The Generalized Burnside Theorem in Noncommutative Deformation Theory

Journal of Generalized Lie Theory and Applications Vol. 5 (2011), Article ID G110109,5pages doi:10.4303/jglta/G110109

Research Article

The Generalized Burnside Theorem in Noncommutative Deformation Theory

Eivind Eriksen

BI Norwegian Business School, Department of Economics, N-0442 Oslo, Norway Address correspondence to Eivind Eriksen, [email protected]

Received 1 October 2009; Accepted 26 January 2011

Abstract LetAbe an associative algebra over a fieldk, and letMbe a finite family of rightA-modules. A study of the noncommutative deformation functorDef_Mof the familyMleads to the construction of the algebraO^A(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.

MSC 2010: 13D10, 14D15 1 Introduction

Letkbe a field and letAbe an associativek-algebra. For any rightA-moduleM, there is a commutative deformation functorDef_M : l→Setsdefined on the categorylof local Artinian commutativek-algebras with residue fieldk. We recall that for an algebraRinl, a deformation ofMtoRis a pair(M_R, τ), whereM_Ris anR-Abimodule (on whichkacts centrally) that isR-flat, andτ:k⊗_RM_R→Mis an isomorphism of rightA-modules.

Leta_rbe the category ofr-pointed Artiniank-algebras forr ≥1, the natural noncommutative generalization ofl. We recall that an algebraRina_ris an Artinian ring, together with a pair of structural ring homomorphisms f:k^r→Randg:R→k^rwithg◦f= id, such that the radicalI(R) = ker(g)is nilpotent. Any algebraRina_r hasrsimple right modules of dimension one, the natural projections{k₁, . . . , k_r}ofk^r.

In [2], a noncommutative deformation functorDef_M : a_r →Setsof a finite familyM={M₁, . . . , M_r}of rightA-modules was introduced, as a generalization of the commutative deformation functorDef_M : l→Setsof a rightA-moduleM. In the caser= 1, this generalization is completely natural, and can be defined word for word as in the commutative case. The generalization to the caser >1is less obvious and has further-reaching consequences, but is still very natural. A deformation ofMtoRis defined to be a pair(M_R,{τ_i}_1≤i≤r), whereM_Ris anR-A bimodule (on whichkacts centrally) that isR-flat, andτ_i:k_i⊗_RM_R→M_iis an isomorphism of rightA-modules for1≤i≤r. We remark thatM_RisR-flat if and only if

M_R∼=

R_ij⊗_kM_j

=

⎛

⎜⎜

⎜⎜

⎜⎝

R₁₁⊗_kM₁R₁₂⊗_kM₂ · · ·R_1r⊗_kM_r R₂₁⊗_kM₁R₂₂⊗_kM₂ · · ·R_2r⊗_kM_r

... ... . .. · · · R_r1⊗_kM₁ R_r2⊗_kM₂ · · ·R_rr⊗_kM_r

⎞

⎟⎟

⎟⎟

⎟⎠ ,

considered as a leftR-module, and that a deformation inDef_M(R)may be thought of as a right multiplication A→End_R(M_R)ofAon the leftR-moduleM_Rthat lifts the multiplicationρ: A→ ⊕_iEnd_k(M_i)ofAon the familyM.

There is an obstruction theory forDef_M, generalizing the obstruction theory for the commutative deformation functor. Hence there exists a formal moduli(H, M_H)forDef_M(assuming a mild condition on_M). We consider thealgebra of observablesO^A(M) = End_H(M_H)∼= (H_ij⊗_kHom_k(M_i, M_j))and the commutative diagram

A ^η //

ρJJJJJJ%%

JJ JJ

J ^OA(M)

π

1≤i≤r⊕ End_k(M_i)

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

given by the versal familyM_H ∈Def_M(H). The algebraB =O^A(M)has an induced right action on the family Mextending the action ofA, and we may considerMas a family of rightB-modules. In fact,Mis the family of simpleB-modules sinceπcan be identified with the quotient morphismB→B/radB.

WhenAis an algebra of finite dimension over an algebraically closed fieldkandMis the family of simple rightA-modules, Laudal proved thegeneralized Burnside theoremin [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). LetAbe a finite-dimensional algebra over a fieldk, and letM= {M₁, M₂, . . . , M_r}be the family of simple rightA-modules. If End_A(M_i) = k for1 ≤ i ≤ r, thenη : A → O^A(M)is an isomorphism. In particular,ηis an isomorphism whenkis algebraically closed.

