Algebras

In this section we give some definitions and basic facts concerning the structure of associa-tive algebras. We assume that the reader is familiar with the basic ring theoretic notions for associative algebras over fields (subalgebras, homomorphisms, ideals, factor algebras, nilpotency, modules, direct sums, tensor products, etc.). Throughout the thesis by an algebra we understand a finite dimensional associative algebra over the field K. Unless otherwise stated, we also assume that the algebra has an identity denoted by 1_A (if the algebra is A) or briefly by 1. Modules are assumed to be finite dimensional unital left A-modules. (The A-module U is called unital if 1Au = u for every u ∈ U.) Let A be an algebra and letM be an A-module (which can beAitself). For subsets B ⊆AandC ⊆M

we denote be BC the K-linear span of {bc|b ∈ B, c ∈ C}. For b, c ∈ A we denote by [b, c] the additive commutatorbc−cb ofb and c. For B, C⊆A we use the notation [B, C]

for the linear span of {[b, c]|b ∈ B, c ∈ C}. For a subset B ⊆ A, C_A(B) stands for the centralizer ofB inA: C_A(B) ={x∈A|[x, b] = 0 for every b∈B}. The center C_A(A) ofA is denoted by Z(A). If K is a field then the algebra of n by n matrices with entries from K is denoted by M_n(K). By a matrix algebra over K we mean a subalgebra of M_n(K) containing the identity matrix for some integern.

2.2.1 Structure of algebras

We recommend the reader familiar with the basic structure theory of algebras to skip this part. Here we briefly recall Wedderburn’s theorems on the structure of algebras. In every finite dimensional algebra A there exists a largest nilpotent ideal Rad(A), called the Jacobson radical (or just radical) ofA. Ais called semisimple if Rad(A) = (0). The factor algebraA/Rad(A) of an arbitrary algebra is semisimple. Ais called simple ifAcontains no proper nonzero ideals. A semisimple algebraA can be decomposed into the direct sum of its minimal idealsA₁, . . . , A_r. We refer to the simple algebras A_i as the simple components of A. A simple algebra A is isomorphic to the algebra M_d(D) of d by d matrices with entries fromD, where Dis a division algebra (or skew field) over A. By this we mean that D contains no zero divisors. IfZ is a subfield of Z(A) containing the identity of A then it is possible (and often convenient) to consider A as an algebra over Z. A is called central over K if Z(A) = K (more precisely, Z(A) = K1_A). The dimension of a central simple K-algebra is always a square.

A moduleU over the semisimple algebraAcan be decomposed as a direct sum of simple A-modules (modules with no proper nonzero submodules). If A is a simple algebra then there is only one isomorphism class of simple A-modules. By A^op we denote the algebra opposite to A. A^op has the same vector space structure as A but the multiplication is reversed. Acan be considered as anA⊗_KA^op-module by the multiplication law (a⊗b)c= acb. The ideal structure of A coincides with the A⊗_K A^op-submodule structure of A.

If A is a central simple K-algebra then A⊗_K A^op ∼= M_d²(K) (where d² = dim_KA) and every simple A⊗_K A^op-module is isomorphic to A with the module structure given above (cf. Corollary 12.3 and Proposition 12.4b in [80]).

Let U be a module for a finite dimensional arbitrary algebra A. By Rad(U) we denote the radical of U which is the intersection of its proper maximal submodules. It is known that Rad(U) = Rad(A)U.

2.2.2 Extending scalars

It is sometimes useful to consider theK⁰-algebra K⁰⊗KA whereK⁰ is a field extensionK. We refer to this construction as extending scalars. (For example ifA≤M_n(K) is the matrix algebra generated by matrices g₁, . . . , g_m then we can think of K⁰ ⊗_KA as the subalgebra ofMn(K⁰) generated by the same matricesg1, . . . , gm considered as matrices overK⁰.) For a subspace B ofAwe considerK⁰⊗_KB embedded intoK⁰⊗_KAin the natural way. Many constructions such as products and commutators of complexes and even centralizers behave well with respect to extension of scalars. For example, [K⁰⊗KB, K⁰⊗KC] =K⁰⊗K[B, C]

and C_K⁰⊗_KA(K⁰⊗_KB) =K⁰ ⊗_KC_A(B).

2.2.3 Idempotents and the primary decomposition

An idempotent of A is a nonzero element e ∈ A with e² = e. Two idempotents e and f are called orthogonal if ef = f e = 0. An idempotent is called primitive if it cannot be decomposed as a sum of two orthogonal idempotents. A system e₁, . . . , e_r of pairwise orthogonal idempotents is called complete if their sum is the identity of A. Primitive idempotents of the center of A are called primitive central idempotents. The primitive central idempotents are pairwise orthogonal and form a complete system in Z(A).

