CHAPTER X

Simple Lie Algebras over an Arbitrary Field

In this chapter we study the problem of classifying the finitedimensional simple Lie algebras over an arbitrary field of characteristic 0. The known methods for handling this involve reductions to the problem treated in Chapter IV of classifying the simple Lie algebras over an algebraically closed field of characteristic 0. One first defines a certain extension field Γ called the centroid of the simple Lie algebra which has the property that can be considered as an algebra over Γ and ( over Γ)_Ω ≡ Ω ⊗_Γ is simple for every extension field Ω of Γ. It is natural to replace the given base field Φ by the field Γ. In this way the classification problem is reduced to the special case of classifying the Lie algebras such that _Ω is simple for every extension field Ω of the base field. If is of this type and Ω is the algebraic closure, then the possibilities for _Ω are known (A_l, B_l, C_l, etc.) and one now has the problem of determining all the such that _Ω is one of the known simple Lie algebras over the algebraically closed field Ω.

This problem can be transferred to the analogous one in which the algebraically closed field Ω is replaced by a finite-dimensional Galois extension P of Γ and _P is one of the split simple Lie algebras over P. This is equivalent to the problem of determining the finite groups of automorphisms of _P considered as an algebra over Φ which are semi-linear transformations in _P over P.

We shall consider this problem in detail for _P one of the classical types A_l – D_l except D₄. Our results will not give complete classifications even in these cases but will amount to a reduction of the problem to fairly standard questions on associative algebras. For certain base fields (e.g., the field of real numbers) complete solutions of these problems are known, so in these cases the classification problem can be solved.

The classification problem for the field of real numbers is quite old. A complete solution was given by Cartan in 1914. Simplifications of the treatment are due to Lardy and to Gantmacher. For an arbitrary base field of characteristic 0 the results for the classical types are due to Landherr and to the present author, for G₂ to the author and for F₄ to Tomber. It is worth noting that most of the results carry over also to the characteristic p ≠ 0 case. This has been shown by the author. References to the literature can be found in the bliography.

1. Multiplication algebra and centroid of a non-associative algebra

Let be an arbitrary non-associative algebra over a field Φ (cf. § 1.1). If a ε , the right (left) multiplication a_R (a_L) is the linear mapping x→xa (x→ax). We define the multiplication algebra () to be the enveloping algebra of all the a_R and a_L, a ∈ . Thus () is the algebra (associative with 1) generated by the a_L and a_R. If is a Lie algebra, then () is the enveloping algebra of the Lie algebra ad . We define the centroid Γ() of to be the centralizer of () in the algebra () of all linear transformations in . Thus the elements of Γ = Γ() are the linear transformations γ such that [γ, A] = 0 for all A ∈ (). Evidently, γ ∈ Γ if and only if [γa_R] = 0 = [γa_L] for all a ∈ , and these conditions can be written in the form

LEMMA 1. If = , then Γ is commutative.

Proof: Let γ, δ ∈ Γ, a, b ∈ . Then (ab)γδ = ((ar)b)δ = (aγ)(bδ) and (ab)γδ = (a(bγ))δ = (aδ)(bγ). If we interchange γ and δ we obtain (ab)δγ = (aδ)(bγ) = (aγ)(bδ). Hence (ab)(rδ — δγ) = 0. Since ² = , any element c of has the form c = Σ a_ib_i. It follows that c(γδ — δγ) = 0 for all c and Γ is commutative.

A non-associative algebra is simple if has no (two-sided) ideals ≠ 0, ≠ and ² ≠ 0. Since ² is an ideal it follows that ² = for simple. Hence the lemma implies that Γ is commutative.

The ideals of a non-associative algebra are just the subspaces which are invariant relative to the right and left multiplications. These are the same as the subspaces which are invariant relative to the multiplication algebra ≡ (). It follows that is simple if and only if is an irreducible algebra of linear transformations. If x ∈ , then the smallest St-invariant subspace containing x is x.

Hence if x ≠ 0 and is simple, then x = . The converse is easily seen also: if ≠ 0 and x = for every x ≠ 0, then is simple. We recall the well-known lemma of Schur: If is an irreducible algebra of linear transformations, then the centralizer of is a division algebra (for proof see, for example, Jacobson [2], vol. II, p. 271). In the special case of = () for simple this and Lemma 1 give

THEOREM 1. The centroid Γ of a simple non-associative algebra is a field.

Since the centroid Γ is a field we can consider as a (left) vector space over Γ by setting γα = αγ, a ∈ , γ ∈ Γ. Then condition (1) can be re-written as

which is just the condition that as vector space over Γ be a non-associative algebra over Γ relative to the product ab defined in over Φ.

A non-associative algebra will be called central if its centroid Γ coincides with the base field. If is simple with centroid Γ, then is central simple over Γ; for we have

THEOREM 2. Let be a simple non-associative algebra over a field Φ and let Γ ( Φ) be the centroid. Consider as algebra over Γ by defining γα = aγ, a ∈ , γ ∈ Γ. Then is simple and central over Γ and the multiplication algebra of over Γ is the same set of transformations as the multiplication algebra of over Φ.

Proof : Since Γ Φ it is clear that a Γ-ideal of is a Φ-ideal so is Γ-simple. Similarly, the centroid of over Γ is contained in Γ; hence = Γ and is central. Let denote the multiplication algebra of over Γ. Then it is clear that = Γ the set of Γ-linear combinations of the elements of . Now let ₀ be the subset of of elements A such that γ A ∈ for all γ in Γ. It is clear that ₀ is a Φ-subalgebra of . If a, x∈ , then γ (ax) = a(γx) = (γa)x which means that γa_L = a_Lγ = (γa)_L. Hence a_L ∈ ₀ and similarly, a_R ∈ ₀. Thus ₀ contains all the left and the right multiplications and consequently We therefore have = Γ = .

We consider next the question of extension of the base field of a non-associative simple algebra. The fundamental result in this connection is the following

THEOREM 3. If is a non-associative central simple algebra over Φ and P is any extension field of Φ, then _P is central simple over P. Next let be an arbitrary non-associative algebra over Φ, let Δ be a subfield (over Φ) of the centroid and suppose the Δ-algebra Γ ⊗ is simple. Then is simple over Φ and Δ = Γ.

Proof: For the proof of the first assertion we shall need a well-known density theorem on irreducible algebras of linear transformations. (See, for example, Jacobson [2], vol. II, p. 272.) A special case of this result states that if is a non-zero irreducible algebra of linear transformations in a vector space over Φ and the centralizer of in () is the set Φ of scalar multiplications, then is a dense algebra of linear transformations in over Φ. This means that if {x₁ x₂, …, x_n} is an ordered finite set of linearly independent elements of and y₁, y₂, …, y_n are n arbitrary elements of , then there exists a T ∈ such that x_iT = y_i, i = 1, 2, …, n. The density theorem is applicable to the multiplication algebra of a central simple non-associative algebra. Now let P be an extension field of Φ and consider the extension algebra _P, for which we choose a basis {u_α | α ∈ I} consisting of elements u_α ∈ . Let x and y be any elements of _P with x ≠ 0. Then we can write is a suitable subset of the basis {u_α} and the ’s and ’s are in P. We may assume also that ₁ ≠ 0. The extension to _P of any element of is in the multiplication algebra of _P over P. Hence there exists an A_i ∈ such that x₁A_i = x_i and x_jA_i = 0 if j ≠ 1, i = 1, 2, …, n. Then xA_i = ₁ x_i and and satisfies xA = y. Thus x э y and since y is arbitrary, x = _P. This implies that _P is simple by the criterion we noted before. Let C be a linear transformation in _P such that [CA] = 0 for all A ∈ . Let u_α be one of the base elements and apply the previous considerations to x = u_α = x₁, y = u_αC. Let Al ∈ satisfy x_iA_i = x_iA_i = 0, i ≠ 1. Then u_αC= x₁ C = x₁ A₁C = x₁ C A₁ = yA₁ = x₁ = u_α. Since ₁ depends on u_α we write ₁ = ρ_α and so we have u_αC = ρ_au_a, ρ_a ∈ P. Next we note that if u_α and u_β are any two base elements, then there exists a Bαβ ∈ such that u_αB_αβ = u_β. This is clear because of the density of . Then ρ_βu_β = uβC = u_αB_αβC = u_αCB_αβ = ρ_αu_αB_αβ = ρ_αu_β. Hence ρ_α = ρ_β = ρ and C is the scalar multiplication by the element p. This shows that the centroid is P, so _P is central over P. This completes the proof of the first assertion.

