The Generalized Burnside Theorem in noncommutative deformation theory

Let A be an associative algebra over a field, and let M be a finite family of right A-modules. Study of the noncommutative deformation functor of the family M leads to the construction of the algebra of observables and the Generalized Burnside Theorem, due to Laudal. In this paper, we give an overview of aspects of noncommutative deformations closely connected to the Generalized Burnside Theorem.


Introduction
Let k be a field and let A be an associative k-algebra. For any right A-module M , there is a commutative deformation functor Def M : l → Sets defined on the category l of local Artinian commutative k-algebras with residue field k. We recall that for an algebra R in l, a deformation of M to R is a pair (M R , τ ), where M R is an R-A bimodule (on which k acts centrally) that is R-flat, and τ : k ⊗ R M R → M is an isomorphism of right A-modules.
Let a r be the category of r-pointed Artinian k-algebras for r ≥ 1, the natural noncommutative generalization of l. We recall that an algebra R in a r is an Artinian ring, together with a pair of structural ring homomorphisms f : k r → R and g : R → k r with g • f = id, such that the radical I(R) = ker(g) is nilpotent. Any algebra R in a r has r simple left modules of dimension one, the natural projections {k 1 , . . . , k r } of k r .
In Laudal [2], a noncommutative deformation functor Def M : a r → Sets of a finite family M = {M 1 , . . . , M r } of right A-modules was introduced, as a generalization of the commutative deformation functor Def M : l → Sets of a right A-module M . In the case r = 1, this generalization is completely natural, and can be defined word for word as in the commutative case. The generalization to the case r > 1 is less obvious and has further-reaching consequences, but is still very natural. A deformation of M to R is defined to be a pair (M R , {τ i } 1≤i≤r ), where M R is an R-A bimodule (on which k acts centrally) that is R-flat, and τ i : considered as a left R-module, and that a deformation in Def M (R) may be thought of as a right multiplication A → End R (M R ) of A on the left R-module M R that lifts the multiplication ρ : There is an obstruction theory for Def M , generalizing the obstruction theory for the commutative deformation functor. Hence there exists a formal moduli (H, M H ) for Def M (assuming a mild condition on M). We consider the algebra of observables When A is an algebra of finite dimension over an algebraically closed field k and M is the family of simple right A-modules, Laudal proved the Generalized Burnside Theorem in Laudal [2], generalizing the structure theorem for semi-simple algebras and the classical Burnside Theorem. Laudal's result be stated in the following form: Theorem (The Generalized Burnside Theorem). Let A be a finite-dimensional algebra over a field k, and let M = {M 1 , M 2 , . . . , M r } be the family of simple is an isomorphism. In particular, η is an isomorphism when k is algebraically closed.
Let A be an algebra of finite dimension over an algebraically closed field k and let M be any finite family of right A-modules of finite dimension over k. Then the algebra B = O A (M) has the property that η B : B → O B (M) is an isomorphism, or equivalently, that the assignment (A, M) → (B, M) is a closure operation. This means that the family M has exactly the same module-theoretic properties, in terms of (higher) extensions and Massey products, considered as a family of modules over B as over A.

Noncommutative deformations of modules
Let k be a field. For any integer r ≥ 1, we consider the category a r of r-pointed Artinian k-algebras. We recall that an object in a r is an Artinian ring R, together with a pair of structural ring homomorphisms f : k r → R and g : R → k r with g • f = id, such that the radical I(R) = ker(g) is nilpotent. The morphisms of a r are the ring homomorphisms that commute with the structural morphisms. It follows from this definition that I(R) is the Jacobson radical of R, and therefore that the simple left R-modules are the projections {k 1 , . . . , k r } of k r .
Let A be an associative k-algebra. For any family M = {M 1 , . . . , M r } of right A-modules, there is a noncommutative deformation functor Def M : a r → Sets, introduced in Laudal [2]; see also Eriksen [1]. For an algebra R in a r , we recall that a deformation of M over R is a pair (M R , {τ i } 1≤i≤r ), where M R is an R-A bimodule (on which k acts centrally) that is R-flat, and τ i : considered as a left R-module, and a deformation in Def M (R) may be thought of as a right multiplication Let us assume that M is a swarm, i.e. that Ext 1 Then Def M has a pro-representing hull or a formal moduli (H, M H ), see Laudal [2], Theorem 3.1. This means that H is a complete r-pointed k-algebra in the pro-categoryâ r , and that M H ∈ Def M (H) is a family defined over H with the following versal property: For any algebra R in a r and