Idempotents can be lifted from the semisimple part A/Rad(A). That is, if ee is an idempotent in A/Rad(A) then there exists an idempotent e of A such that e ∈ e. Evene complete systems of pairwise orthogonal idempotents can be lifted: assume that ee1, . . . ,eer

are pairwise orthogonal idempotents ofA/Rad(A) such thatee₁+. . .+ee_r = 1_A/Rad(A). Then there exist e_i ∈ ee_i (i = 1, . . . , r) such that e₁, . . . , e_r are pairwise orthogonal idempotents of A and e1+. . .+er = 1A.

If we lift a complete system ee₁, . . . ,ee_r of pairwise orthogonal primitive central idempo-tents of Rad(A) as above, the we obtain a decomposition

A=e1Ae1+. . .+erAer+N⁰,

as a direct sum of vector spaces, where N⁰ is a subspace of Rad(A) and for every i ∈ {1, . . . , r}, the subspace A_i = e_iAe_i is a subalgebra of A with identity element e_i. Furthermore, A_i are primary algebras. (An algebra B is primary if B/Rad(B) is simple.) The decomposition above is called the primary decomposition of A, see Theorem 49.1 of [67].

2.2.4 Separability and the Wedderburn–Malcev theorem

It is obvious that K⁰⊗_KRad(A) is a nilpotent ideal ofK⁰⊗_KA. However, there are cases where Rad(K⁰⊗KA) can be bigger thanK⁰⊗KRad(A). A general sufficient condition for Rad(K⁰⊗_KA) = K⁰⊗_KRad(A) is that K⁰ is a (not necessarily finite) separable extension of K. We say that A is separable over K if for every field extension K⁰ of K the K⁰ -algebraK⁰⊗KA is semisimple. (Note that in Chapter 10 of [80] a more general definition of separable algebras over an arbitrary ring is given. The simple definition given here for algebras over a field is equivalent to the general one, see Corollary 10.6 of [80]). Separability of finite dimensional algebras generalizes the notion of separability of finite field extensions:

by Proposition 10.7 of [80], A is separable iff the centers of the simple components of A are separable extensions of K. From this characterization it follows immediately that A is separable over K if and only if K⁰ ⊗K A is semisimple where K⁰ denotes the algebraic closure ofK. Obviously, over a perfect ground field K the notion of separability coincides with semisimplicity. It is immediate that ifA is separable thenK⁰⊗A is separable as well for an arbitrary field extension K⁰ of K. Direct sums, homomorphic images and tensor products of separable algebras are separable as well (cf. Section 10.5 of [80]).

An extremely useful result where separability plays a role is the Wedderburn–Malcev Principal Theorem (See Section 11.6 of [80] for a general form): Assume that A/Rad(A) is separable. Then there exists a subalgebra D ≤ A such that D ∼= A/Rad(A) and A=D+ Rad(A) (direct sum of vector spaces). Furthermore, ifD₁ is another subalgebra such that D1 ∼= A/Rad(A) then there exists an element w ∈ Rad(A) such that D1 = (1 +w)⁻¹D(1 +w). We shall refer to such subalgebras as Wedderburn complements inA.

We shall make use of the following consequence of the Principal Theorem. It states that separable subalgebras of A/Rad(A) can be lifted to A.

Corollary 2.1. Let A be a finite dimensional K-algebra and B ≤ A be a subalgebra of A which is separable over K and assume that De is a separable subalgebra of A/Rad(A) containing B +Rad(A). Then there exists a subalgebra D of A such that B ≤ D and D∼=D.e

Proof. Working in the pre-image of De at the natural projection A → A/Rad(A) we may assume that De =A/Rad(A). Then, by the first part of the principal theorem there exists a subalgebra D1 ≤A such thatD1 ∼=De andA=D1+ Rad(A). Letπ be the projection of A onto D₁ corresponding to this decomposition and B₁ =π(B+ Rad(A)). By comparing dimensions it is clear thatB₁+ Rad(A) =B+ Rad(A). By the second part of the principal theorem, applied to the algebraB+ Rad(A), there exists an elementw∈Rad(A) such that (1−w)⁻¹B(1−w) =B₁. Now the subalgebra D= (1−w)D₁(1−w)⁻¹ has the required property.

2.2.5 Tori

A toralK-algebra or torus over K is a finite dimensional commutative K-algebra which is separable overK. LetK⁰stand for the algebraic closure ofK. ThenT is a torus if and only ifK⁰⊗T is isomorphic to the direct sum of copies ofK⁰. LetT ≤M_n(K) be a commutative matrix algebra. Then T is a torus if and only if the matrices in T can be simultaneously diagonalized over K⁰. By this we mean that there exists a matrix b ∈ M_n(K⁰) such that b⁻¹T b ⊆ Diag_n(K⁰), where Diag_n(K⁰) is the matrix algebra consisting of the diagonal n byn matrices. (The diagonalization can be obtained by decomposingK⁰⊗V into a direct sum of irreducible K⁰ ⊗T-modules.) By a maximal torus of the algebra A we mean a torus which is not properly contained in any other toral subalgebra of A. Note that by Corollary 2.1, maximal tori of A/Rad(A) can be lifted to maximal tori in A.