Now suppose is a non-associative algebra over Φ such that Γ is simple over Δ, Δ a subfield containing Φ of the centroid Γ of . We can consider as algebra over Δ (δa = aδ, a ∈ , δ ∈ Δ) and we show first that is simple over Δ. Thus let be a Δ ideal in . Then the subset of elements of the form ∑ γ_i ⊗ b_i, γ_i ∈ Γ, b_i ∈ is a Δ-ideal in Γ . The properties of tensor products over fields imply that if ≠ 0, , then the ideal indicated is a proper non-zero Δ-ideal in Γ . Hence = 0 or = . Also ≠ 0 since (Γ )² ≠ 0. This shows that is simple over Δ. The proof of Theorem 2 shows that the multiplication algebra of over Φ is the same set as the multiplication algebra of over Δ. Since we have x = for all x ≠ 0 it is now clear that is simple over Φ also and Γ is a field. Consider again as algebra over Δ. Then one checks that the mapping

γ_i ∈ Γ, a_i ∈ is a Δ-algebra homomorphism of Γ onto . Suppose Γ ⊃ Δ and let γ ∈ Γ, 0 Δ so that 1, γ are Δ-independent. Then a relation γ ⊗ a₁ + 1 ⊗ a₂ = 0, a_i ∈ , implies that the a_i= 0. Choose a₁ = a ≠ 0, a₂ = —γa. Then γ ⊗ a₁ + 1 ⊗ a₂ ≠ 0 and the image of this element under (2) is γα — γα — 0. Hence if Γ ≠ Δ, then (2) has a non-zero kernel as well as a non-zero image. This contradicts the assumption that Γ is simple over Δ. Hence Γ = Δ and the proof is complete.

It is immediate that a dense algebra of linear transformations in a finite-dimensional vector space is the complete algebra (). Thus if (x₁, …, x_n) is a basis and A is any linear transformation, then contains T such that x_iT = x_iA. Hence A = T ∈ . If is non-associative simple with centroid Γ, then we have seen that the multiplication algebra is a dense algebra of linear transformations in considered as a vector space over Γ. Hence if is finite-dimensional over Φ and consequently over Γ, then is the complete algebra of linear transformations in over Γ. We therefore have the following

THEOREM 4. Let be a finite-dimensional simple algebra with centroid Γ and multiplication algebra . Then is the complete set of linear transformations in considered as a vector space over Γ.

Let the dimensionality [ : Φ] of over Φ be n and let [Γ : Φ] = r, [ : Γ] = m. Then it is well known that n = rm and we now see that [ :Γ] = m²r.

Again let be arbitrary and let a→α^θ be an isomorphism of onto a second non-associative algebra . Then θ is a 1 :1 linear mapping of onto and (ab)^θ = a^θb^θ which implies that

Thus

This implies that the mapping X→θ^-1Xθ is an isomorphism of the multiplication algebra () onto (). It is clear also that γ → γ^θ ≡ θ^-1γθ is an isomorphism of the centroid Γ() onto Γ(). If a ∈ we have (αγ)^θ = (α^θθ-1γ)^θ = a^θγ^θ. In particular, if is simple and is considered as a vector space over Γ, then we have

Next let D be a derivation in . Then D is linear and (ab)D — (aD)b + a(bD), which gives

This implies that the inner derivation X → [X, D] in () maps () into itself. Consequently, this induces a derivation γ → γ^α = [γD] in Γ(). By definition of γ^α we have (aγ)D = (aD)γ + αγ^α so that for simple we have

We can state the results we have just noted in the following convenient form.

THEOREM 5. Let and be central simple non-associative algebras over a field Γ and let Φ be a subfield of Γ. Then any isomorphism Φ of over Φ onto over Φ is a semi-linear transformation of over Γ onto over Γ. Any derivation D in over Φ satisfies (γα)D = γ^dα + r(aD), γ ∈ Γ, a ∈ where d is a derivation in Γ as a field over Φ.

We add one further remark to this discussion. Suppose (u₁, …, u_n) is a basis for over Γ and that u_iu_j = Σ γ_ijku_k, γ_ijk ∈ Γ, gives the multiplication table. A 1 : 1 semi-linear transformation θ of onto maps the u_i into the basis , if i = 1, …, n for over Γ. If θ is a Φ isomorphism we have where θ acting on γ_ijk is the automorphism in Γ associated with the semi-linear mapping θ of . In many cases which are of interest one can choose a basis so that the γ_ijk ∈ Φ, that is, = _0P where ₀ is a non-associative algebra over Φ. Then we have which implies that and are isomorphic as algebras over Γ.

2. Isomorphism of extension algebras

Let be a non-associative algebra over Φ and let P be a finitedimensional Galois extension field of Φ, G = {1, s, …, u} the Galois group of P over Φ. In this section we shall obtain a survey of the isomorphism classes of non-associative algebras such that _P p. We recall that if is a vector space over Φ then can be identified with a subset of _P. This subset is a Φ-subspace of _P which generates _P as space over P and has the property that any set of elements of which is Φ-independent is necessarily P-independent. Moreover, these properties are characteristic. If (ρ) is a basis for P over 0, then any element of _P has a unique representation in the form If s ∈ G, then s defines a semi-linear transformation U_s in _P by the rule

It is easy to check that U_s is independent of the choice of the basis in P over Φ, that the automorphism in P associated with U_s is s (that is, and

Hence the U_s form a group isomorphic to G. If we take ρ₁ = 1, then the elements of have the form ρ₁x₁, x₁ ∈ and it is clear from (7) that these elements are fixed points for every U_s, s ∈ G. It is easy to show directly that is just the set of fixed points relative to the U_s This will follow also from the following basic lemma:

LEMMA 2. Let be a vector space over a field P which is a finite-dimensional Galois extension of a field Φ. Suppose that for each s in the Galois group G of P over Φ there is associated a semi-linear transformation U_s in with associated automorphism s in P such that U₁ = 1, U_sU_t = U_st. Let be the set of fixed points relative to the U_s, s ∈ G. Then is a Φ-subspace of such that = _p.

