1. Let (j_i j_i+1) be a second pair with j_l > j_i+1. There are essentially two cases: I. l > k + 1, II l = k + 1.

I. Set u_jk = u, u_jk+1 = v, u_jl = w, u_il+1 = x. Then the induction hypothesis permits us to write for the right hand side of (11)

If we start with (j_lj_{l + 1}) we obtain

This is the same as the value obtained before.

II. Set u_jk = u, u_jk+1 = v, u_jl = w, u_ij+1 = w. If we use the induction hypothesis we can change the right hand side of (11) to

Similarly, if we start with

we can wind up with

Hence we have to show that σ annihilates the following element of

Now, it follows easily from the properties of σ in Φ1 ⊕ … ⊕ _n-i that if where a, b ∈ ₁, then

Hence σ applied to (14) gives

Since [[vw]u] + [v[uw]] + [[uv]w] = [[vw]u] + [[wu]v] + [[uv]w] = 0, (16) has the value 0. Hence in this case, too, the right hand side of (11) is uniquely determined. We now apply (11) to define σ for the monomials of degree n and index i. The linear extension of this mapping to the space _{n, i} gives a mapping on satisfying our conditions. This completes the proof of the lemma.

We can now prove the following

THEOREM 3 (Poincaré-Birkhoff- Witt). The cosets of 1 and the standard monomials form a basis for = /.

Proof : Lemma 1 shows that every coset is a linear combination of 1 + and the cosets of the standard monomials. Lemma 2 gives a linear mapping σ of into satisfying (9) and (10). It is easy to see that every element of the ideal is a linear combination of elements of the form

Since σ maps these elements into 0, σ = 0 and so σ induces a linear mapping of = / into Since (9) holds, the induced mapping sends the cosets of 1 and the standard monomial u_i1 ⊗ … ⊗ u_in into 1 and u_i1u_i2 … u_in respectively. Since these images are linearly independent in , we have the linear independence in of the cosets of 1 and the standard monomials. This completes the proof.

COROLLARY 1. The mapping i of into is 1: 1 and

Proof : If (u_j) is a basis for over Φ, then 1 = 1 + and the cosets u_ji = u_j + are linearly independent. This implies both statements.

We shall now simplify our notations in the following way: We write the product in in the usual way for associative algebras: xy. We write 1 for the identity in and we identify with its image i in . This is a subalgebra of _L since the identity mapping is an isomorphism of into _L. Also generates and the Poincaré- Birkhoff-Witt theorem states that if {u_j | j ∈ J}, J ordered, is a basis for , then the elements

form a basis for . In particular, if has the finite basis u₁, u₂, …, u_n then the elements

(u⁰_i = 1) form a basis for . The defining property of can be re-stated in the following way: If θ is a homomorphism of into _L, an algebra, then θ can be extended to a unique homomorphism θ (formerly θ′) of into . In particular, a representation R of can be extended to a unique representation R of . This implies that any module for can be considered in one and only one way as a right -module in which is as defined for as -module. Conversely, the restriction to of a representation of is a representation of and any right -module defines a right -module on restricting the multiplication to . In the sequel we shall pass freely from -modules to -modules and conversely, without comment.

The Poincaré-Birkhoff-Witt theorem (hereafter referred to as the P-B-W theorem) gives a characterization of the universal enveloping algebra in the following sense: Let be a subalgebra of _L, an algebra having the property that if {u_j | j ∈ J} is a certain ordered basis for , then the elements 1 and the standard monomials u_i1 u_i2 … u_ir, i₁ i₂ … i_r, form a basis for . Then (and the identity mapping) is a universal enveloping algebra for . Thus we have a homomorphism of the universal enveloping algebra into which is the identity on . The condition shows that this is 1: 1 and surjective. Hence can be taken as a universal enveloping algebra. Now suppose that is a subalgebra of and let be the universal enveloping algebra of . We may choose an ordered basis for so that and is an ordered basis for . Let be the subalgebra of generated by (or by the u_k.) Then it is clear that 1 and the standard monomials u_klu_k2 … u_ks, k₁ k₂ … k_s, form a basis for . Hence can be taken to be the universal enveloping algebra of .

Next assume is an ideal in and let the notations be as before. Let be the ideal in generated by . By Theorem 1, part 4., we know that = / and the mapping a + → a + define a universal enveloping algebra for /. By Corollary 1, a + → a + is 1:1 so we may identify / with the subalgebra ( + )/ of _L. This subalgebra is the set of cosets a + and it has the basis . Hence by the P-B-W theorem, the cosets 1 + and u_l1u_l2 … u_ll + , l_l l₂ … l_l, form a basis for . Hence if is the subspace spanned by the elements 1, and the standard monomials u_l1u_l2 … u_ll then . We note next that any standard monomial of the form

where s 1 and t 0, is in and these monomials together with 1 and u_l1 … u_lt l₁ l₂ … l_t, constitute all the elements of the standard basis for . It follows that the elements (19) form a basis for .

The main results on subalgebras and ideals we have noted can be stated as follows:

COROLLARY 2. Let be a Lie algebra, its universal enveloping algebra. If is a subalgebra of , then the subalgebra of generated by can be taken to be the universal enveloping algebra of . If is an ideal in , then / can be identified with ( + )/ where is the ideal in generated by and is the universal enveloping algebra of /. Moreover, the set of standard monomials forms a basis for , if (u_j) is an ordered basis for such that (u_k) is a basis for .

If we take = in this corollary we see that the ideal ₀ generated by in has the basis consisting of the standard monomials u_i1u_i2 …, u_ir, i₁ i₂ … i_r. Since these elements together with 1 form a basis for we have the following:

COROLLARY 3. Let ₀ be the ideal in generated by . Then = .

Since the imbedding of in is 1 : 1 and since any algebra has a faithful representation, we have:

COROLLARY 4. Any Lie algebra has a faithful representation by linear transformations.

The main properties of given in Theorem 1 which remain to be re-stated are:

2′. generates .

3′. Any homomorphism of a Lie algebra ₁ into a Lie algebra ₂ can be extended to a unique homomorphism of the universal enveloping algebra ₁ of ₁ into the universal enveloping algebra ₂ of ₂.

5′. There exists a unique anti-automorphism π of satisfying aπ = —a for a ∈ .

6′. There exists a homomophism δ (the diagonal mapping) of into ⊗ such that aδ = a ⊗ 1 + 1 ⊗ a, a ∈ .

7′. Any derivation D of has a unique extension to a derivation D of .

We prove next:

COROLLARY 5. The diagonal mapping δ of into ⊗ is 1:1.

Proof: The decomposition gives a decomposition where 1 ⊗ ₀ is the subspace of elements is the subspace of elements and is the subspace of elements . We have . Since ₀ ⊗ ₀ is an ideal in ⊗, it is easy to prove by induction on r that

(u_i1u_i2 … u_ir)δ ≡ u_ir … ⊗ 1 + 1 ⊗ u_i1 … u_ir     (mod ₀ ⊗ ₀)

where (u_i) is a basis for . Is follows that the set of images under δ of the standard basis 1, u_i1 … u_ir, i₁ i₂ … i_r, is linearly independent. Hence δ is 1:1.

Examples. (1) Let have the basis u, v with [uv] = u. Then has the basis vⁱu^k and we have the commutation relation

Hence

The elements of are the polynomials

where the . Multiplication for such polynomials is defined in the usual way except that we have

which is a consequence of (21). This is a type of ring of difference polynomials.

(2) Let be abelian with basis (u_i). Then is commutative and has the basis 1 and the monomials u_i1u_i2 … u_in, i₁ i₂ … i_n. This means that the u_j are algebraically independent and is the polynomial algebra in these u_j.