The generalized Burnside theorem
Let k be a field and let A be a finite-dimensional associative k-algebra. Then the simple right modules over A are the simple right modules over the semi-simple quotient algebra A/ rad(A), where rad(A) is the Jacobson radical of A. By the classification theory for semi-simple algebras, it follows that there are finitely many non-isomorphic simple right A-modules.
We consider the noncommutative deformation functor Def M : a r → Sets of the family M = {M 1 , M 2 , . . . , M r } of simple right A-modules. Clearly, M is a swarm, hence Def M has a formal moduli (H, M H ), and we consider the commutative diagram By a classical result, due to Burnside, the algebra homomorphism ρ is surjective when k is algebraically closed. This result is conveniently stated in the following form: Theorem 1 (Burnside's Theorem). If End A (M i ) = k for 1 ≤ i ≤ r, then ρ is surjective. In particular, ρ is surjective when k is algebraically closed.

Proof. There is an obvious factorization
is an isomorphism by the classification theory for semi-simple algebras. Since End A (M i ) is a division ring of finite dimension over k, it is clear that End A (M i ) = k whenever k is algebraically closed.
Let us write ρ : A/ rad A → ⊕ i End k (M i ) for the algebra homomorphism induced by ρ. We observe that ρ is surjective if and only if ρ is an isomorphism. Moreover, let us write J = rad(O A (M)) for the Jacobson radical of O A (M). Then we see that J = (rad(H) ij ⊗ k Hom k (M i , M j )) = ker(π) Since ρ(rad A) = 0 by definition, it follows that η(rad A) ⊆ J. Hence there are induced morphisms The conclusion in Burnside's Theorem is therefore equivalent to the statement that gr(η) 0 is an isomorphism. Proof. It is enough to prove that η is injective and that gr(η) q is an isomorphism for q = 0 and q = 1, since A and O A (M) are complete in the rad(A)-adic and J-adic topologies. By Burnside's Theorem, we know that gr(η) 0 is an isomorphism. To prove that η is injective, let us consider the kernel ker(η) ⊆ A. It is determined by the obstruction calculus of Def M ; see Laudal [2], Theorem 3.2 for details. When A is finite-dimensional, the right regular A-module A A has a decomposition series That is, A A is an iterated extension of the modules in M. This implies that η is injective; see Laudal [2], Corollary 3.1. Finally, we must prove that gr(η) 1 : rad(A)/ rad(A) 2 → J/J 2 is an isomorphism. This follows from the Wedderburn-Malcev Theorem; see Laudal [2], Theorem 3.4 for details.

Properties of the algebra of observables
Let A be a finite-dimensional algebra over a field k, and let M = {M 1 , . . . , M r } be any family of right A-modules of finite dimension over k. Then M is a swarm, and we denote the algebra of observables by B = O A (M). It is clear that is semi-simple, and it follows that M is the family of simple right B-modules. In fact, one may show that M is a swarm of B-modules, since B is complete and B/(rad B) n has finite dimension over k for all positive integers n. Proof. Since M is a swarm of A-modules and of B-modules, we may consider the commutative diagram w w n n n n n n n n n n n n ⊕ 1≤i≤r End k (M i ) The algebra homomorphism η B induces maps B/ rad(B) n → C/ rad(C) n for all n ≥ 1. Since k is algebraically closed and B/ rad(B) n has finite dimension over k, it follows from the Generalized Burnside Theorem that B/ rad(B) n → C/ rad(C) n is an isomorphism for all n ≥ 1. Hence η B is an isomorphism.
In particular, the proposition implies that the assignment (A, M) → (B, M) is a closure operation when k is algebraically closed. In other words, the algebra B = O A (M) has the following properties: (1) The family M is the family the simple B-modules.
(2) The family M has exactly the same module-theoretic properties, in terms of (higher) extensions and Massey products, considered as a family of modules over B as over A.
Moreover, these properties characterizes the algebra B = O A (M) of observables.

Examples: Representations of ordered sets
Let k be an algebraically closed field, and let Λ be a finite ordered set. Then the algebra A = k[Λ] is an associative algebra of finite dimension over k. The category of right A-modules is equivalent to the category of presheaves of vector spaces on Λ, and the simple A-modules corresponds to the presheaves {M λ : λ ∈ Λ} defined by M λ (λ) = k and M λ (λ ′ ) = 0 for λ ′ = λ. The following results are well-known: (1) If λ > λ ′ in Λ and {γ ∈ Λ : λ > γ > λ ′ } = ∅, then Ext 1 Let us first consider the following ordered set. We label the elements by natural numbers, and write i → j when i > j: In this case, the simple modules are given by M = 5.2. The diamond. Let us also consider the following ordered set, called the diamond. We label the elements by natural numbers, and write i → j when i > j: In this case, the simple modules are given by M = {M 1 , M 2 , M 3 , M 4 }. Since