Proof: One verifies directly that is a Φ-subspace of . If x ∈ and ρ ∈ P, then since Let (ρ₁, ρ₂, …, ρ_n) be a basis for P over Φ. Then it is well known that the n × n matrix whose rows are s ∈ G, is non-singular. It follows from this that x: is a P-linear com B_i nation of the n elements which belong to . Thus the P-space P spanned by is . Next let x₁, …, x_r be elements of which are Φ-independent. Assume there exist _i ∈ P, not all 0, such that Then we may take r minimal for such relations and we may suppose ₁ = 1. Evidently we have r > 1 and we may assume that Then and we can choose s in G so that We can therefore obtain a relation and this is non-trivial and shorter than the relation This contradiction shows that the x_j’s are P-inde- pendent. Hence = _P.

If Thus the U_s are the transformations we constructed before for _P. We return to the situation considered before in which we were given = _P and we defined the U_s by (7). We saw that , the Φ-subspace of fixed elements relative to the U_s On the other hand, we have Let {e_α} be a basis for over Φ and let b ∈ . Then Since the x_i and b are in this relation implies that these elements are Φ-dependent and since the x_i are Φ-independent we have Hence b ∈ and = so that is just the set of fixed elements relative to the U_s Our results establish the following

THEOREM 6. Let be a vector space over P, a finite-dimensional Galois extension of the field 0. Let {U_s | s ∈ G} be a (finite) set of semi-linear transformations in over P such that: (1) U₁ = 1, U_sU_t = U_st, (2) the automorphism in P associated with U_s is s. Let be the Φ-space of fixed elements relative to the U_s. Then = _P and the correspondence {U_s}→ is a B_i jection of the set of finite sets of semi-linear transformations satisfying (1) and (2) and the set of Φ-subspaces of satisfying = _P.

Now assume that is a non-associative algebra over Φ. Then it is easy to check that the U_s defined by (7) are automorphisms of = _P considered as an algebra over Φ. Conversely, if the group {U_s} of semi-linear transformations in is given and every U_s is an automorphism of over Φ then the set of fixed elements is an algebra over Φ. Hence the correspondence of Theorem 6 induces a bijection of the set of groups {U_s}, U_s an automorphism of over Φ and the set {} of Φ-subalgebras such that _p=. Let and be two of the Φ-algebras such that _P = _P and suppose A is an isomorphism of onto . Let {V_s} be the group of semi-linear transformations associated with , {U_s} that with . The isomorphism A has a unique extension to an automorphism A of over P. The mapping A^-lU_sA is a semi-linear transformation in with associated automorphism s in P. Also A^-1U_sA = 1 and The set of fixed points relative to the A^-lU_sA is the space . Hence A^-lU_s A = V_s, s ∈ G. Conversely, let {U_s}, {V_s} be groups of semi-linear mappings as in Theorem 6 such that the U_s, V_s are automorphisms of as non-associative algebra over Φ and assume there exists a P-auto- morphism A of such that

Then A = for the associated non-associative algebras and . Hence and are isomorphic as algebras over Φ.

We therefore have the following

THEOREM 7. Let be a non-associative algebra over a finite dimensional Galois extension P of a field Φ. Then the correspondence of Theorem 6 between sets of semi-linear transformations {U_s} in satisfying (1) and (2) and Φ-subspaces such that ^P = induces a bijection of the set of {U_s} such that every U_s is a Φ-algebra isomorphism and the which are Φ-subalgebras of . The corresponding Φ-subalgebras are isomorphic if and only if there exists an automorphism A of over P such that (9) holds for the associated groups.

If is a non-associative algebra over Φ such that _{P P}, then can be identified with its image in = _P. In this way we see that Theorem 7 gives a survey of the isomorphism classes of algebras such that _{P P} via certain similarity classes of groups of automorphisms of over Φ. This is the type of result we wished to establish.

In the sequel we shall be concerned with finite-dimensional non-associative algebras over a field Φ of characteristic 0 and we shall be concerned with the question of equality _Ω = _Ω for two such algebras, Ω being the algebraic closure of the base field Φ. Let (e₁, …, e_m), (f₁, …, f_m) be bases for over Φ and over Φ respectively. Then the equality _Ω = _Ω implies that . Let be ∑ the subfield over Φ generated by the ρ_ij. Since every ρ_ij is algebraic over Φ and there are only a finite number of these, Σ is a finite-dimensional extension of Φ. Since Φ is algebraic and of characteristic zero, Σ is contained in a subfield P of Φ which is a finite-dimensional Galois extension of Φ. Since the ρ_ij ∈ P it is clear that and the converse inequality holds since Also it is clear that the f_i and e_i the both form linearly independent sets over P. Hence the algebraic closure of the base field Φ implies _P = for _P a suitable finite-dimensional Galois extension of Φ.

3. Simple Lie algebras of types A-D

We now take up the main problem of this chapter: the classification of the finite-dimensional simple Lie algebras over any field Φ of characteristic 0. If is such an algebra and Γ is its centroid, then Γ is a finite-dimensional field extension of Φ and is finitedimensional central simple over Γ. Conversely, if is finitedimensional central simple over any finite-dimensional extension field Γ of Φ, then is finite-dimensional simple over Φ. If and ₁ and ₂ are two isomorphic simple Lie algebras over Φ, then the respective centroids Γ₁ and Γ₂ are isomorphic so both algebras may be considered as central simple over the same field Moreover, Theorem 5 shows that we have a semi-linear transformation θ of ₁ over Γ onto 22 over Γ such that θ is a Φ-isomorphism of ₁ onto ₂ as non-associative algebras over Φ. These results reduce the classification problem for simple Lie algebras over Φ to the following problems: (1) the classification of the finite-dimensional field extensions Γ of Φ, (2) the classification of the finitedimensional central simple Lie algebras over the Γ in (1), and (3) determination of conditions for the existence of a semi-linear transformation θ of two central simple Lie algebras over Γ such that θ is a Φ-algebra isomorphism. To make this concrete we consider the important special case: Φ the field of real numbers. Here any algebraic extension field Γ of Φ is either Φ itself or is isomorphic to the field of complex numbers. If Γ is the field of complex numbers, then Γ is algebrically closed and we know the classification of the finite-dimensional simple Lie algebras over Γ. We recall that the algebras in this classification (A_l, B _l etc.) all have the form _0Γ, ₀ an algebra over Φ (Theorem 4.2). It follows from the remark following Theorem 5 that the algebras in our list (which are not Γ-isomorphic) are not isomorphic over Φ. Thus, to complete the classification over the reals it rem A_i ns to classify the central simple algebras over this field.

Now suppose is a finite-dimensional central simple Lie algebra over Γ which is any field of characteristic 0. If Φ is the algebraic closure of Γ, then _Ω is simple. Conversely, if is finite-dimensional over Γ and _Ω is simple, then is evidently simple so its centroid Γ′ is a finite-dimensional field extension of Γ. This can be considered as a subfield of Ω. Then _Γ′ is simple since is simple. Hence is central by Theorem 3. Thus is central simple over Γ if and only if _Ω is simple over Ω.

Since Ω is algebraically closed of characteristic 0 we know the possi B_i lities for _Ω. They are the Lie algebras , in the Killing Cartan list. If _Ω is the Lie algebra X in this list then we shall say that is of type X. Usually the subscript l will be dropped and we shall speak simply of of type A, type B, etc. For each type X we shall choose a fixed Lie algebra ₀ of this type. For example, we can take ₀ to be the split Lie algebra over Γ of type X. Then our problem is to classify the Lie algebras such that _Ω = _0Ω. For a particular there exists a finite-dimensional Galois extension P such that and we have seen that to determine the which satisfy this condition for a particular P, then we have to look at the automorphisms of _0P or over Γ. We shall study the cases of ₀ of types, A_l, B_l, C_l, D_l, l > 4. In this section we shall give some constructions of Lie algebras of types A to D. In the next section we give the conditions for isomorphism of these Lie algebras and in § 5 we shall prove that every Lie algebra of type A_l, B_l, C_l, D_l, l > 4 can be obtained in the manner given here.

