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**The Generalized Burnside Theorem in Noncommutative Deformation Theory ^{*}**

**Eivind Eriksen
**

BI Norwegian Business School, Department of Economics, N-0442 Oslo, Norway

^{*}This article is a part of a Special Issue on Deformation Theory and Applications (A. Makhlouf, E. Paal and A. Stolin, Eds.).

- Corresponding Author:
- Eivind Eriksen

**E-mail:**[email protected]

**Received Date:** October 01, 2009; **Accepted Date:** January 26, 2011

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

Let A be an associative algebra over a field k, and letMbe a finite family of right A-modules. A study of the noncommutative deformation functor DefM of the familyMleads to the construction of the algebra OA(M) of observables and the generalized Burnside theorem, due to Laudal (2002). In this paper, we give an overview of aspects of noncommutative deformations closely connected to the generalized Burnside theorem.

Let k be a field and let A be an associative k-algebra. For any right A-moduleM, 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 (MR, τ), where MR is an R-A bimodule (on which k acts centrally) that is R-flat, and τ : k ⊕R MR → 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 ar is an Artinian ring, together with a pair of structural ring homomorphisms
f : k^{r} → R and g : R → k^{r} with gof = id, such that the radical I(R) = ker(g) is nilpotent. Any algebra R in a_{r}
has r simple right modules of dimension one, the natural projections {k_{1}, . . . , k_{r}} of kr.

In [2], a noncommutative deformation functor Def_{M} : ar → Sets of a finite family M = {M1, . . . , Mr} of
right A-modules was introduced, as a generalization of the commutative deformation functor DefM : 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 caser > 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} : k_{i} ⊕_{R}M_{R} → Mi is an isomorphism of right A-modules
for 1 ≤ i ≤ r. We remark that M_{R} is R-flat if and only if

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 : A → ⊕_{i} End_{k}(M_{i}) of A on the
familyM.

There is an obstruction theory for DefM, 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 and the commutative diagram

given by the versal family M_{H} ∈ Def_{M}(H). The algebra B = O^{A}(M) has an induced right action on the family
Mextending the action of A, and we may considerMas a family of right B-modules. In fact,Mis the family of
simple B-modules since π can be identified with the quotient morphism B → B/radB.

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 [2], generalizing the structure theorem for
semi-simple algebras and the classical Burnside theorem. Laudal’s result is stated in the following form.

**Theorem** (The generalized Burnside theorem). Let A be a finite-dimensional algebra over a field k, and letM =
{M1,M2, . . . , Mr} be the family of simple right A-modules. If EndA(Mi) = k for 1 ≤ i ≤ r, then η : A →
OA(M) 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 letMbe any finite family of right
A-modules of finite dimension over k. Then the algebra B = O^{A}(M) has the property that ηB : 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 Mhas 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.

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 gof = 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 right R-modules are the projections {k_{1}, . . . , k_{r}} of k_{r}.

Let A be an associative k-algebra. For any familyM = {M_{1}, . . . , M_{r}} of right A-modules, there is a noncommutative
deformation functor Def_{M} : a_{r} → Sets, introduced by Laudal [2]; see also Eriksen [1]. For an algebra
R in ar, we recall that a deformation ofMover 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} : k_{i}⊕_{R}M_{R} → Mi is an isomorphism of right A-modules for 1 ≤ i ≤ r.
Moreover, are equivalent deformations over R if there is an isomorphism of R-A bimodules such that We may prove that M_{R} is R-flat if and only if

considered as a left R-module, and a deformation in Def_{M}(R) may be thought of as a right multiplication A →
EndR(MR) of A on the left R-moduleMR that lifts the multiplication ρ : A→⊕_{i} End_{k}(M_{i}) of A on the familyM.

Let us assume that M is a swarm, that is, Ext^{1}_{A}(M_{i},M_{j}) has finite dimension over k for 1 ≤ i, j ≤ r. 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-category âr, and that M_{H} ∈ Def_{M}(H) is a family defined over
H with the following versal property: for any algebra R in ar and any deformation MR ∈ DefM(R), there is
a homomorphism φ : H → R such that DefM(φ)(MH) = MR. The formal moduli (H,MH) is unique up to
non-canonical isomorphism. However, the morphism φ is not uniquely determined by (R,MR).

