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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

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

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.

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 of simple A-modules and let
H = (Hij) be the formal noncommutative moduli of then

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

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

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

**Definition 2.2.** The deformation functor Def V : is defined by

Def_{V} (S) =

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

Let Then and as such it has an obvious structure as
left module. The (right) A-module structure is determined by the k-algebra homomorphism which is completely determined by
the morphisms Let MS be the deformation of V to S given by the k-algebra homomorphisminducing as above Let be a small morphism. We may lift *σ _{ij}(a)* in the diagram

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

Because π is a small morphism, we have I^{2} = 0 and thus and implying that is a Hochschild 2-cocycle whose class is called the obstruction for lifting M_{S} to R.

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

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

Consider the r-pointed matrix k-algebra where all products for all whereis the two-sided ideal generated by {t_{ij}}.For any
covariant functor is a k-vector space which is called the tangent space of
F. The procategory is the category of all k-algebras with morphisms such that
R/ rad for all

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

**Definition 3.2 **(Non-commutative generalized Massey products). Let be
the deformation functor of the family Then we have an isomorphism as k-vector
spaces, whereLet Let S_{2} = A sequence of elementsdefines a deformation Let B'_{2} be the set of all monomials of degree 2 in the t_{ij}(l_{ij}) and
consider

Then we have that

is then called a defining system for the second order Massey products Choose bases for the dual spaces Then

Put

Put and let π_{2} be the induced morphism. Choose a monomial basis for
ker π_{2} and put where is the set of all monomials of degree less than or equal
to 1. Then

Assume that S_{N−1} has been constructed such that can be lifted to Also assume that monomial bases have been constructed. Put

Write

with

and pick a monomial basis where we may assume that for or for some Then for every monomial with
degree less than N we have a unique relation in R_{N}

and we have that

We call a defining system for the Massey products

To continue, we put

and is the natural morphism. We choose a monomial
basis for ker π_{N} and we put and we continue by induction.

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

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

Consider the 2-pointed k-algebra

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

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

where the bi-module structure on Hom_{k}(V_{i}, V_{j}) is given by Notice that by Ad we mean the trivial derivations

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

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

which is trivial.

For the rest we put a = 1, that is V_{1} = V_{1}(1) and compute Let

Then the infinitesimal liftings are given by

Now S_{2} = S/ rad^{2} and the obstruction for lifting to R_{3} = S/ rad^{3} is

In general, so

But in A, thus with

We see that this can be lifted toon giving the result

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

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.

- EriksenE (2005)An introduction to noncommutative deformations of modules. Lect. Notes Pure Appl Math 243: 90–125.
- LaudalOA (2003)Noncommutative algebraic geometry. Rev Mat Iberoamericano19: 509–580.
- LaudalOA (1979) Formal moduli of algebraic structures. Lecture Notes in Math pg: 754.
- SchlessingerM (1968) Functors of Artin rings. Trans Amer Math Soc130: 208–222.
- SiqvelandA, The Noncommutative Moduli ofrk 3Endomorphisms.Report series, Buskerud University College, 26: 1-132.

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