The starting point of our constructions is the fundamental Wedderburn structure theorem on simple associative algebras: Any finite-dimensional simple associative algebra is isomorphic to an algebra of all the linear transformations of a finite-dimensional vector space over a finite-dimensional division algebra Δ. An equivalent formulation is that Δ_n the algebra of n × n matrices over a finite-dimensional division algebra Δ. (For a proof see: Jacobson, Structure of Rings, p. 39, or Artin, Nes B_i tt, and Thrall, Rings with Minimum Condition, p. 32.) If the base field Γ is algebraically closed then the only finite-dimensional division algebra over Γ is Γ itself, so in this case Γ_n for some n. If the base field is the field of real numbers, then it is known that there are just three possi B_i lities for the field of complex numbers or Δ the division algebra of quaternions. (Theorem of Frobenius, cf. Dickson [1], p. 62, or Pontrjagin [1], p. 175.) The center of a finite-dimensional simple associative algebra is a field and the centroid consists of the mappings This can be identified with the center. If is central ( = Γ) and Ω is the algebraic closure of Γ, then _Ω is finite-dimensional simple over Ω. Hence _Ω Ω_n for some n. Since this shows that the dimensionality of any finite-dimensional central simple associative algebra is a square.

Let be a finite-dimensional central simple associative algebra over Γ. Consider the derived algebra of the Lie algebra _L. If Ω is the algebraic closure of On the other hand, we know that if n = l + 1, l 1, then is the Lie algebra of (l + 1) × (l + 1) matrices of trace Φ and this is the simple Lie algebra A_t over Ω. Since it follows that is a central simple Lie algebra of type A_l.

Next let be a finite-dimensional simple associative algebra with an involution J. By definition, J is an anti-automorphism of period two in . Hence —J is an automorphism of _L. The set of J-skew elements (a^J = –a) is the subset of fixed elements relative to the automorphism —J. Hence this is a subalgebra of _L. The anti-automorphism –J induces an automorphism in the center of which is either the identity or is of period two. In the first case J is of first kind and in the second J is an involution of second kind.

We assume first that J is of second kind. More precisely, we assume that = P Γ(q) a quadratic extension of the base field Γ and that be the space of J-symmetric elements (a^J = a). Any a ∈ has the form and Hence the mapping x→qx is a 1:1 linear mapping of onto so we have the dimensionality relation: and We recall also that [ :P] is a square n², so [ : Γ] = n². Let (a₁, a_n²) be a basis for over Γ. Then every element of has the form It follows that the a_l form a basis for (or _L) over P. This implies that _P = _L over P. Let Ω be the algebraic closure of Γ which we may assume to be an extension of the field P. Then Let be the derived algebra Then so is the simple Lie algebra A_l Hence is central simple of type A_l.

We have now given two constructions of Lie algebras of type A. We summarize our results in the following

THEOREM 8. Let be a finite-dimensional central simple associative algebra over Γ, ≠ Γ. Then the derived algebra is a central simple Lie algebra of type A _l, l 1. Let be a finite-dimensional simple associative algebra with center P a quadratic extension of the base field Γ and suppose possesses an involution f of second kind. Suppose also that ≠ P. Let (, J) be the Lie algebra of skew elements of . Then the derived algebra (, J)′ is a central simple Lie algebra of type A_l l 1.

Before proceeding to the discussion of the Lie algebras (, J), J of first kind we quote some well-known results on involutions in algebras of linear transformations (cf. Jacobson [3], pp. 80-83). Let be a finite-dimensional vector space over a division algebra Δ and let be the algebra of linear transformations in over Δ. Then it is known that has an involution A→ A^J if and only if Δ has an anti-automorphism of period one or two. If the period is one, then implies that Δ is commutative. Conversely if Δ is commutative, then one can take If is given, then one can define a non-degenerate hermitian or skew hermitian form (x, y) in over Δ relative to d → d. Such a form is defined by the conditions

for x, x₁, x₂, y, y₁, y₂ , d Δ,

according as the form is hermitian or skew-hermitian, and nondegeneracy means that (x, z) = 0 for all x implies z = 0. If then we obtain a symmetric or skew B_i linear form. If (u₁, u₂, …, u_n) is a basis for over Δ, then defines a non-degenerate hermitian form and is a non-degenerate skew hermitian form if then non-degenerate skew B_i linear forms exist for if and only if is of even dimensionality over Γ. Let (x, y) be any nondegenerate hermitian or skew hermitian form associated with in Δ. If A ∈ we let A^J denote the adjoint of A relative to (x, y), that is, A^J is the linear transformation in such that (xA,y) = (x, yA^J) for x, y ∈ . Then it is easily seen that A → A^J is an involution in . Moreover, one has the fundamental theorem that every involution J of is obtained in this way.

In particular, suppose Δ = Γ is an algebraically closed field. Since Δ = Γ, the only anti-automorphism of Δ as algebra over Γ is the identity mapping. Hence (x, y) is either a non-degenerate symmetric B_l linear form or is a non-degenerate skew B_l linear form. The latter can occur only in the even-dimensional case. The Lie algebra (, J) determined by the form is the Lie algebra B_l if the form is symmetric and dim = 2l + 1. The Lie algebra (, J) is C_l if the form is skew and dim = 2l and it is D_l if the form is symmetric and dim = 2l.

Now let be a finite-dimensional central simple associative algebra over Γ which has an involution J of first kind. If Ω is the algebraic closure of Γ, then The extension J of J to a linear transformation in is an involution in _Ω and (_Ω, J) = (, J)_P. Since and J is an involution in Ω_n the result above shows that (_Ω, J) is one of the Lie algebras B_l, C_l or D_l. We assume now that according as (_Ω, J) is B_l, C_l or D_l. Then the algebras (_Ω, J) are simple and we see that (, J) is central simple over Φ of types C_l or D_l. We recall that n is the dimensionality of the space considered before and that For B_l we have n = 2l + 1 and For C_l, n = 2l and . We can now state the following

THEOREM 9. Let be a finite-dimensional central simple associative algebra over Γ of dimensionality n² and suppose has an involution J of first kind. Let (, J) be the Lie algebra of J-skew elements of . If n = 2l + 1, then [(, J) ι Γ] = l(2l + 1) and is central simple of type B_l for l 2. If n = 2l, then [ : Γ] = l(2l + 1) or 1(2l - 1). In the former case assume l 3. Then is central simple of type C_l. If n = 2l and [ : Γ] = l(2l — 1) then we assume l 4. Then is central simple of type D_l.

