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A standard example in noncommutative deformation theory 1 | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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A standard example in noncommutative deformation theory 1

Arvid SIQVELAND*

Buskerud University College, P. O. Box 235,3603 Kongsberg, Norway

*Corresponding Author:
Arvid SIQVELAND
Buskerud University College, P. O. Box 235,3603 Kongsberg, Norway
E-mail: [email protected]

Received date: December 12, 2007; Revised date: March 10, 2008

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Abstract

In this paper we generalize the commutative generalized Massey products to the noncommutative deformation theory given by O. A. Laudal. We give an example illustrating the generalized Burnside theorem, one of the starting points in this noncommutative algebraic geometry.

Introduction

In [2], O. A. Laudal defines a noncommutative algebraic geometry based on noncommutative deformation theory, see also [1].

One of the main ingredients in this theory is the following:

Theorem 1.1 (the generalized Burnside theorem). Let A be a finite dimensional k-algebra, k an algebraically closed field. Consider the family image of simple A-modules and let H = (Hij) be the formal noncommutative moduli of image thenimage

Affine deformations

Definition 2.1. An r-pointed artinian k-algebra is a k-algebra S together with morphisms imagesuch that imageand such that image for some n > 0. ker(ρ) is called the radical of S and denoted rad(S).

Let image be the idempotents. Ifimage it follows that every r-pointed k-algebra can be written as the matrix algebra image

Let V = {V1, . . . , Vr} be a family of right A-modules. Let image be an r-pointed artinian k-algebra.

Definition 2.2. The deformation functor Def V : imageis defined by

DefV (S) = image

Definition 2.3. A morphism image between to r-pointed artinian k-algebras is called small if ker π · rad(R) = rad(R) · ker(π) = 0.

Let image Thenimage and as such it has an obvious structure as left image module. The (right) A-module structure is determined by the k-algebra homomorphismimage which is completely determined by the morphisms image Let MS be the deformation of V to S given by the k-algebra homomorphismimageinducing as above image Letimage be a small morphism. We may lift σij(a) in the diagram

image

by composing with a section of the right hand vertical map, adding any k-linear morphism image Choosing the k-linear lifting σR this way, there is a k-linear mapimage For this to be an A-module structure commuting with R, we need the conditions; for every image Because this holds for S, we get an element

image

Because π is a small morphism, we have I2 = 0 and thusimage andimage implying thatimage is a Hochschild 2-cocycle whose classimageimage is called the obstruction for lifting MS to R.

Theorem 2.1. o(π,MS) = 0 if and only if there exists a lifting image of MS. The set of isomorphism classes of such liftings is a torsor under image

Proof. If 0 = o(π,MS), thenimage is the desired lifting.

Generalized Massey Products

Consider the r-pointed matrix k-algebra image where all productsimage for allimage whereimageis the two-sided ideal generated by {tij}.For any covariant functor image is a k-vector space which is called the tangent space of F. The procategory image is the category of all k-algebras with morphismsimage such that R/ radimage for allimage

Definition 3.1. A procouple image is called a prorepresentable hull or formal moduli for F, if the induced mapimageis smooth, and the tangent map image is a bijection.

Definition 3.2 (Non-commutative generalized Massey products). Let image be the deformation functor of the family image Then we have an isomorphism as k-vector spaces, image whereimageLet image Let S2 =image A sequence of elementsimagedefines a deformationimage Let B'2 be the set of all monomials of degree 2 in the tij(lij) and consider

image

Then we have that

image

image is then called a defining system for the second order Massey productsimage Choose basesimage for the dual spacesimage Then

image

Put

image

Put image and let π2 be the induced morphism. Choose a monomial basis image for ker π2 and putimage whereimage is the set of all monomials of degree less than or equal to 1. Then image

Assume that SN−1 has been constructed such that image can be lifted toimageimage Also assume that monomial basesimage have been constructed. Put

image

Write

image

with

image

and pick a monomial basis image where we may assume that forimage orimage for someimage Then for every monomialimage with degree less than N we have a unique relation in RN

image

and we have that

image

We call image a defining system for the Massey products

image

To continue, we put

image

and image is the natural morphism. We choose a monomial basis image for ker πN and we put image and we continue by induction.

Theorem 3.1. The functor DefV has a prorepresenting Hull image uniquely determined by a set of matric Massey products

image

image

Proof. It follows from Laudal’s classical article [3], and it is possible to generalize from Schlessinger [4], that image where

image

and image is a basis forimage

Example

Consider the 2-pointed k-algebra

image

This k-algebra has geometric points, i.e simple A-modules, given by the line and the point respectively

image

We are going to compute the local formal moduli image of the modulesimagefor a fixed a, following the algorithm given in [2]. We start by computing the tangent spaces: In general we have

image

where the bi-module structure on Homk(Vi, Vj) is given by image Notice that by Ad we mean the trivial derivationsimage

Any derivation δ is determined on a generator set. In this particular example, we choose as generators

image

image

image

image

image

image

Thus if a = 1 we choose as basis the one point set image

image which is trivial.

For the rest we put a = 1, that is V1 = V1(1) and compute image Let

image

Then the infinitesimal liftings are given by

image

Now S2 = S/ rad2 and the obstruction for lifting to R3 = S/ rad3 is

image

In general, image so

image

But imagein A, thusimage withimage

image

We see that this image can be lifted toimageon image giving the result

image

In general terms this says that A is a scheme for its 1-dimensional simple modules.

Acknowledgement

I would like to thank the organizers of the AGMF workshop 2007. It is a great opportunity to meet different aspects of the field of noncommutative geometry. Also, I am thanking the referee for useful comments and corrections.

References

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