LetAbe an algebra of finite dimension over an algebraically closed fieldkand letMbe any finite family of right A-modules of finite dimension overk. Then the algebraB=O^A(M)has the property thatη_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_Mhas exactly the same module-theoretic properties, in terms of (higher) extensions and Massey products, considered as a family of modules overBas overA.

2 Noncommutative deformations of modules

Letk be a field. For any integerr ≥ 1, we consider the categorya_rofr-pointed Artiniank-algebras. We recall that an object ina_r is an Artinian ring R, together with a pair of structural ring homomorphisms f : k^r → R andg: R →k^rwithg◦f = id, such that the radicalI(R) = ker(g)is nilpotent. The morphisms ofa_rare the ring homomorphisms that commute with the structural morphisms. It follows from this definition thatI(R)is the Jacobson radical ofR, and therefore that the simple rightR-modules are the projections{k₁, . . . , k_r}ofk^r.

LetAbe an associativek-algebra. For any familyM={M₁, . . . , M_r}of rightA-modules, there is a noncom- mutative deformation functorDef_M : a_r → Sets, introduced by Laudal [2]; see also Eriksen [1]. For an algebra Rina_r, we recall that a deformation ofMoverRis a pair(M_R,{τ_i}_1≤i≤r), whereM_Ris anR-Abimodule (on whichkacts centrally) that isR-flat, andτ_i:k_i⊗_RM_R→M_iis an isomorphism of rightA-modules for1≤i≤r. Moreover,(M_R,{τ_i})∼(M_R,{τ_i})are equivalent deformations overRif there is an isomorphismη:M_R→M_R ofR-Abimodules such thatτ_i=τ_i◦(1⊗η)for1≤i≤r. We may prove thatM_RisR-flat if and only if

M_R∼=

R_ij⊗_kM_j

=

⎛

⎜⎜

⎜⎜

⎜⎝

R₁₁⊗_kM₁R₁₂⊗_kM₂ · · ·R_1r⊗_kM_r R₂₁⊗_kM₁R₂₂⊗_kM₂ · · ·R_2r⊗_kM_r

... ... . .. · · · R_r1⊗_kM₁ R_r2⊗_kM₂ · · ·R_rr⊗_kM_r

⎞

⎟⎟

⎟⎟

⎟⎠ ,

considered as a leftR-module, and a deformation inDef_M(R)may be thought of as a right multiplicationA → End_R(M_R)ofAon the leftR-moduleM_Rthat lifts the multiplicationρ:A→ ⊕_iEnd_k(M_i)ofAon the familyM. Let us assume thatMis aswarm, that is,Ext¹_A(M_i, M_j)has finite dimension overkfor1≤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 completer-pointedk-algebra in the pro-categoryˆa_r, and thatM_H ∈ Def_M(H)is a family defined over H with the following versal property: for any algebraR ina_r and any deformationM_R ∈ Def_M(R), there is a homomorphismφ : H → R such thatDef_M(φ)(M_H) = M_R. The formal moduli(H, M_H)is unique up to non-canonical isomorphism. However, the morphismφis not uniquely determined by(R, M_R).

WhenMis a swarm with formal moduli(H, M_H), right multiplication on theH-AbimoduleM_Hby elements inAdetermines an algebra homomorphism

η:A−→End_H(M_H).

We write O^A(M) = End_H(M_H) and call it the algebra of observables. Since M_H is H-flat, we have that End_H(M_H)∼= (H_ij⊗_kHom_k(M_i, M_j)), and it follows thatO^A(M)is explicitly given as the matrix algebra

⎛

⎜⎜

⎜⎜

⎜⎝

H₁₁⊗_kEnd_k(M₁) H₁₂⊗_kHom_k(M₁, M₂)· · ·H_1r⊗_kHom_k(M₁, M_r) H₂₁⊗_kHom_k(M₂, M₁) H₂₂⊗_kEnd_k(M₂) · · ·H_2r⊗_kHom_k(M₂, M_r)

... ... . .. · · · H_r1⊗_kHom_k(M_r, M₁)H_r2⊗_kHom_k(M_r, M₂)· · · H_rr⊗_kEnd_k(M_r)

⎞

⎟⎟

⎟⎟

⎟⎠ .

Let us writeρ_i :A →End_k(M_i)for the structural algebra homomorphism defining the rightA-module structure onM_ifor1≤i≤r, and

ρ:A−→ ⊕

1≤i≤rEnd_k(M_i)

for their direct sum. SinceHis a completer-pointed algebra inˆa_r, there is a natural morphismH →k^r, inducing an algebra homomorphism

π:O^A(M)−→ ⊕

1≤i≤rEnd_k(M_i).