4. Conditions for isomorphism

Let ₁ and ₂ be finite-dimensional central simple Lie algebras over the field Γ of characteristic 0. Suppose Φ is a subfield of Γ and that ₁ and ₂ are isomorphic as algebras over Φ: _{1 Φ 2}. Then we know that a Φ-isomorphism θ of ₁ onto ₂ is a semi- linear transformation of ₁ over Γ onto ₂ over Γ. We denote the associated automorphism in Γ by θ also so that we have (γa)^θa^θ if . Let (a₁, ···, a_m) be a basis for ₁ over Γ with the multiplication table . Then is a basis for ₂ over Γ and Let Ω be the algebraic closure of Γ. Then it is a well-known result of Galois theory that the automorphism θ in Γ has an extension to an automorphism θ in Ω. The a_l form a basis for _1Ω over Ω and the form a basis for _2Ω over Ω. It follows that the mapping is Φ-isomorphism of _1Ω onto _2Ω. Thus On the other hand, _1Ω is simple over the algebraically closed field Ω. Hence it has a basis over Ω whose multiplication coefficients are in the prime field and so are in Φ. This implies (remark following Theorem 5) that We have therefore proved the following

LEMMA 3. Let ₁ and ₂ be finite-dimensional central simple Lie algebras over a field Γ of characteristic zero. Let Φ be a subfield of Γ and Ω the algebraic closure of Γ. Then

This result implies that the only Φ-isomorphisms which can exist for the Lie algebras of Theorems 8 and 9 are those between the algebras defined in Theorem 8 and between algebras of the same type (B_l, C_l, D_l) of Theorem 9. We shall call the Lie algebras of the form of Theorem 8 Lie algebras of type A_I, those of the form (, J)′, J of second kind, Lie algebras of type A_II. For the latter class we assume l > 1 if the type is A_l. This amounts to assuming that [ : P] = n² > 4. We shall suppose also from now on that l > 4 for the algebras of type D_l We consider next the enveloping associative algebras of the Lie algebras of Theorems 8, 9.

LEMMA 4. Let Φ, Γ, Ω be as in Lemma 3. (1) Let be a finitedimensional central simple associative algebra over Γ of dimensionality n² > 1. Then the enveloping associative algebra of over Φ is . (2) Let be a finite-dimensional simple associative algebra with center P a quadratic extension of Γ and with an involution J of second kind. Assume [ : P] = n² > 4. Then the enveloping associative algebra of = (, J)′ over Φ is . (3) Let be finite-dimensional central simple over Γ with an involution J of first kind and[ : Γ] = n² > 1. Then the enveloping associative algebra of = (, J) over Φ is .

Proof: We note first that in all cases the enveloping associative algebra of over Φ is the same as that of over its centroid Γ. Thus since is a vector space over Hence the two enveloping associative algebras indicated coin C_i de with the set of sums of products This remark shows that we may as well assume the base field Φ = Γ and we shall now do this. In the cases 1 and 3 we introduce the algebra and we consider _Ω which is a subalgebra of Ω_nL. We know that _Ω is the Lie algebra of matrices A_l, B_l, C_l or A as defined in § 4.6. In all cases an elementary direct calculation with the bases given in § 4.6 shows that the Ω-subalgebra generated by _Ω is Ω_n, that is, where the * denotes the enveloping associative algebra. If * denotes the enveloping associative algebra over Φ of (in ), then it clear that the Ω-subspace of Ω_n spanned by * is . Hence * contains a basis for Ω_n over Ω. Since the elements of this basis are contained in it follows that they constitute a basis for . Thus we have * = . The argument just used cannot be applied readily to the case 2 (Lie algebras of type A_II) since in this case We therefore proceed in a somewhat different manner. Let P = Φ(q) where q^J = –q, as before. Then q ∈ (, J) and Hence it suffices to show that Since Φ = Γ and = ′ we have [:Φ] = n² and Hence, it suffices to show that * contains an element Assume the contrary that Let b_i, i = 1,2,3, be skew elements. Then it is immediate that {b₁ b₂ b₃} ≡ b₁ b₂ b₃ + b₃ b₂ b₁ is skew. If the b_i ∈ ′, then

We now consider the algebra ( over P)_Ω = Ω_n We have seen that this algebra has a basis consisting of elements of and that has a basis of elements in = ′. The multilinear character of {b₁ b₂ b₃} now implies that If we take b_i = b this implies that Thus we must have tr b³ = 0 for all b satisfying tr b = 0. This is impossible since n > 2. For example, we can take so that tr b = 0, tr b³ = 6 ≠ 0. This contradiction shows that We are now ready to prove our main isomorphism theorems. In all of these Φ, Γ and Ω are as in the foregoing discussion.

THEOREM 10. Let and be finite-dimensional central simple associative algebras over Γ and let θ be an isomorphism of over Φ onto over Φ. Assume [ : Γ] = n² > 1 and [ : Γ] > 1. Then if n = 2, θ can be extended in one and only one way to an isomorphism of over Φ onto over Φ and if n > 2 then θ can be extended in one and only one way to either an isomorphism or the negative of an anti-isomorphism of over Φ onto over Φ.

Proof. We have By Lemma 3, hence and m = n. Thus we may assume that _Ω = Ω_n = _Ω so that Ω_n has a basis (a₁, …, a_n²) such that the ax form a basis for over Γ. We may assume also that (a₁, …, a_n², …, 1) is a basis for over Γ. Now θ is a semi-linear mapping of over Γ onto , over Γ whose automorphism in Γ we denote by θ. Then is a basis for over Γ. If θ is extended to the automorphism θ in Ω, then is an automorphism θ′ in over Φ which is a semi-linear transformation with associated automorphism θ in Ω. Let (e_ij), i,j =1, be the U_s ual matrix units for Ω_n Then the mapping is an automorphism θ′′ of the associative algebra Ω_n over Φ whose associated automorphism in Ω is θ. θ′′ induces an automorphism θ′′ in the Lie algebra and since θ′ and θ′′ have the same associated automorphism in is an automorphism in over Ω By Theorem 9.5 this has the form if n = 2 and it either has this form or the form if n > 2. Since the mapping is an automorphism of is an anti-automorphism of Ωη, η can be realized by an automorphism of Ω_n over Ω if n 2 and either by an automorphism or by the negative of an anti-automorphism of Ω_n over Ω for n > 2. Then θ′ = θ′′ can be extended to an automorphism ζ of Ω_n over Φ if n = 2, or to an automorphism ζ or the negative of an anti-automorphism ζ of Ω_n over Φ if n > 2. Since θ′ is the mapping it is clear that θ′ coin C_i des with the given θ on . Since the enveloping associative algebra over Φ of is and that of is it follows that ζ maps onto and consequently θ can be extended to an isomorphism or the negative of an anti-isomorphism of over Φ and over Φ. Since generates over Φ it is clear that the extension is unique.

This result shows that if and are isomorphic as algebras over Φ, then and are either isomorphic or anti-isomorphic as Φ-algebras. The converse is clear since any isomorphism of onto induces an isomorphism of i onto and if θ is an antiisomorphism of onto then -θ induces an isomorphism of onto . Our result also gives a description of the group of automorphisms of . If n = 2 or if has no anti-automorphism, then the group of automorphisms of i over Φ can be identified with the group of automorphisms of over Φ. If n > 2 and has an anti-automorphism j, then it is easy to prove by a field extension argument that the automorphism a→ — a^J in is not of the form a→ a^θ, θ an automorphism of . It follows that the group of automorphisms of over Φ is isomorphic to a subgroup of index two of the group of automorphisms of over Φ.

If we take Φ = Γ, then it is a known result of the associative theory that every automorphism of over Φ is inner. This could also be deduced from the form of the automorphisms of by a field extension argument. Then it follows that the automorphisms of over Φ are of the form or of the form where J is a fixed anti-automorphism in .

THEOREM 11. Let _i, i = 1, 2, be a finite-dimensional simple associative algebra over Γ with center a quadratic field P_i = Γ(q_i) and involution J_i of second kind such that Assume [_i : P_i] = and let Φ be a sub field of Γ. Then any Φ-isomorphism θ of (_i, J₁)′ onto (₂, J₂)′ can be extended in one and only one way to a Φ-isomorphism of ₁ onto ₂. The Lie algebra (₁, J₁)′ is not isomorphic to any Lie algebra of type A_r.