3. Filtration and graded algebra

An algebra is said to be graded if where _i is a subspace and . An example is the tensor algebra where we take . Another example is the algebra of polynomials in algebraically independent elements in which _i is the space of homogeneous elements of degree i. If is any graded algebra, then the elements of _i are called homogeneous of degree i and every a ∈ has a unique representation and a_i = 0 for all but a finite number of i. The a_i are called the homogeneous parts of a. A left (right) ideal of is called homogeneous if . This is equivalent to saying that b ∈ if and only if its homogeneous parts b_i ∈ .

An algebra is said to be filtered if for each non-negative integer i there is defined a subspace ⁽ⁱ⁾ such that (1) i j: (2) (3) . If is graded and we set then this defines a filtration in and becomes a filtered algebra. Another standard way of obtaining a filtered algebra is the following. Let be any algebra and let be a subspace of which generates as an algebra. This implies that we can write where ⁱ is the subspace spanned by all products of i elements taken out of . Set ⁽ⁱ⁾ = . Then it is clear that these ⁽ⁱ⁾ define a filtration in . This applies in particular to the universal enveloping algebra of a Lie algebra and the space = of generators of .

An important notion associated with a filtered algebra associated graded algebra . One obtains this algebra by forming the vector space

where we take ^(-1) = 0. A multiplication in is defined component-wise by

if and then a_ia_j ≡ b_ib_j (mod ) Hence (24) gives a single-valued product for an element of _i and an element of _j with result in _i+j. This is extended by addition to . It is easy to see that this gives a graded algebra with _i as the space of homogeneous elements of degree i.

Let b ∈ and suppose . The element = b + ^(n-1) is a homogeneous element of degree n in and is called the leading term of b. If b = 0 we take the leading term to be 0. If b and c have the same leading term of degree n, then b — c ∈ ^(n-1). Any homogeneous element of is a leading term. If is a leading term of b and is a leading term of c, then either is the leading term of bc. These remarks imply that if is a left ideal in , then the set of sums of the leading terms of the elements of is a homogeneous left ideal in . We can now prove the following

THEOREM 4. Let be a filtered algebra with as the associated graded algebra. If has no zero divisors ≠ 0, then has no zero divisors ≠ 0. If is left (right) Noetherian, then is left (right) Noetherian (definition below).

Proof: Let . Suppose is the leading term of b, that of c. Then, by definition, and so . Then is the leading term of bc and this is ≠ 0. Hence bc ≠ 0. The statement that a ring is left Noetherian means that the ascending chain condition holds for left ideals of the ring. As is well known, this is equivalent to the property that every left ideal is finitely generated. Suppose has this property. Let be a left ideal in and let be the associated homogeneous left ideal in consisting of the sums of the leading terms of the elements b ∈ . By hypothesis, has generators and we may assume that _i is the leading term of b_i ∈ . We assert that the b_i generate , that is, every b ∈ has the form . We assume that the leading term of b is of degree n and we may suppose that the result holds for elements with leading terms of degree less than n. Now . By dropping some of the terms (equating homogeneous parts) we obtain where _j is homogeneous and is homogeneous of degree n. Then _j is the leading term of an element C_j and the leading term of Hence so that d is in the left ideal generated by the b_j. Hence b is in the left ideal generated by the b_j.

We prove next the following ring theoretic result which we shall apply to the universal enveloping algebra of a finite-dimensional Lie algebra.

THEOREM 5 (Goldie-Ore). Let be a ring (associative with 1) without zero-divisor ≠ 0 satisfying the ascending chain condition for left ideals. Then has a left quotient division ring (definition below).

Proof: Goldie’s part of this result is that any two non-zero elements a,b ∈ have a non-zero common left multiple m = b′a = a′b. This is equivalent to saying that the intersection of the principal left ideals . To prove this we consider (following Lesieur and Croisot) the sequence of ideals By the ascending chain condition there exists a k such that Then x_i ∈ . Since has no zero divisors ab^k+1 ≠ 0 so not every x_i = 0.

If x_h is the first one of these which is not zero we have ab^k+1 = . Cancellation of b^h gives

Hence has the (left) common multiple property. Now Ore has shown that any ring without zero divisors ≠ 0 has a left quotient division ring if and only if every pair of non-zero elements a, b have a non-zero common left multiple m = a′b = b′a. We recall that is called a left quotient division ring for if: (1) is a division ring; (2) is a subring of ; (3) every element of has the form a^-1b, a, b ∈ (cf. Jacobson [1], p. 118.)

We now apply our results to the universal enveloping algebra of a Lie algebra . As before, we employ the filtration of defined by

Let = G() be the associated graded algebra. It is easy to see from the definition of G() that, since generates generates . It follows that if {u_j\j ∈ J, ordered, i_s a basis for , then the cosets generate . We have and and . Hence . Thus the generators commute and consequently is a commutative algebra. It follows that every element of is a linear combination of the elements . It is easily seen from the definition and the Poincaré-Birkhoff-Witt theorem that the different “standard” monomials indicated here form a basis for . This means that the are algebraically independent and is the ordinary algebra of polynomials in these elements. The general results we have derived and the properties of polynomial rings now give the following theorem on .

THEOREM 6

1. The universal enveloping algebra of any Lie algebra has no zero divisors ≠ 0.

2. If is finite-dimensional, then satisfies the ascending chain condition for left or right ideals and has a left or right quotient division ring.

Proof:

1. Since is a polynomial ring in algebraically independent elements over a field, has no zero divisors ≠ 0. Consequently has no zero-divisors ≠ 0, by Theorem 4.

2. If has the finite basis u₁, u₂ …, u_n, then is the polynomial algebra in . This satisfies the ascending chain condition on ideals by the Hilbert basis theorem. Hence H is left and right (Noetherian) by Theorem 4. Hence has a left and right quotient division ring by the Goldie—Ore theorem.

4. Free Lie algebras

The notion of a free algebra (free Lie algebra) generated by a set can be formulated in a manner similar to that of the definition of a universal enveloping algebra of a Lie algebra. We define this to consist of a pair consisting of an algebra (Lie algebra ) and a mapping i of X into () such that if θ is any mapping of X into an algebra (Lie algebra ), then there exists a unique homomorphism θ′ of () into () such that θ = iθ′. It is easy to construct a free algebra generated by any set X. For this purpose one forms a vector space with basis X and one forms the tensor algebra based on . The mapping i is taken to be the injection of X into . Now let θ be a mapping of X into an algebra . Since X is a basis for , θ can be extended to a unique linear mapping of into and this can be extended to a unique homomorphism θ of into . Hence and the injection mapping of X into is a free algebra generated by X.

It is somewhat awkward to give a direct construction for a free Lie algebra generated by X. Instead, one obtains the desired Lie algebra by using the free algebra generated by X. Let denote the subalgebra of the Lie algebra _L generated by the subset X. Let θ be a mapping of X into a Lie algebra and let be the universal enveloping algebra of , which (by the Poincaré-Birkhoff- Witt theorem) we suppose contains . Then θ can be considered as a mapping of X into , so this can be extended to a homomorphism θ of into . Moreover, θ is a homomorphism of _L into _L and since θ maps X into a subset of , the restriction of θ to the subalgebra of generated by X is a homomorphism of into . We have therefore shown that θ can be extended to a homomorphism of into . Since X generates , θ is unique. Hence and the injection mapping of X into is a free Lie algebra generated by X.