By construction, there is a right action ofO^A(M)on the familyMextending the right action ofA, in the sense that the diagram

A ^η //

ρJJJJJJ%%

JJ JJ

J ^OA(M)

π

1≤i≤r⊕ End_k(M_i)

commutes. This makes it reasonable to callO^A(M)the algebra of observables.

3 The generalized Burnside theorem

Letkbe a field and letAbe a finite-dimensional associativek-algebra. Then the simple right modules overAare the simple right modules over the semi-simple quotient algebraA/rad(A), whererad(A)is the Jacobson radical ofA. By the classification theory for semi-simple algebras, it follows that there are finitely many non-isomorphic simple rightA-modules.

We consider the noncommutative deformation functorDef_M : a_r→ Setsof the familyM={M₁, M₂, . . . , M_r}of simple rightA-modules. Clearly, Mis a swarm, hence Def_M has a formal moduli(H, M_H), and we consider the commutative diagram

A ^η //

ρJJJJJJ%%

JJ JJ

J ^OA(M)

π

1≤i≤r⊕ End_k(M_i).

By a classical result, due to Burnside, the algebra homomorphismρis surjective whenkis algebraically closed. This result is conveniently stated in the following form.

Theorem 1(Burnside’s theorem). IfEnd_A(M_i) =k for1≤i≤r, thenρis surjective. In particular,ρis surjective whenkis algebraically closed.

Proof. There is a factorization A → A/rad(A) → ⊕_iEnd_k(M_i) ofρ. If End_A(M_i) = k for1 ≤ i ≤ r, thenA/rad(A) → ⊕_i End_k(M_i)is an isomorphism by the classification theory for semi-simple algebras. Since End_A(M_i)is a division ring of finite dimension overk, it is clear thatEnd_A(M_i) =kwheneverkis algebraically closed.

Let us writeρ:A/radA→ ⊕_i End_k(M_i)for the algebra homomorphism induced byρ. We observe thatρis surjective if and only ifρis an isomorphism. Moreover, let us writeJ = rad(O^A(M))for the Jacobson radical of O^A(M). Then we see that

J=

rad(H)_ij⊗_kHom_k

M_i, M_j

= ker(π).

Sinceρ(radA) = 0by definition, it follows thatη(radA)⊆J. Hence there are induced morphisms gr(η)_q: rad(A)^q/rad(A)^q+1→J^q/J^q+1

for allq ≥0. We may identifygr(η)₀withρ, since O^A(M)/J ∼= ⊕_i End_k(M_i). The conclusion in Burnside’s theorem is therefore equivalent to the statement thatgr(η)₀is an isomorphism.

Theorem 2(The generalized Burnside theorem). LetAbe a finite-dimensional algebra over a fieldk, and letM= {M₁, M₂, . . . , M_r}be the family of simple rightA-modules. IfEnd_A(M_i) =kfor1≤i≤r, thenη:A→ O^A(M) is an isomorphism. In particular,ηis an isomorphism whenkis algebraically closed.

Proof. It is enough to prove thatηis injective and thatgr(η)_qis an isomorphism forq= 0andq= 1, sinceAand O^A(M)are complete in therad(A)-adic andJ-adic topologies. By Burnside’s theorem, we know thatgr(η)₀is an isomorphism. To prove thatηis injective, let us consider the kernelker(η)⊆A. It is determined by the obstruction calculus ofDef_M; see Laudal [2, Theorem 3.2] for details. WhenAis finite-dimensional, the right regularA-module A_Ahas a decomposition series

0 =F₀⊆F₁⊆ · · · ⊆F_n=A_A

withF_p/F_p−1a simple rightA-module for1≤p≤n. Namely,A_Ais aniterated extensionof the modules inM. This implies thatηis injective; see Laudal [2, Corollary 3.1]. Finally, we must prove thatgr(η)₁: rad(A)/rad(A)²

→J/J²is an isomorphism. This follows from the Wedderburn-Malcev theorem; see Laudal [2, Theorem 3.4], for details.