Proof: We may choose a basis (a₁, …, a_n²) for ₁ over P₁ so that (a₁, …, a_n²_–1) is a basis for (₁, J₁)′ over Γ and for over P₁ If Ω is the algebraic closure of Γ (chosen to contain P₁ and P₂), then Similarly, we have Assume there exists a Φ- isomorphism of Then we know that and we may assume that ₁ and ₂ are Φ-subalgebras of Ω_n such that any P_i basis for _i is a basis for Ω_n. Let θ be a Φ-isomorphism of onto Then θ is semi-linear in ₁ over Γ onto ₂ over Γ with associated automorphism θ in Γ. If is a basis for ₁ over Γ, then is a basis for ₂ over Γ and both of these are bases for over Ω. If the automorphism θ in Γ is extended to an automorphism θ in Ω, then the mapping is an automorphism in over Φ whose associated automorphism in Ω is θ. The proof of Theorem 10 shows that this can be extended to an automorphism or the negative of an anti-automorphism of the associative algebra Ω_n over Φ. Since the enveloping associative algebra of (_i, J_i)′ over Φ is _i it follows that the isomorphism θ on _i over Φ can be extended to an isomorphism or the negative of an anti-isomorphism of ₁ over Φ onto ₂ over Φ. If the second possi B_i lity holds let ζ denote the anti-isomorphism. Then J₁ζ is an isomorphism of ₁ onto ₂ and if then Hence J₁ζ is an isomorphism of ₁ which coin C_i des with θ on Thus in every case we can extend θ to an isomorphism of ₁ onto ₂. This extension is unique since ₁ is the enveloping associative algebra of This proves our first assertion. Next let ₁ = and suppose we have a Φ-isomorphism θ of ₁ onto ₂ = where is central simple associative over Γ. The argument just used for ₁ and ₂ shows that θ can be extended to an isomorphism θ of ₁ onto . Since the centroids of the Lie algebras and consist of the multiplications (x → γx) in these Lie algebras by the elements γ ∈ Γ it follows that θ maps Γ into itself. Hence P₁ = Γ(q₁) is mapped into a subfield of which properly contains Γ. On the other hand, P₁ is the center of ₁ so this is mapped into the center Γ of . This contradiction proves that ₁ = cannot be isomorphic to .

We consider next the Lie algebras of types B, C and D which we defined in Theorem 9. The main isomorphism result on these is the following

THEOREM 12. Let _i, i = 1,2, be a finite-dimensional central simple associative algebra over Γ such that _i has an involution J_i of first kind. If and In The respective cases indicated we assume that Then if Φ is a sub field of Γ, any Φ-isomorphism θ of (₁, J₁ onto (₁, J₂) can be extended in one and only one way to a Φ-isomorphism of₁onto ₂.

Proof. If Ω is the algebraic closure of Φ, then _iΩ Ω_ni and [(_i, J_i)_Ω is the Lie algebra according as and It follows that if and we may suppose that is either the Lie algebra of skew symmetric matrices in Ω_n (types B or D) or the Lie algebra of matrices satisfying Q^-1A′Q = — A, Q a skew symmetric matrix with entries in the prime field (type C). Let be a basis for (₁, J₁)over Γ and hence for _Ω over Ω. If θ is a Φ-isomorphism of (₁, J₁) onto (₂, J₂), θ is semi-linear with associated automorphism θ in Γ and is a basis for (₂, J₂) over Γ and for _Ω over Ω. If θ is extended to the automorphism θ in Ω then is an automorphism θ′ in _Ω over Φ with θ as its associated automorphism in Ω. If (e _ij) is a usual matrix basis for Ω_n over Ω then is an automorphism θ′′ of Ω_n over Ω whose associated automorphism in Ω is θ. Moreover, since Q has entries in Φ, θ′ ′ maps _Ω into itself and so it induces an automorphism θ′ ′ in _Ω over Φ. It follows that θ′(θ′′)^-1 is an automorphism of _Ω over Ω. In all cases this has the form X→ M¹XM and so it can be extended to an inner automorphism of Ω_n over Ω. It follows that Ω′ can be extended to an automorphism ζ of Ω_n over Ω. Since the restriction of ζ to (₁, J₁) is the given θ and since the enveloping associative algebra of (_i, J_i) is _i, ζ maps ₁ isomorphically on ₂ and coin C_i des with θ on (₁, J₁). Thus θ can be extended to a Φ-isomorphism of₁ onto ₂. Since (₁, J₂) generates₁ this extension is unique.

5. Completeness theorems

Let be a finite-dimensional central simple Lie algebra of type Δ over Γ. Then there exists a finite-dimensional Galois extension field P of Γ such that Thus we may suppose that is a Γ-subalgebra of such that the P-space spanned by is and elements of which are Γ-independent are P- independent. We have seen also that for each s in the Galois group G of P over Γ there corresponds an automorphism U₈ of over Γ such that is a basis for over Γ, hence for over P, then U_s is the mapping We have U₁ = 1, U_sU_t = U_st and which is the set of elements is the set of fixed elements for the U_s, s ∈ G. We have seen in the last section that U_s has an extension U_s in the enveloping associative algebra P_n of such that U_s is an automorphism of P_n over Γ if n = 2 and U_s is either an automorphism of P_n over Γ or the negative of an anti-automorphism of P_n over Γ if n > 2. Every y ∈ P_n is a sum of products and if ρ ∈ P and U_s is an automorphism of P_n, then If U_s is the negative of an anti-automorphism, then Hence in either case U_s is semi-linear with automorphism s in P. Since U_sU_t and U_st have the same effect on the set of generators of P_n we have U_sU_t = U_st Similarly U₁ = 1 is valid in P_n. If U_s and U_t are the negatives of anti-automorphisms then is an automorphism. It follows immediately from this that the subset H of elements s ∈ G such that U_s is an automorphism of P_n over Γ is a subgroup of index one or two in G.

Case I. H=G. The subset of P_n of fixed elements under the U_s is a Γ-subalgebra of P_n such that _P = P_n (Theorem 7). Hence is finite-dimensional central simple over Γ. Evidently and so On the other hand, [ : Γ] = n² -1 and is a Lie algebra of type A_I as defined on p. 303.

Case II. H ≠ G. Then H has index two in G. We know also that n > 2 in this case. The subset of P of elements such that is a quadratic subfield Γ(q) over Γ. It is clear from the form of the U_s that satisfies xU_s = x, s ∈ H, if and only if all the ρi ∈ Γ(q). Thus _{r (q)}, the set of Γ(q)-linear combinations of the a_i, is the set of elements of which are fixed for all the U_s, s ∈ H, and H is the Galois group of P over Γ(q). It follows from case I that where is central simple over Γ(q) and is the enveloping associative algebra of _Γ(q). Now let t ∈ G, H. Since H is of index two in G, an element is in if and only if Now U_t = − J where J is an anti-automorphism of P_n over Γ. Since the conjugate of ρ under the automorphism ≠ 1 of Γ(q) over Γ, J = −U_t maps _Γ(q) into itself. Hence J induces an anti-automorphism in the enveloping algebra . Hence J induces the automorphism ρ → in Γ(q), so J is of second kind. Since aU_t − a for a a ∈ , a^J −a and Δ (, J). Thus and . Comparison of dimensionalities over Γ shows that . Hence S is a simple Lie algebra of type A_II.