We note next that (and the injection mapping) is the universal enveloping algebra of . Thus let θ be a homomorphism of into a Lie algebra _L, an algebra. Then there exists a homomorphism θ of into which coincides with the restriction of θ to X. Then θ is a homomorphism of _L into _L and so the restriction θ′ of θ to is a homomorphism of into_L. Since xθ′ = xθ for X ∈ X and X generates , it is clear that θ′ coincides with the given homomorphism θ of into_L. Thus we have extended θ to a homomorphism of into . Since generates it is clear that the extension is unique. Hence is the universal enveloping algebra of .

The two results which we have established can be stated in the following

THEOREM 7 (Witt). Let X be an arbitrary set and let denote the free algebra (freely) generated by X. Let denote the subalgebra of _L generated by the elements of X. Then is a free Lie algebra generated by X and is the universal enveloping algebra of .

For the sake of simplicity we shall now restrict our attention to the case of a finite set X = {x₁, x₂, …, x_r}. Then

and , and we write . The algebra is graded with (m-times) as the space of homogeneous elements of degree m. A basis for this space is the set of monomials of the form x_i1x_i2 … x_im, i_j = 1, 2, …, r; hence dim _m = r^m.

An element a ∈ is called a Lie element if a ∈ . We proceed to obtain two important criteria that an element of be a Lie element. We observe first that it is enough to treat the case in which the given element a is homogeneous. Thus consider the collection of linear combinations of the Lie elements of the form

i_j = 1, 2, …, r, m = 1, 2, …. The Jacobi identity shows that this subspace is a subalgebra of _L. Since it contains the xi it coincides with . We therefore see that every element of is a sum of homogeneous Lie elements. Hence an element is a Lie element if and only if its homogeneous parts are Lie elements. We see also that if a is a Lie element which is homogeneous of degree m then a is a linear combination of elements of the form (26).

Let ′ denote the ideal in . An element of is in ′ if and only if it is a linear combination of monomials . Since the different monomials of this type form a basis for ′ we have a linear mapping σ of ′ into such that

We consider also the adjoint representation of . Since is the universal enveloping algebra of , this representation can be extended to a homomorphism θ of into the algebra of linear transformations in the space . We have

This implies that

If a, b ∈ then

Thus the restriction of σ to ′ is a derivation in . We can use this result to obtain the following criterion.

THEOREM 8 (Dynkin-Specht-Wever). If Φ is of characteristic 0, then a homogeneous element a of degree m > 0 is a Lie element if and only if aσ = ma where a is the linear mapping of ′ into defined by (27).

Proof : The condition is evidently sufficient (even for characteristic p, provided p m). Now let a be a Lie element which is homogeneous of degree m. We use induction on m. We have seen that a is a linear combination of terms of the form (26). Hence it suffices to prove that aσ = ma for a as in (26). We have

which is the required result.

Our next criterion for Lie elements does not require the reduction to homogeneous elements. This is

THEOREM 9 (Friedrichs). Let be the free algebra generated by the x_i over a field of characteristic 0. Let δ be the diagonal mapping of that is, the homomorphism of into such that Then a ∈ is a Lie element if and only if

Proof: We have which implies that the set of elements a satisfying is a subalgebra of _L. This includes the x_i; hence it contains . Let y₁, y₂, … be a basis for . Then since is the universal enveloping algebra of the elements arbitrary, k_i 0 (y_i^o = 1) form a basis for . Hence the products

form a basis for ft ⊗ ft. We have

where * represents a linear combination of base elements of the form with The second through the (m + l)-st term do not occur in the expressions of this type for any other base element . It follows that in order that aδ shall be a linear combination of the base elements of the form 1 and 1 it is necessary that in the expression for a in terms of the chosen basis only base elements with one k_i = 1 and all the other k_j = 0 occur with nonzero coefficients. This means that a is a linear combination of the y_i; hence a ∈ . Hence if and only if a ∈

5 The Campbell-Hausdorff formula

We shall now use our criteria for Lie elements to derive a formula due to Campbell and to Hausdorff for the product of exponentials in an algebra. For this purpose we need to extend the free algebra to the algebra of formal power series in the x_i. More generally, let be any graded algebra with _i as the subspace of homogeneous elements of degree i, i = 0, 1, 2, …. Let be the complete direct sum of the spaces _i. Thus the elements of are the expressions such that if and only if a_i = b_i, i = 0, 1, 2, …. Addition and scalar multiplication are defined component-wise. We introduce a multiplication in by where

It is easy to check that is an algebra. Moreover, the subset of of elements such that is a subalgebra which may be identified with . The subset ⁽ⁱ⁾ of elements of the form a_i + a_i+1 + … is an ideal in and . We define a valuation in by setting |0| = 0, |a = 2^-i if a ≠ 0 and . Then we have the following properties:

This valuation makes a topological algebra. Convergence of series , is defined in the usual way. The non-archimedean property (iii) of the valuation implies the very simple criterion that x₁ + x₂ + … converges if and only if |x_i| → 0. It is clear that the subalgebra is dense in

If the characteristic of the base field is 0 and , then

and

are well-defined elements of (that is, the series indicated converge). A direct calculation shows that

Moreover, if , then

and

We consider all of this, in particular, for . The resulting algebra is called the algebra of formal power series in the x_i. We shall also apply the construction to the algebra ⊗ . Since , where _i is the subspace of homogeneous elements of degree i, . We have , Hence, if we set , then and . It follows that ⊗ is graded with as subspace of homogeneous elements of degree k. We can therefore construct the algebra .

If a , then and are elements of which we denote as a ⊗ 1 and 1 ⊗ a, respectively. The mappings are isomorphisms and homeomorphisms of into . We have [a ⊗ 1, 1 ⊗ b] = 0 for any . Also, in the characteristic 0 case we have exp (a ⊗ 1) = exp a ⊗ 1, exp (l ⊗ a) = l ⊗ exp a and similar formulas for the log function.

Let denote the subset of of elements of the form b₁ + b₂ + … where b_i is a Lie element in _i. It is clear that is a subalgebra of . The diagonal isomorphism δ of into ⊗ has an extension to an isomorphism of into . We note first that if then . This is immediate by induction on i, using the formulas . Hence if , then is a well-defined element of . We denote this as aδ and it is clear that δ: a → aδ is an isomorphism and homeomorphism of into . It is clear also from Friedrichs’ theorem applied to the a_i that in the characteristic 0 case , is in if and only if .

We assume now that r = 2 and we denote the generators as x, y and write . Assume also that the characteristic is 0. Consider the element exp x exp y of . We can write this as 1 + z, where , and we can write

where w = log (1 + z). We shall prove

PROPOSITION 1. The element w = log (exp x exp y) is a Lie element, that is,

Proof. Consider (exp x exp y)δ. We have

Hence and so

This shows that log (1 + z) satisfies Friedrichs’ condition for a Lie element. Hence log

Now that we know that w = log (1 + z) is a Lie element, we can obtain an explicit formula for this element by using the Specht- Wever theorem. We have

and

Hence

Since this is a Lie element, if we apply the operator σ to the terms of degree k we obtain k times the homogeneous part of degree k in this element. It follows that we have the following expression of log (1 + z) as a Lie element.