4 Properties of the algebra of observables

LetAbe a finite-dimensional algebra over a fieldk, and letM={M₁, . . . , M_r}be any family of rightA-modules of finite dimension overk. ThenMis a swarm, and we denote the algebra of observables byB =O^A(M). It is clear that

B/rad(B)∼=⊕

i End_k(M_i)

is semi-simple, and it follows thatMis the family of simple rightB-modules. In fact, we may show thatMis a swarm ofB-modules, sinceBis complete andB/(radB)ⁿhas finite dimension overkfor all positive integersn. Proposition 3. Ifkis an algebraically closed field, thenη_B:B→ O^B(M)is an algebra isomorphism.

Proof. Since_Mis a swarm ofA-modules and ofB-modules, we may consider the commutative diagram

A ^ηA //

ρJJJJJJ%%

JJ JJ

J ^B=OA(M)

η^B //_C=O^A_(M)

wwnnnnnnnnnnn

1≤i≤r⊕ End_k(M_i).

The algebra homomorphismη^B induces mapsB/rad(B)ⁿ →C/rad(C)ⁿfor alln≥1. Sincekis 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)ⁿis an isomorphism for alln≥1. Henceη^Bis an isomorphism.

In particular, the proposition implies that the assignment(A,M) →(B,M)is a closure operation whenkis algebraically closed. In other words, the algebraB=O^A(M)has the following properties:

(1) the familyMis the family of the simpleB-modules;

(2) the familyMhas exactly the same module-theoretic properties, in terms of (higher) extensions and Massey products, considered as a family of modules overBas overA.

Moreover, these properties characterize the algebraB=O^A(M)of observables.

5 Examples: representations of ordered sets

Letkbe an algebraically closed field, and letΛbe a finite ordered set. Then the algebraA=k[Λ]is an associative algebra of finite dimension overk. The category of rightA-modules is equivalent to the category of presheaves of vector spaces onΛ, and the simpleA-modules correspond to the presheaves{M_λ:λ∈Λ}defined byM_λ(λ) =k andM_λ(λ) = 0forλ=λ. The following results are well known:

(1) ifλ > λinΛand{γ∈Λ:λ > γ > λ}=∅, thenExt¹_A(M_λ, M_λ)∼=k; (2) if{γ∈Λ:λ≥γ≥λ}is a simple loop inΛ, thenExt²_A(M_λ, M_λ)∼=k; (3) in all other cases,Ext¹_A(M_λ, M_λ) = Ext²_A(M_λ, M_λ) = 0.

5.1 A hereditary example

Let us first consider the following ordered set. We label the elements by natural numbers, and writei→jwhen i > j:

1

?

??

??

?? ²

3



4.

In this case, the simple modules are given byM ={M₁, M₂, M₃, M₄}, and we can easily compute the algebra O^A(M)of observables sinceExt²_A(M_i, M_j) = 0 for all1≤i, j≤4. We obtain

O^A(M) =

H_ij⊗_kHom_k

M_i, M_j∼=H∼=

⎛

⎜⎜

⎝ k 0 0k 0k 0k 0 0k k 0 0 0k

⎞

⎟⎟

⎠.

It follows from the generalized Burnside theorem thatη:A→ O^A(M)is an isomorphism. Hence we recover the algebraA∼=O^A(M)∼=H.

5.2 The diamond

Let us also consider the following ordered set, calledthe diamond. We label the elements by natural numbers, and writei→jwheni > j:

1



?

??

??

??

?

2

?

??

??

?? ³



4.

In this case, the simple modules are given byM = {M₁, M₂, M₃, M₄}. Since Ext²_A(M₁, M₄) ∼= k, we must compute the cup-products

Ext¹_A(M₁, M₂)∪Ext¹_A(M₂, M₄)−→Ext²_A(M₁, M₄), Ext¹_A(M₁, M₃)∪Ext¹_A(M₃, M₄)−→Ext²_A(M₁, M₄)

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

O^A(M) =

H_ij⊗_kHom_k

M_i, M_j∼=H∼=

⎛

⎜⎜

⎝ k k k k 0k 0k 0 0k k 0 0 0k

⎞

⎟⎟

⎠.

Note thatH₁₄ is two-dimensional at the tangent level and has a relation. Also in this case, it follows from the generalized Burnside theorem that η : A → O^A(M) is an isomorphism. Hence we recover the algebra A ∼= O^A(M)∼=H.

References

[1] E. Eriksen,An introduction to noncommutative deformations of modules, in Noncommutative Algebra and Geometry, C. de Concini, F. van Oystaeyen, N. Vavilov, and A. Yakovlev, eds., vol. 243 of Lect. Notes Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, USA, 2006, 90–125.

[2] O. A. Laudal,Noncommutative deformations of modules, Homology Homotopy Appl., 4 (2002), 357–396.