We have therefore proved the following

THEOREM 13. Any central simple Lie algebra of type A_i is isomorphic either to a Lie algebra a finite-dimensional central simple associative algebra or to an algebra where is finitedimensional simple associative with an involution J of second kind.

We consider next the Lie algebras of types B, C and D in the following

THEOREM 14. Let be a central simple Lie algebra of type B_l, Then is isomorphic to a Lie algebra where is a finite-dimensional central simple associative algebra, J an involution of first kind in .

Proof: There exists a finite-dimensional Galois extension P of Γ such that _P is the Lie algebra of J-skew matrices in P_n where J is the involution X→X^′, the transpose of X in P_n, or the involution X→Q^−lX′Q where Q′ = −Q and the entries of Q are in the prime field. Also we have n 5 if n = 2l +, n 6 if n = 3l and the involution is X→Q^−lX′Q, and n 10 in the remaining case. For each s in the Galois group G of P over Γ we have the automorphism U_s of _P over Γ: where (a₁, … a_m) is a basis for _P over Γ and for _P over P. is the subset of _P of elements which are fixed relative to the U_s The conditions on n insure that U_s can be extended to an automorphism U_s of the enveloping associative algebra P_n of _P (Theorem 12). The extension U_s is semi-linear in P_n with associated automorphism s, U₁ = 1 and U_s U_st = U_st hold in P_n Hence the subset of P_n of elements which are fixed relative to the U_s, s ∈ G, is a subalgebra of P_n such that ^P = P_n Hence is finitedimensional central simple over Γ. If X∈ _P, then X^J = −X and XU_s ∈ _p. Hence X^JU_s = −XU_s = (XU_s)^J. Thus JU_s = U_sJ holds in _P. Since P_n is the enveloping associative algebra of _P = (P_n, J) it follows that JU_s = U_sJ in P_n also. This implies that J maps into itself and hence J induces an involution in over Γ which is of first kind since Γ is the center of . If a ∈ then a^J = −a so and . On the other hand, . Hence . This completes the proof.

6. A closer look at the isomorphism conditions

We have seen in §4, that if J_i = 1,2, is an involution (either kind) in a finite-dimensional simple associative algebra _i over Φ, then implies that₁ and ₂ are isomorphic. It therefore suffices to consider one algebra = ₂ = ₂ and consider the condition for isomorphism of and where J and K are involutions in . We have seen that any isomorphism θ of into can be realized by an automorphism of . If a ∈ then a^Jθ = −a^θ a^θK. Thus Jθ = θK holds in . Since the enveloping Φ-algebras of and are we have Jθ = θK in or K = θ We shall call the involutions J and K of cogredient if there exists an automorphism θ of such that K = θ⁻¹ Jθ. We have seen that cogredience is a necessary condition for isomorphism of and . Conversely, if J and K are cogredient and K =θ⁻¹ Jθ where θ is an automorphism then θ maps into and onto . Hence θ induces an isomorphism of the Lie algebra onto . Thus if and only if J and K are cogredient. We see also that the group of automorphisms of the Lie algebra can be identified with the subgroup of the group of automorphisms θ of the algebra such that θJ = Jθ

Now let = the algebra of linear transformations in the finitedimensional vector space over the finite dimensional division algebra Δ. Let be an involution in Δ and let (x, y) be a hermitian or skew hermitian form corresponding to this involution. Then the mapping A→A*, A ∈ , A* the adjoint of A, is an involution in . We recall that A* is the unique linear transformation such that (xA, y) = (x,yA*) and we have where ∈ = 1 or −1 according as the form is hermitian or skew hermitian. Suppose a→ a′ is a second involution in Δ and (x,y)₁ a second hermitian or skew hermitian form corresponding to this so we have (y, x) = ∈₁(x, y)′₁ where ∈₁ = ±1. Suppose the involution determined by this form is the same as that given by (x, y). Thus if (x, y A*) = (xA, y) then (x, yA*) 1 = (xA,y)₁. Let u and v be arbitrary vectors in . Then x → (x, u)U is a linear transformation in and one checks that its adjoint relative to (x,y) is x→∈(x, v)u. Hence we have

Since x, y, u, v are arbitrary this shows that (x, y)₁ = (x, y)ρ, ρ ≠ 0 in Δ. If α∈Δ (x, αy)₁ = (x, y)α′ gives hence Also we have . Hence . Conversely, let ρ be any element of Δ satisfying . Then a direct verification shows that is an involution in Δ and (x, y)₁ ≡ (x, y)ρ is hermitian or skew hermitian relative to this involution. If (x, y) is skew hermitian and then (x, y)ρ is hermitian. Hence if Δ contains a skew element ≠ 0, then a skew hermitian form can be replaced by a hermitian one which gives the same involution A→ A* in . If Δ contains no such elements then for all ρ ∈ Δ and this implies that Δ is a field. Hence we may restrict our attention to hermitian forms and to alternate forms (Δ a field). Two such forms (x, y) and (x, y) give the same involution in if and only if is the involution of (x, y).

It is known that any automorphism of has the form A → S⁻¹ AS where S is a semi-linear transformation in over Δ (Jacobson [3], p. 45). If θ is the automorphism in Δ associated with S, then one checks that is hermitian or alternate with involution Moreover, if A ∈ then

Thus (xA, y)₁ = (x, y(S(S^−l AS)*S^−l)₁ and the involution in determined by (x, y)₁ is A → S(S⁻¹ AS)*S⁻¹. Thus if we call A → A*, J, and A→ S⁻¹ AS, θ then the new involution is K = θJθ^−l which is cogredient to J. Because of this relation it is natural to extend the usual notion of equivalence of forms in the following manner:

Two hermitian forms (x, y) and (x, y)₁ are said to be S-equivalent if there exists a 1:1 semi-linear transformation S with associated isomorphism θ such that (x, y)₁ = (xS,yS)^θ-1.

It is well known that any two non-degenerate alternate forms are equivalent in the ordinary sense. Hence the involutions in determined by any two such forms are cogredient. Moreover, these are not cogredient to any involution determined by a hermitian form. Our results imply also that the non-degenerate hermitian forms (x, y) and (x,y)i define cogredient involutions in if and only if (x, y)₁ = (xS, yS)^θ-1ρ where S is semi-linear with automorphism θ and ρ is symmetric relative to .

7. Central simple real Lie algebras

We shall now apply our results and known results on associative algebras to classify the central simple Lie algebras of types A-D (except D₄) over the field Φ of real numbers. By Frobenius’ theorem, the finite-dimensional division algebras over Φ are: Φ; the complex field the quaternion division algebra Δ with basis 1, i,j, k such that

Δ has the standard involution Since the automorphisms in Δ are all inner, every involution in Δ is either standard or it has the form where The dimensionality of the space of skew elements under the standard involution is three; under the dimensionality is one. We denote the automorphism ≠ 1 in P over Φ by

By the Wedderburn theorem, the finite-dimensional simple associative algebras over Φ are the full matrix algebras Φ_n, P_n and Δ_n. These can be identified with the algebras (Φ, n), (P, n) and (Δ, n) of linear transformations in over Φ, P and Δ respectively. The algebras are central and have only inner automorphisms. The center of (P, n) P_n is P. In addition to inner automorphisms, (P, n) has the outer automorphisms X→S^-1XS where S is a semi-linear transformation with associated automorphism in P.