It is easy to calculate the first few terms of this series and obtain

6. Cohomology of Lie algebras The standard complex

In this section we shall give the Cartan-Eilenberg definition of the cohomology groups of a Lie algebra and we shall show that this is equivalent to the definition given in § 3.10. To obtain the Cartan-Eilenberg definition one begins with the field Φ which one regards as a trivial module for , that is, one sets . This module and all -modules can be considered as right modules for the universal enveloping algebra of , since every representation of has a unique extension to a representation of . The term -module will mean “right -module” throughout our discussion. We recall that a module is called free if it is a direct sum of submodules isomorphic to the module . One seeks a sequence of free -modules X₀, X₁, X₂ … with -homomorphism ∈ of X₀ into Φ and -homomorphisms d_i-1 of X_i into X_i-1 such that the sequence

is exact, that is, ∈ is surjective, the kernel, Ker ∈ = image, Im d₀, and for every i 1, Ker d_i-1 = Im d_i. A sequence (41) is called a free resolution of the module Φ. Now let be an arbitrary -, hence -, module. Let Hom , denote the set of -homomorphisms of X_i into . The usual definitions of addition and scalar multiplication can be used to make Hom (X_i, ) a vector space over Φ. If η_i ∈ Hom (X_i ), then d_iη_i ∈ Hom (X_i+1, ). We therefore obtain a mapping of Hom (X_i, ) into Hom . This mapping is linear. Moreover, the exactness of (41) implies that , and this implies that . Hence Im . One defines the i-th cohomology group of relative to the module to be the factor space Ker and Ker δ₀ for i = 0. The Cartan-Eilenberg theory shows that these groups are independent of the particular resolution (41) for Φ.

We shall not go into the details of this theory here but shall be content to give the construction of what appears to be the most useful resolution (41). The space we shall obtain is called the standard complex for . It will turn out that the standard complex has an algebra structure as well as a vector space structure. This has the consequence that in suitable circumstances one can define a cohomology algebra in place of the cohomology groups.

We now suppose we have a representation R of a Lie algebra by derivations in an algebra . Let be the universal enveloping algebra of . Then we shall define a new algebra (, , R), the algebra of differential operators of the representation R of . The space of (, , R) will be ⊗ . Since the mapping of elements v ∈ into their left multiplications is an anti-homomorphism, the mapping is a representation of acting in We now form the tensor product of this representation and the given representation R in . The resulting representation maps l into the linear transformation sending a ⊗ u into al^R ⊗ u – a ⊗ lu. We can extend this representation to a representation of , which we shall now write as πL where π is the anti-homomorphism of such that , and L is an anti-homomorphism of into the algebra of linear transformations of ⊗ . We write the image of v ∈ under L as L_v. Then we have (a ⊗) u)(l(πL)) = al^R ⊗ u – a ⊗ lu; hence,

Next if b ∈ , we define L_b to be the linear mapping in ⊗ such that

The mapping b → L_b is an anti-homomorphism of into the algebra of linear transformations in ⊗ . It follows from (42) and (43) that

Hence we have

which shows that the set of linear transformations L_b is an ideal in the Lie algebra of linear transformations of the form L_b + L_l.

It follows that the enveloping algebra of this set of transformations is the set of mappings of the form .

We have the canonical mapping of ⊗ into sending into since l^R is a derivation. Hence and, consequently, . This implies that so that, if , then . Thus the mapping is a vector space isomorphism. Since the set of mappings of the form is an algebra we can use this mapping to convert ⊗ into an algebra by specifying that our mapping is an algebra antiisomorphism. The resulting algebra whose space is ⊗ is the algebra (, , R) which we wished to define.

It is immediate from our definitions that the subset of (, , R) of elements of the form a ⊗ 1 is a subalgebra isomorphic to . We identify this with and write a for a ⊗ 1. Similarly, the set of elements of the form 1 ⊗ u, u ∈ , is a subalgebra isomorphic to . We identify this with and write u for 1 ⊗ u. The mapping of into sends a = a ⊗ 1 into L₁L_a = L_a and sends u = 1 ⊗ u into L_uL₁ = L_u. The formula (44) gives the following basic commutation formula in (, , R):

Since every element of has the form every element of (, , R) has the form . Also if and only if .

We establish next a “universal” property of the algebra (, , R) in the following

PROPOSITION 2. Let be an algebra and let θ_i be a homomorphism of into , θ₂ a homomorphism of into such that

holds. Then there exists a unique homomorphism θ of (, , R) into such that bθ = bθ₁, uθ = uθ₂.

Proof: We form which we consider as an algebra of pairs , with component addition and multiplication. We have the homomorphisms π₁, π₂ of this algebra onto (, , R) and respectively defined by (x, d)π₁ = x, (x, d)π₂ = d. Let be the subalgebra of the direct sum generated by the elements . The mapping π₁ induces a homomorphism of onto (, , R) and the mapping π₂ induces a homomorphism of onto the subalgebra of generated by the elements . By (45) and (46), we have the following relation in

This permits us to carry out a collecting process such as indicated in the discussion of the construction of the algebra (, , R) to write the elements of in the form . The property of implies that we have a vector space homomorphism of (, , R) onto sending into . The existence of this mapping implies that π₁ is an isomorphism of onto (, , R). Then is a homomorphism of (, , R) into such that and . The uniqueness of θ follows from the fact that and generate (, , R).

Remark. If is a set of generators for then the conclusion of Proposition 2 will hold provided that (46) holds for all b ∈ . This follows from the fact that the set of b satisfying this for all l ∈ is a subalgebra of —as can be verified directly.

We consider next an extension of the notion of a derivation:

DEFINITION 2. Let and be algebras, S₁ and S₂ homomorphisms of into . Then a linear mapping d of into is called an (S₁, S₂)-derivation if

It is easy to check that the conditions on d are just those which insure that the mapping

is a homomorphism of into the two-rowed matrix algebra ₂. (A special case of this remark was used in the proof of 7 of Th. 1.) If is a subalgebra of and s₁ is the inclusion mapping, then we call d an s₂-derivation and if s₂ is also the inclusion mapping, then we have a derivation of into . One can verify directly, or use (49) to see, that if d is an (s₁ s₂)-derivation, then the kernel of d is a subalgebra. It follows that d = 0 if d = 0 for a set of generators of . Also if d_i and d₂ are (s₁ s₂)-derivations then d₁ d₂ is an (s₁, s₂)-derivation. Hence it is clear that d₁ = d₂ if xd₁ = xd₂ for every x in a set of generators.

We consider again the algebra (, , R) and we can prove the following result on derivations for this algebra.

PROPOSITION 3. Let be an algebra, s₁ s₂ homomorphisms of (, , R) into . Let d₁ be an (s_l s₂)-derivation of into and d₂an (s_i, s₂)-derivation of into (s_i, s₂ are the restrictions to , respectively of s_i, s₂ on (, , R).) Suppose that for b ∈ , l ∈ :

Then there exists a unique (s₁, s₂)-derivation d of (, , R) into such that bd = bd₁, ud = ud₂.

Proof: Consider the mappings and θ₁ and θ₂ of and , respectively, such that

Since d_i is an (s₁, s₂) -derivation, θ₁ and θ₂ are homomorphisms into the matrix algebra ₂. A direct calculation shows that the condition (46) is a consequence of (50). Hence Proposition 2 implies that there exists a homomorphism θ of (, , R) into ₂ such that bθ = . It is clear that θ has the form

Set y = xd; then since θ is a homomorphism, d is an (s₁, s₂) -derivation of (, , R) into . We have bd = bd₁, ud = ud₂, as required. The uniqueness is clear.

The proof of Proposition 3 and the remark following Proposition 2 show that it is enough to suppose that (50) holds for all b in a set of generators of . We shall need this sharper form of Proposition 3 later on.

Let be a vector space over Φ and let E() be the exterior algebra (or Grassmann algebra) over . We recall that E is the difference algebra of the tensor algebra with respect to the ideal generated by the the elements x ⊗ x. It follows immediately that if s is a linear mapping of into an algebra and (xs)² = 0 for every x ∈ , then s can be extended to a unique homomorphism of E into . The canonical mapping of onto E is an isomorphism on Φ1 ⊗ . One identifies Φ1 + with its image. It is clear that generates E. We shall denote the multiplication in E simply as ab (rather than the more complicated a Λ b which is customary in differential geometry.) If x, y ∈ , then xy + yx = (x + y)² — x² — y² = 0. The algebra E is graded with ^m as the space E_m of homogeneous elements of degree m. If is an ordered basis for , then the set of monomials of the form is a basis for E_m. In particular, if dim = n, then E_m = 0 if m > n and dim if m n. Hence dim E = 2ⁿ. The exterior algebra has an automorphism η such that xη = — x for all x ∈ . If x_m ∈ E_m then x_mη = (–1)^m x_m.