WhenMis a swarm with formal moduli (H,MH), right multiplication on the H-A bimodule MH by elements in A determines an algebra homomorphism

We write OA(M) = EndH(MH) and call it the algebra of observables. Since MH is H-flat, we have that and it follows that OA(M) is explicitly given as the matrix algebra

Let us write for the structural algebra homomorphism defining the right A-module structure on Mi for 1 ≤ i ≤ r, and

for their direct sum. Since H is a complete r-pointed algebra in âr, there is a natural morphism H → k^{r}, inducing
an algebra homomorphism

By construction, there is a right action of OA(M) on the familyMextending the right action of A, in the sense that the diagram

commutes. This makes it reasonable to call OA(M) the algebra of observables.

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 DefM : ar → Sets of the family M = {M_{1},M_{2}, . . . ,
M_{r}} of simple right A-modules. Clearly, M is a swarm, hence DefM 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). then ρ is surjective. In particular, ρ is surjective
when k is algebraically closed.

Proof. There is a factorization then is an isomorphism by the classification theory for semi-simple algebras. Since
EndA(Mi) 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 for the algebra homomorphism induced by ρ. We observe that ρ is
surjective if and only if is an isomorphism. Moreover, let us write J = rad(O^{A}(M)) for the Jacobson radical of
O^{A}(M). Then we see that

Since ρ(radA) = 0 by definition, it follows that η(radA) ⊆ J. Hence there are induced morphisms

for all q ≥ 0. We may identify gr(η)0 with The conclusion in Burnside’s theorem is therefore equivalent to the statement that gr(η)0 is an isomorphism.

**Theorem 2** (The generalized Burnside theorem). Let A be a finite-dimensional algebra over a field k, and letM=
{M_{1},M_{2}, . . . , M_{r}} be the family of simple right A-modules. If End_{A}(M_{i}) = k for 1 ≤ i ≤ r, then η : A→O^{A}(M)
is an isomorphism. In particular, η is an isomorphism when k is algebraically closed.

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 DefM; see Laudal [2, Theorem 3.2] for details. When A is finite-dimensional, the right regular A-module
AA has a decomposition series

with F_{p}/F_{p−1} a simple right A-module for 1 ≤ p ≤ n. Namely, A_{A} is an iterated extension of the modules inM.
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.

Let A be a finite-dimensional algebra over a field k, and letM= {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, we may show that M is a
swarm of B-modules, since B is complete and B/(radB)^{n} has finite dimension over k for all positive integers n.

Proposition 3. If k is an algebraically closed field, then ηB : B →O^{B}(M) is an algebra isomorphism.

Proof. SinceMis a swarm of A-modules and of B-modules, we may consider the commutative diagram

The algebra homomorphism η^{B} induces maps Since k is algebraically
closed and B/rad(B)n has finite dimension over k, it follows from the generalized Burnside theorem that 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 familyMis the family of 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 characterize the algebra B = OA(M) of observables.

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 correspond to the presheaves {Mλ : λ ∈ Λ} defined by Mλ(λ) = k and Mλ(λ’) = 0 forλ’ = λ. The following results are well known:

**A hereditary example**

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 = {M_{1},M_{2},M_{3},M_{4}}, and we can easily compute the algebra
OA(M) of observables since Ext^{2}_{A}(Mi,Mj) = 0 for all 1 ≤ i, j ≤ 4. We obtain

It follows from the generalized Burnside theorem that is an isomorphism. Hence we recover the algebra

**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 = {M1,M2,M3,M4}. Since we must compute the cup-products

in order to compute H. These cup-products are non-trivial; see Laudal [2, Remark 3.2] for details. Hence we obtain

Note that H_{14} is two-dimensional at the tangent level and has a relation. Also in this case, it follows from the
generalized Burnside theorem that is an isomorphism. Hence we recover the algebra A ≅ OA(M) ≅
H.

- Eriksen E, Concini C, Oystaeyen FV, Vavilov N, Yakovlev A (2006) An introduction to noncommutative deformations of modules, Noncommutative Algebra and Geometry. Lect Notes Pure Appl Math Chapman & Hall/CRC, Boca Raton, FL, USA X 243: 90–125.
- Laudal OA (2002) Noncommutative deformations of modules. Homology Homotopy Appl 4: 357–396.

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