All our algebras have involutions and (P, n) has involutions of second kind. The involutions of ((Φ,n) have the form X→ X* where X* is the adjoint of X relative to a non-degenerate symmetric or skew B_i linear form in over Φ. Involutions determined by skew forms are not cogredient to any determined by a symmetric form. Any two non-degenerate skew B_i linear forms are equivalent so these give a single cogredience class of involutions. If (x, y) and (x, y)₁ are non-degenerate the criterion of the last section shows that the involution determined by (x, y) is cogredient to that of (x, y)₁ if and only if (x, y) is equivalent to a multiple of (x, y)₁. Since (x, y) is equivalent to any positive multiple of (x, y) it follows that the involution given by (x, y) is cogredient to that of (x, y)i if and only if (x, y) is equivalent to ± (x, y)₁. If (x, y) is nondegenerate symmetric it is well known that there exists a basis (u₁, u₂, …, u_n) for such that

The number p is an invariant by Sylvester’s theorem. Accordingly, we obtain [n/2] + 1 cogredience classes of involutions corresponding to the values p = 0,1, …, [n/2]. A simple calculation using the canonical basis (13) shows that the dimensionality over Φ of the space of skew elements determined by (x, y) is (n − n)/2. This implies that the Lie algebra (, J) of these elements is of type B or D according as n is odd or even. The Lie algebra determined by a skew bilinear form (x, y) is of type C.

We consider next the involutions of second kind in (P, n). Such an involution is the adjoint mapping relative to a non-degenerate hermitian form (x, y) in over P. A basis (u₁, u₂, …, u_n) can be chosen so that (13) holds. If (x, y) is S-equivalent to (x, y)₁ in the sense that there exists a semi-linear transformation S with automorphism in Psuch that (x, y) = (xS, yS)₁ then we have Then (x, y) and (x, y)₁ are equivalent in the usual sense. Since (x, y) is equivalent to γ(x, y), γ real and positive, it follows that (x, y) and (x, y)₁ determine cogredient involutions if and only if (x, y) is equivalent to ±(x, y)₁. This and Sylvester’s theorem for hermitian forms implies again that the number of non-cogredient involutions of second kind in (P, n) is [n/2] + 1.

The discussion we have just given carries over verbatim to hermitian forms in over Δ for which the involution in Δ is the standard one. A basis satisfying (13) can be chosen (cf. Jacobson [2], vol. II, p. 159). The number of cogredience classes of involutions given by the standard hermitian forms is [n/2] + 1. If we use a canonical basis for which (13) holds it is easy to calculate that the dimensionality of ((Δ, n), J) determined by the associated involution J is n(2_n + 1). Since Δ_P P₂, (Δ, n)_P P_2n. Since ((Δ, n), J)_P has dimensionality n(2n + 1) over P it follows that is central simple of type C_n.

It remains to consider the hermitian forms (x, y) in over Δ whose involutions in Δ are of the form If we replace (x, y) by (x, y)q^-1 we obtain a skew hermitian form relative to the standard involution. We prefer to treat these.

LEMMA 5. If q₁ and q₂ are non-zero skew in Δ (relative to the standard involution), then there exists a non-zero a in Δ such that

Proof: We note first that if b₁ and b₂ are elements of Δ not in Φ which have equal traces and norms, then there exists an isomorphism of Φ(b₁) onto Φ(b ₂) mapping b₁ into b₂. This can be extended to an inner automorphism of Δ (Jacobson, Structure of Rings, p. 162). It follows that b₁ and b₂ are similar, that is, for some c in Δ. In particular, if N(q₂) = N(q₁), then since the traces T(q₁) = 0 = T(q₂), there exists a c such that We note also that since the norm of any non-zero element is positive, q₁ is necessarily similar to a suitable positive multiple of q₂. We now see that it suffices to prove the lemma for q₂ = γq₁, γ > 0. Then we consider the quadratic field Φ(q₁) = Φ(q₂). We have γ = N(c) for some c in this field. Hence as required.

The usual method of obtaining a diagonal matrix for a hermitian or skew hermitian form now gives the following

LEMMA 6. If (x, y) is a non-degenerate skew hermitian bilinear form relative to the standard involution in Δ, then there exists a basis (u₁, u₂, …, u_n) for such that the matrix ((u_i, u_j)) =

where q is any selected non-zero skew element of Δ.

This result implies that there is just one cogredience class of involutions in (Δ, n) given by skew hermitian forms in over Δ.

If we use (14) we can calculate the dimensionality for the Lie algebra defined by the involution. This is n(2n – 1) which implies that the Lie algebra is central simple of type D_n

We can now list a set of representatives for the isomorphism classes of central simple Lie algebras of types A-D over Φ, as follows.

Let (P, n, p) denote the Lie algebra of matrices X ∈ P_n such that Then the Lie algebras (P, n, p) for p = 0,1, …, [n/2] constitute our list.

Type B. Let (Φ, n, p) be the Lie algebra of matrices X ∈ Φ_n such that Then our list is the set of Lie algebras (P, n, p) with n odd, n 5.

Type C. Let (Δ, n, p) be the Lie algebra of matrices X ∈ Δ_n satisfying and let (Φ,2_n, Q) the Lie algebra of matrices in Φ_2n such that Q^-1X^′Q = –X where Q is any skew symmetric matrix. Then the list is: (Δ, n, p), n 3, p = 0, …, [n/2] and (Φ, 2n, Q), n 3.

Type D. Let (Δ, n, Q) be the Lie algebra of matrices in Δ_n satisfying Q^-1X^′Q = –X where Q is a skew hermitian matrix. The list is; (Δ, n, Q), n > 5, and (Φ, 2n, S_P), n > 5, and p = 0,1, ···,n.

Exercises

1.   Determine the groups of automorphisms of the simple real Lie algebras of types A-D, except D₄.

2.   Determine an invariant non-degenerate symmetric bilinear form for every central simple real Lie algebra of type A-D. use this to enumerate the compact Lie algebras in the list (cf. § 4.7).

3.   Show that the Lie algebras S(Δ, 4, Q) and S(Φ, 8, S₁) in the notation of § 7 are isomorphic, (This shows that Theorem 12 is not valid for Lie algebras of type D₄, since Δ₄ and Φ₈ are not isomorphic.)

4.   A (generalized) quaternion algebra over an arbitrary field Φ is defined to be an algebra with basis l,i, j, k such that 1 is the identity and the multiplication table for i, j, k is

Show that an algebra Δ is a quaternion algebra over Φ if and only if Δ_Ω Ω₂ for Ω the algebraic closure of Φ.

5. A non-associative algebra over an arbitrary field Φ is called a Cayley algebra if has an identity 1, contains a quaternion subalgebra Δ containing 1 and every element of can be writen in one and only one way in the form Δ + bu, a, b in Δ and u an element of such that

where μ is a non-zero element of Φ. Show that is alternative (cf. § 4.6) and that the mapping x = a + bu → – bu is an involution in . Prove that where N(x) ∈ Φ and satisfies N(xy) = N(x)N(y). Let (x,y) = ½ [N(x + y) — N(x) — N(y)]. Show that (x, y) is a non-degenerate symmetric bilinear form and that two Cayley algebras are isomorphic if and only if their forms (x, y) are equivalent. Prove that a non-associative algebra is a Cayley algebra if and only if _Ω is the split Cayley algebra of § 4.6.

6.  Use Theorem 9.7 and Exercise 5 to prove that two Cayley algebras over a field of characteristic Φ are isomorphic if and only if their derivation algebras are isomorphic.

7.   Prove that a central simple Lie algebra over a field of characteristic Φ is of type G if and only if it is isomorphic to the derivation algebra of a Cayley algebra.