We are interested in derivations and η-derivations of E() into algebras containing E as a subalgebra. The η-derivations will be called anti-derivations of E into . We shall need the following criterion.

PROPOSITION 4. Let be an algebra which contains the exterior algebra E = E() as a subalgebra. Let d be a linear mapping of into . Then d can be extended to a derivation (anti-derivation) of E into if and only if x(xd) + (xd)x = 0. (x(xd) — (xd)x = 0) for all x ∈

Proof: The conditions are necessary since x² = 0 in E implies x(xd) + (xd)x = 0 or x(xd) – (xd)x = 0 according as d is a derivation or an anti-derivation. Conversely, suppose the conditions hold. In the first case we consider the mapping

and in the second, the mapping

of into the matrix algebra ₂. In both cases one checks that (xθ)² = 0. Hence θ can be extended to a homomorphism of E. The rest of the argument is like that of Proposition 3.

It is clear that when the condition is satisfied the extension is unique.

We apply this first to the following situation. Let be a Lie algebra and a module for . Let E = E() be the exterior algebra defined by . If so that x(xl) + (xl)x = 0 in E. Hence there exists a unique derivation d_l in E such that xd₁ = xl, x ∈ . We have since is an -module. Since all the mappings considered here are derivations and generates E, we have in E. Consequently, l → d_i is a representation of by derivations in E().

We consider the special case of this in which = and the representation is the adjoint representation. We form the algebra where A denotes the extension of the adjoint representation to a representation by derivations in E(). There is a slight difficulty here in our notations since we now have two copies of , one contained in , the other in E. We shall therefore denote the elements of the copy in by and we write . We denote the corresponding elements in E by l and we denote this copy as . In order to avoid ambiguity we shall denote the Lie composition in —which does not coincide with the Lie product in E_l —by [l₁ ° l₂]. Then we have the Lie isomorphism of onto ; hence we have . Since and generate and E, respectively, generates X. The following relations connecting these generators are decisive:

l_i ∈ . The last of these is a consequence of (45).

The last relation in (51) implies that if x₁ ∈ E_i, then . It follows easily from this that for in . Hence . If we set , then we have . Also since X = it follows from the properties of tensor products that This shows that X is a graded algebra with X_i = E_i as the space of homogeneous elements of degree i.

PROPOSITION 5. There exists a unique automorphism η of X = (E(), , A) and a unique η-derivation d of X into X such that

Moreover, ηd + dη = 0 and d² = 0.

Proof: Evidently we have a homomorphism η₁ of E into X such that lη₁ = –l, l ∈ and a homomorphism η₂ of into X such that η = . The condition (46) for b = l₁, l = ₂, θ₁ = η₂ θ₁ = η₂ reads which is equivalent to . This holds by (51). Hence Proposition 2 can be applied to prove the existence and uniqueness of η. It is evident that the mapping d₂ = 0 is an η-derivation of into X. Since the condition of Proposition 4 is satisfied and so there exists an anti-derivation d₁ of E() into X such that ld₁ = , l ∈ . We now check (50) for b = l₁ l = ₂, s₂ = η, d₁, d₂ = d₂ The left-hand side is

The right-hand side is . Hence (50) holds for these elements, so by the remark following Proposition 3, there exists a unique η-derivation d in X satisfying the conditions (52). We have X₀d = 0 and where . This can be established by induction since X_i = X_i-1X₁. Also it is immediate that . Hence and . Hence . Since η² = 1 this shows that d² is a derivation. On the other hand, (52) implies that ld² = 0 = d². Hence d² = 0. This concludes the proof.

Let d_i-1, i = l, 2, …, denote the restriction of d to X_i. Then d_i-1 maps X_i into X_i-1 and d_id_i-l = 0. Since d = 0, d and the d_i commute with the right multiplications by elements of . Hence if we consider X_i in this way as a -module, then d_i is a -homomorphism. We have and by the properties of tensor products. Hence any Φ-basis for E_i is a set of free generators for X_i relative to . Thus every X_i is a free -module. We have X₀ = . If ′ denotes the ideal in generated by the , then is an ordered basis for , then we know that the elements form a basis for . If then we define a mapping ∈ of into Φ by . This mapping is a -homomorphism of X₀ = into Φ considered as -module where . Evidently ∈ is surjective.

Let b ∈ X₁. Then the basis for . Then and bd₀∈ = 0. Conversely, let ∈ X₀ and assume ∈ = 0. Then and = bd₀ for . We have therefore shown that is exact. It remains to show that Ker d_i-1 = Im d_i, i 1. For this purpose we consider first the case in which is abelian; hence is the polynomial algebra in the which commute and are algebraically independent. Also in the abelian case . Hence the elements of commute with those of E(). Consequently, X is the tensor product of E and in the sense of algebras.

Now let where , are subspaces, hence subalgebras of the abelian Lie algebra . Let . It is easy to see that F can be identified with the subalgebra of E generated by , with the subalgebra of generated by and Y with the subalgebra of X generated by . Similar statements hold for . These results follow easily by looking at bases. Similarly, we leave it to the reader to check that X = YZ where Y and Z are the subalgebras we have indicated and that we have a vector space isomorphism of Y ⊗ Z onto X sending y ⊗ z into yz, y ∈, Y, z ∈ Z. This result implies that if μ is a linear mapping in Y and v is a linear mapping in Z then we have a unique linear mapping λ in X such that .

We now regard the augmentation ∈ of X₀ into Φ as a mapping of X₀ into the subalgebra Φ1 of X and we extend this to a linear mapping in X such that . This is an algebra homomorphism of X onto the subalgebra Φ. We now prove

PROPOSITION 6. There exists a linear mapping D in X = X(), abelian, such that Dd + dD = 1 — ∈.

Proof: Suppose first that dim = 1 and (u) is a basis for . Then X has the basis . We have . Hence if we define D to be the linear mapping such that , then one checks that dD + Dd = 1 — ∈ as required. We note also that is the automorphism previously defined in . Moreover, we have the relations ∈d = 0 = d∈ and ∈η = ∈ = η∈. Now suppose where is one-dimensional. Then X = YZ, Y = X(), Z = X() where Y and Z are the algebras defined before. It is clear that Y and Z are invariant under d, ∈ and η, and that their restrictions are just the corresponding mappings in X() and X(). Let D₂ be the mapping just defined in Z and let D₁ be any mapping in Y such that dD₁ + D₁d = 1 — ∈ in Y. There exists a unique linear mapping D in X such that

Then we have

Hence,

The inductive step we have just established implies the result in the finite-drmensional case by ordinary induction and in the infinitedimensional case by transfinite induction or by Zorn’s lemma.

This proposition implies the exactness of Thus let satisfy xd = 0. Then x = x(l — ∈) = x(dD + Dd) = (xD)d ∈ Im d. This implies that Ker d_i-1 = Im d_i, i l.

We consider now the general case of an arbitrary Lie algebra and we introduce a filtration in X using the space of generators . Thus we define . Since is a subcomplex. Also it is clear that X^(j) is a sum of homogeneous subspaces relative to the grading in X. We can form the difference complex X^(j)/X(j-l) which has a grading induced by that of X. If is an ordered basis for then the cosets relative to X^(j-l) of the base elements h + k = j form a basis for X^(j) / X_(j-1). We identify these cosets with the base elements. Then we can say that d in X^(j) / X_(j-1) is determined by the rule

where the ^ indicates that u_jq is omitted, and the order of the subscripts in the product of the ’s is non-decreasing. This differentiation does not make use of the structure of . Thus it is the same as that which one obtains in the abelian Lie algebra. In the latter case the space spanned by the monomials on the left-hand side of (54) satisfying h + k = j fixed form a subcomplex of X and X is a direct sum of these subcomplexes for j = 0, 1, …. It follows that if x is in one of these subcomplexes and is in X_i-1, i 1, and xd = 0, then x = yd, y ∈ X_i and where y is the given complex containing x. The result we wished to prove on the exactness of (41) will now follow by induction on the index j in X^(j) by the following

LEMMA 3. Let Y be a subcomplex of a graded complex such that Set . Suppose Ker d_i-1 = Im d_i, i holds in Y and in Z. Then this holds also in X.

Proof: Let satisfy xd = 0. Then (x + Y)d = 0. Hence there exists x′ + Y such that (x′ + Y)d = x + Y. Thus x′d x + y, y ∈ Y. Then yd = x′d² — xd = 0 and so there exists a y′ ∈ Y such that y = y′d. Hence x = x′d — y = x′d — y′d = (x′ — y′)d.

We have now completed the proof of the following

THEOREM 10. Let X = (E(), , A) be the algebra of differential operators determined by the exterior algebra of E() and the extension of the adjoint representation of to E. Let d be the anti-derivation in X defined in Proposition 5 and let X_i = E_i. Then is a free resolution of the module Φ.

It remains to show that the cohomology groups defined by this resolution coincide with those of § 3.10. Let there be a -homomorphism φ of X_i into a (right) -module . Let (l₁, l₂ …, l_i) be an ordered set of i elements of and define a mapping f of × … × (i times) into by f (l₁, …, l_i) = (l₁l₂ … l_i)φ. where l₁l₂ … l_i ∈ E_i X_i. It is clear that this f is multilinear and alternate and it is easy to see that if f is any multilinear alternate mapping of x … x into then there exists a linear mapping φ of E_i() into such that (l₁l₂ … l_i)φ = f(l₁, …, l_i). This φ has a unique extension to a -homomorphism of X_i into . We therefore have a 1 : 1 linear isomorphism of the space of -homomor- phisms of X_i into and the space of multilinear alternate mappings of the i-fold product set × … × into . If f is a mapping of the latter type we define fδ by

We have

and by (45)

Hence

It follows that

which are the same definitions as given in § 3.10.

7. Restricted Lie algebras of characteristic p

In many connections in which Lie algebras arise naturally one encounters in the characteristic p ≠ 0 case structures which are somewhat richer than ordinary Lie algebras. For example, let be an arbitrary non-associative algebra and let () be the set of derivations of . Then we know that () is a subalgebra of the Lie algebra of linear transformations in . We note also that one has the Leibniz formula

for any derivation D. This can be established by induction on k. Now assume the base field of , hence of () is of characteristic p and take k = p in (57). Then the binomial coefficients = 0 if . Hence (57) reduces to

which implies that D^p is a derivation. Thus () is closed under the mapping D → D^P as well as the Lie algebra compositions. Similarly, let be an associative algebra and a → an antiautomorphism in . Let be the subset of skew elements relative to a → . Then we know that is a subalgebra of _L. Moreover, if the characteristic is p then = — a implies that — a^p. Hence a^p ∈ , so again we have a Lie algebra closed under the p-mapping. The systems () and just considered are examples of restricted Lie algebras which we shall define abstractly.

For this purpose we need some relations connecting a^p with the compositions in a Lie algebra _L, associative of characteristic p. We recall first the following two identities in ϕ[λ, μ], λ, μ algebraically independent indeterminates, Φ of characteristic p:

The first of these is well-known and the second is a consequence of the first and the identity . These relations imply corresponding relations for commuting elements a, b in any associative algebra. In particular we may take a = b_R, b = b_L the right and left multiplications determined by an element b ∈ . These give

or,

Also it is clear that

and we shall use (61) to prove that

where is_i(a, b) is the coefficient of λ^i-1 in

λ an indeterminate. To prove this we introduce the polynomial ring [λ] and we write

where S_i(a, b) is a polynomial in a, b of total degree p. If we differentiate (65) with respect to λ we obtain

By (61), this gives

Thus we see that is_i(a, b) is the coefficient of λ^i-1 in a(ad (λa + b))^p-1. On the other hand, substitution of λ = 1 in (65) gives the relation (63). It is clear that S_i(a, b) is obtained by applying addition and commutation to a, b and so is in the Lie subalgebra of _L generated by a, b. For example,

These considerations lead to the following

DEFINITION 4. A restricted Lie algebra of characteristic p ≠ 0 is a Lie algebra of characteristic p in which there is defined a mapping a → ^[p] such that

where is_i(a, b) is the coefficient of λ^i-1 in a(ad (λa + b))^p-1 and

If is an associative algebra of characteristic p, then the foregoing discussion shows that defines a restricted Lie algebra in which the vector space compositions are as in , [ab] = ab – ba and a^[p] = a^p. We use the notation _L for this restricted Lie algebra (as well as for the ordinary Lie algebra). A homomorphism S of a restricted Lie algebra into a second restricted Lie algebra is, by definition, a mapping satisfying (a + b)^s = a^s + b^s, (ab)^s = αa^s, [ab]^s = [a^sb^s], (a^[p])^s = (a^s)^[p]. Ideals and subalgebras are defined in the obvious way. A representation of is a homomorphism of into the restricted Lie algebra the algebra of linear transformations of a vector space over Φ. If R is a representation acting in the space , then is an -module relative to xa ≡ xa^R, x ∈ , a ∈ . The module product xa satisfies the usual conditions in a Lie module and the additional condition that xa^[p] = (… (xa)a) … a, p a’s.

We consider now the following two basic questions: (1) Does every restricted Lie algebra have a 1 : 1 representation? (2) What are the conditions that an ordinary Lie algebra be restricted relative to a suitable definition of a^[p]? In connection with (2) it is clear that a necessary condition is that for every a ∈ the derivation (ad a)^p is inner; for, in a restricted Lie algebra (ad a)^p = ad a^[p]. We shall show that this condition is sufficient. In fact, we shall see that it will be enough to have (ad u_i)^p inner for every u_i in a basis {u_i} of . We note also that if is restricted relative to two p-mappings a → a^[p]₁ and a → a^[p]₂ then f(a) = a^[p]₁ – a^[p]₂ is in the center of since [b, f(a)] = [ba^[p]₁] – [ba^[p]₂] = b(ad a)^p – b(ad a)^p = 0. It is clear from (ii) and (i) that

A mapping of one vector space of characteristic p ≠ 0 into a second one having these properties is called a p-semi-linear mapping. Conversely, if is restricted relative to a → a^[p]₁ and a → f(a) is p-semi-linear mapping of into the center of , then is also restricted relative to a → a^[p]₂ ≡ a^[p]₂ + f(a). The kernel of a p-semi-linear mapping is a subspace. Hence if f(u_i) = 0 for u_i in a basis, then f = 0. It follows that if two p-mappings a → a^[p]₁ and a → a^[P]₂ making restricted coincide on a basis, then they are identical.

Suppose now that is a Lie algebra with the basis {u_i | i ∈ I} where I is an ordered set and let be the universal enveloping algebra of . The Poincaré-Birkhoff-Witt theorem states that the “standard monomials” form a basis for . We have the filtration of defined by . It is easy to see (induction on k and the usual “straightening’′ argument) that the monomials such that k₁ + k₂ + … + k_r k form a basis for ^(k). We assume now that for each base element u_i there exists a positive integer n_i and an element z_i in the center of such that

is in . Then we have the following

lEMMA 4. The elements of the form

such that i₁ < i₂ … < i_r, h_j 0, 0 λ_j < n_ij, r = 1, 2, 3, … form a basis for Vi.

Proof: We show first that every element is a linear combination of the elements (69). If k = k_i + k₂ + … + k_r, then we employ an induction on k. If every l_j λ n_ij then the result is clear. Hence we may assume k_j n_ij for some j. Then we may replace by U_ij + z_ij and obtain

The elements and the second term on the right are in ^(k-1). Hence these elements are linear combinations of the elements (69). Since the set of elements in (69) is closed under multiplication by any Zi, the result asserted is clear. We show next that the elements (69) are linearly independent. We replace z_ij by . This gives

if Hence the element , where * is a linear combination of standard monomials belonging to ^(k-1). It follows that if we have a non-trivial linear relation connecting the elements in (69), then we have such a relation for terms for which the “degree” k = is fixed. This gives a relation with the same coefficients in the corresponding standard monomials . Since the standard monomials are linearly independent we must have h_jn_ij + λ_j, = k_j j = 1, …, r, for the elements of (69) which appear in the relation with non-zero coefficients. Since λ_j < n_ij this implies that h_j, λ_j are determined by the equation h_jn_ij + λ_j = k_j. Hence there is just one term in the relation. This is impossible in view of (70). Hence the elements of (69) are linearly independent and so form a basis for .

We can now prove the following

THEOREM 11. Let be a Lie algebra of characteristic p ≠ 0 with ordered basis {u_i} such that for every u_i, (ad u_i)^p is an inner derivation. For each u_i let u^[p]_i be an element of such that (ad u_i)^p = ad u_i^[P]. Then there exists a unique mapping a → a^[p] of into such that u_i^[p] is as given and is a restricted Lie algebra relative to the mapping a → a^[p].

Proof: Let be the universal enveloping algebra of . Since commutes with every l ∈ . Hence Z_i is in the center of and u_i^p = z_i + u_i^[p] where . We can therefore conclude from the lemma that the elements

such that , form a basis for . Let denote the ideal in generated by the Z_i. Then it is clear that the subset of our basis consisting of the elements (71) with some h_j > 0 is a basis for . Hence the cosets of the elements , form a basis for the algebra = /. Since the canonical homomorphism is a homomorphism of _L into _L, the restriction to is a homomorphism of onto . Since the are linearly independent, is an isomorphism of onto . We note next that is a subalgebra of considered as a restricted Lie algebra. Thus we have . It follows from (63) and (62) that which proves the assertion. The isomorphism of and permits us to consider as a restricted Lie algebra by defining . Then we have so that as required. We now write l^[p] = l^[p]_i and the result is proved.

We recall the result of § 3.6 that if is a finite-dimensional Lie algebra with a non-degenerate Killing form, then every derivation of is inner. It is clear also that the center of is 0. Hence if the characteristic is p, then for every a ∈ there exists a unique element a^[p] such that the derivation (ad a)^[p] = ad a^[p]. It follows that we can introduce a p-operator in in one and only one way so that is a restricted Lie algebra. We therefore have the following

COROLLARY. If is a finite-dimensional Lie algebra of characteristic p ≠ 0 with non-degenerate Killing form, then there is a unique p-mapping in which makes restricted.

We suppose next that is an arbitrary restricted Lie algebra and we prove the following

THEOREM 12. Let be a restricted Lie algebra of characteristic p ≠ 0, the universal enveloping algebra, the ideal in generated by the elements a^p – a^[p], . Then the mapping is an isomorphism of into the restricted Lie algebra _L. If S is a homomorphism of into the restricted Lie algebra _L, an algebra (associative with 1), then there exists a unique homomorphism of into such that → a^s. If is of finite dimensionality n, then dim = pⁿ.

Proof: Since is a homomorphism of the restricted Lie algebra into the restricted Lie algebra _L. If {u_i is a basis for over Φ, then it is clear from the rules for p-powers and the operation a → a^[p] that a^p – a^[p] is a linear combination of the elements u^p_i – u^[p]_i. Hence is also the ideal generated by the elements u^p_i — u^[p]_i. The proof of the preceding theorem shows that the cosets are linearly independent; hence a → is an isomorphism of into _L. Now let S be a homomorphism of into a restricted Lie algebra _L, an algebra. Then we have a homomorphism of into sending a ∈ into a^s. under this mapping a^p → (a^s)^p = (a^[p])^s. Thus a^p – a^[p] is in the kernel and so is in the kernel. Consequently, we have an induced homomorphism of into such that → a^s. If has the finite basis u₁,u₂, …, u_n then we have seen that the cosets of the elements form a basis for . This proves the last statement of the theorem.

The algebra = / of the theorem will be called the u-algebra of the restricted Lie algebra . It plays the same role for as restricted Lie algebra as is played by for considered as an ordinary Lie algebra. In particular, a representation of as restricted Lie algebra defines a representation of and conversely. Since has a faithful representation it follows that every restricted Lie algebra has a faithful representation. Moreover, if is finitedimensional, then is finite-dimensional and so has a faithful representation acting in a finite-dimensional space. Consequently this holds also for .

8. Abelian restricted Lie algebras

An abelian restricted Lie algebra is a vector space with a mapping a → a^p (we use this notation for the p-operator from now on) in such that

The theory of these algebras is a special case of the theory of semi-linear transformations. This is equivalent to a theory of modules over certain types of non-commutative polynomial domains (cf. Jacobson, Theory of Rings, Chapter 3). In the present instance the polynomial ring is the set of polynomials α₀ + tα₁ + … + t^mα_m, α_i ∈ Φ, t an indeterminate such that αt = tα^p. If Φ is perfect it can be shown that the ring has no zero divisors and every left ideal and right ideal in the ring is a principal ideal. The study of this ring and its modules is a natural tool for studying abelian restricted Lie algebras. However, we shall not undertake this here. Instead we shall derive one or two basic results on the algebras without using the polynomial rings.

THEOREM 13. Let be a finite-dimensional abelian restricted Lie algebra over an algebraically closed field of characteristic p. Assume that the p-mapping in is 1 : 1. Then has a basis (a₁, a₂, …, a_n) such that a^p_i = aⁱ

Proof: Let a ≠ 0 be in and let m be the smallest positive integer such that Then a, a^p, …, a^pm-1 are linearly independent and every a^pk is a linear combination of these elements. If α₁ = 0, set . Then which implies that a^pm-1 = contrary to the choice of m. Hence α₁ ≠ 0. This implies that β is a linear combination of a^p, a^p2, …. We shall now show that there exists a b = such that b^p = b. This will be the case if the β_i satisfy the following system of equations:

and not all the β_i are 0. Successive substitution gives

Since αa non-zero solution for₁ ≠ 0 and Φ is algebraically closed this has a non-zero solution for β_m. Then the remaining β_i can be determined from (73) for i m — 1 and the equation for i = m will hold by (74). Now suppose we have already determined a_l, a₂, …, a_r which are linearly independent and satisfy a^p_i = a_i Then is a subalgebra of Suppose a is an element of such that Since a is a linear combination of a^p, a^p₂, … it follows that . Thus we have shown that /₁ is a restricted Lie algebra satisfying the hypotheses of the theorem. Hence if ≠ ₁, then we can find a b such that b^p = b (mod ₁). Thus We can determine δ_i so that δ_i^p – δ_i + r_i = 0, i = 1, 2, …, r. Then satisfies . Hence the result follows by induction.

THEOREM 14. Let be a commutative restricted Lie algebra of finite-dimensionality over an algebraically closed field. Suppose the p-mapping a → a^p is 1:1 and let be a finite-dimensional module for . Then is completely reducible into one-dimensional sub- modules. If (a₁, a₂, …, a_n) is a basis such that a_i^p = a_i and a = then every weight in has the form in the prime field.

Proof: Let the basis (a₁, a₂, …, a_n) be as indicated and let a_i → A_i in the given finite-dimensional representation. Then A^p_i = A_i so the minimum polynomial of A_i is a factor of λ^p — λ = . Thus the minimum polynomial of A_i has distinct roots and these are in the prime field. It follows that there exists a basis (x₁, x₂, …, x_n) for such that x_iA_i = m_ijx_i, m_ij in the prime field. Since the A_i commute we can find a basis which has this property simultaneously for the A_i, i = 1,2, …, n. Then so that is a direct sum of the irreducible invariant subspaces Φx_j and the weights are .

Remark. It is easy to extend the first part of Theorem 14 to arbitrary base fields of characteristic p, that is, complete reduci- bility holds if the p-mapping is 1:1. On the other hand, it has been shown by Hochschild ([4]) that if all the modules for a restricted Lie algebra (everything finite-dimensional) are completely reducible, then is abelian with non-singular p-mapping.

Exercises

1. Let Hence the result follows by induction be a Lie algebra over a field of characteristic zero, the universal enveloping algebra. Show that every element of is a linear combination of powers of elements of .

2 (Witt). Let be the free Lie algebra with r (free) generators x₁, x₂, …, x_r over a field of characteristic 0, the universal enveloping algebra of . Let the space of homogeneous elements of degree n in . Show that

where μ is the Mobius function.

3. Let be finite-dimensional of characteristic 0, the nil radical of . Show that the collection of linear transformations exp , is a group under multiplication.

4. Show that if Z is a nilpotent linear transformation in a finite dimensional vector space over a field of characteristic 0, then exp Z is unimodul ar (det exp Z = 1). Show that if Z is skew relative to a non-degenerate symmetric or skew bilinear form, then exp Z is orthogonal relative to this form.

5. If is an algebra there exists a unique automorphism of period two in ⊗ such that . Show that if is the universal enveloping algebra of a Lie algebra, then δ is contained in the subalgebra of -fixed elements of ⊗ .

6. Let be the universal enveloping algebra of a Lie algebra and let ^* be the conjugate space of . If , then there exists a unique linear function on ⊗ such that . Define by the diagonal mapping of into ⊗ . This makes * an algebra. Show that this algebra is commutative and associative.

7. Prove the following analogue of Friedrichs’ theorem in the characteristic p ≠ 0 case: an element a of satisfies aδ = a ⊗ 1 + 1 ⊗ a if and only if a is in the restricted Lie algebra generated by the x_i.

8. Let be a finite-dimensional Lie algebra, ₁ and ₂ ideals in which are contragredient modules for relative to the adjoint representation. Let (u₁, …, u_n), (u¹, …, uⁿ) be dual bases for _i and ₂ as in the definition of a Casimir element of a representation (§ 3.7). Show that is in the center of the universal enveloping algebra of .

9. Let the notations be as in 8. Let R be a finite-dimensional representation for (hence for ) and let . Show that the element

is in the center of . More generally, show that if θ is a permutation of 1, 2, …, r, then

is in the center of .

10. Determine the center of the universal enveloping algebra of the three- dimensional split simple algebra over a field of characteristic zero.

11. Let Φ be a field, be the algebra of polynomials in indeterminates the closure of regarded as a graded algebra in the usual way (an element is homogeneous of degree k if it is a homogeneous polynomial of degree k in the usual sense). Φ < t_i > is called the algebra of formal power series in the t_i and its quotient field P is the field of formal power series in the t_i. Φ < t_i > has a valuation as defined in § 5. Thus if a = a_k + a_{k + 1} + a_{k + 2} + … a_j homogeneous of degree j, a_k ≠ 0 then |a| = 2^-k. This valuation has a unique extension to a valuation in P satisfying |ab| = |a||b|. Let P_n be the algebra of n × n matrices with entries in P. Define . Show that this defines a valuation in P_n and show that if D₁, D₂, …, D_r are matrices with entries in Φ, then there exists a unique continuous homomorphism of the algebra of § 5 into P_n mapping .

12. Let be a finite dimensional Lie algebra over a field Φ of characteristic 0 and let P be defined as in 11. Let D₁, D₂, …, D_r be derivations in over Φ and denote their extensions to p over P by D₁, D₂, …, D_r again. Show that G = exp t₁D₂ exp t₂D₂ … exp t_rD_r is a well defined automorphism in _p and that G = exp D where D is in the subalgebra generated by the D_i in the Lie algebra of derivations of _P.

13. Let be a finite-dimensional restricted Lie algebra which every element is nilpotent: a^pk = 0 for some k > 0. Show that the u-algebra of has the form where is the radical.

14. A polynomial of the form is called a p-polynomial and it is called regular if a_m ≠ 0. Let be a restricted Lie algebra (possibly infinite-dimensional) with the property that for every a ∈ there exists a regular p-polynomial μ_a(λ) such that μ_a(a) = 0. Show that if c is an element of such that all the roots of μ_c(λ) are in the base field Φ, then c is in the center of . Hence show that is abelian if Φ is algebraically closed. Show that any finite dimensional nonabelian restricted Lie algebra over an algebraically closed field contains an element z ≠ 0 such that z^p = 0.

15. Let be restricted with the property that a^p = αa, α fixed and ≠0. Prove that is abelian.

16. use 14 and 15 to prove that if a^p2 = a in a restricted Lie algebra, then is abelian. Conjecture: If , then is abelian.

17. Prove that if is restricted of characteristic three and a³ = 0 for all a, then any finitely generated subalgebra of is finite-dimensional.

Conjecture (probably false but probably true under additional hypotheses): If is finitely generated and every element of (restricted of characteristic p) is algebraic in the sense that there exists a non-zero p-polynomial μ_α (λ) such that , then is finite-dimensional.

18. Call a derivation D of a restricted Lie algebra restricted if a^pD = (aD)(ad a)^n-1. Note that every inner derivation is restricted. Show that a derivation is restricted if and only if it can be extended to a derivation of the μ-algebra. Show that if has center 0, then every derivation is restricted.

19. Let be an abelian finite-dimensional restricted Lie algebra over a perfect field. Show that where ₀ is the space of nilpotent elements (a^pk = 0) and

20. as in 19, base field infinite and perfect. Assume ₀ = 0. Show that is cyclic in the sense that there exists an element b such that

21. Let be the group algebra over a field of characteristic p of the cyclic group of order p. Then has the basis with x^p = 1. Show that the derivation algebra of has a basis , where xD_i = x^{i + l}. Verify that

Prove that is simple. is called the Witt algebra.

22. Generalize 21 by considering the derivation algebra of the group algebra of the direct product of r cyclic groups of order p, Φ of characteristic p. These derivation algebras are simple too.

23. Prove the following identity for Lie algebras of characteristic p ≠ 0 :

24. Let be a nilpotent Lie algebra of linear transformations in a finitedimensional vector space over a field of characteristic p ≠ 0 such that ^P = 0. Show that if . Use this to prove that if the base field is algebraically closed, then the weight functions are linear.

25. Let be the Lie algebra of n × n triangular matrices of trace 0 over a field of characteristic . Prove complete. (Hint: Show that every derivation is restricted if is considered as a restricted Lie algebra and study the effect of a derivation on a diagonal matrix with distinct diagonal entries. (Φ may be assumed infinite.))