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An example of noncommutative deformations 1 | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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An example of noncommutative deformations 1

Eivind ERIKSEN

Oslo University College, Postboks 4, St. Olavs plass, N-0130 Oslo, Norway, E-mail: [email protected]

Received date: December 12, 2007; Accepted date: March 27, 2008

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Abstract

We compute the noncommutative deformations of a family of modules over the first Weyl algebra. This example shows some important properties of noncommutative deformation theory that separates it from commutative deformation theory.

Introduction

Let k be an algebraically closed field and let A be an associative k-algebra. For any left A-module M, there is a commutative deformation functor equation defined on the category l of local Artinan commutative k-algebras with residue field k. We recall that for an object equation a deformation of M over R is a pair (MR, equation ), where MR is an A-R bimodule (on which k acts centrally) that is R-flat, and equation is an isomorphism of left A-modules. Moreover, equation as deformations in equation if there is an isomorphism equation of A-R bimodules such that equation

In [2], Laudal introduced noncommutative deformations of modules. For any finite family equation of left A-modules, there is a noncommutative deformation functor equation equation defined on the category ap of p-pointed Artinian k-algebras. We recall that an object R of ap is an Artinian ring R, together with a pair of structural ring homomorphisms equation and equation, such that equation and the radical I(R) = ker(g) is nilpotent. The morphisms of ap are ring homomorphisms that commute with the structural morphisms.

A deformation of the family equation over R is a (p + 1)-tuple equation where MR is an A-R bimodule (on which k acts centrally) such that equation as right R-modules, and equation is an isomorphism of left A-modules for equation By definition,

with the natural right R-module structure, and k1, . . . , kp are the simple left R-modules of dimension one over k. Moreover, equation as deformations in equation if there is an isomorphism equation bimodules such that equation

There is a cohomology theory and an obstruction calculus for equation see Laudal [2] and Eriksen [1]. We compute the noncommutative deformations of a family equation of modules over the first Weyl algebra using the constructive methods described in Eriksen [1].

An example of noncommutative deformations of a family

Let k be an algebraically closed field of characteristic 0, let equation and let D = Diff (A) be the first Weyl algebra over k. We recall that equation. Let us consider the family equation of left D-modules, where equation We shall compute the noncommutative deformations of the family equation.

In this example, we use the methods described in Eriksen [1] to compute noncommutative deformations. In particular, we use the cohomology equation of the Yoneda complex

equation

for equation where equation is a free resolution of Mi, and an obstruction calculus based on these free resolutions. We recall that equation

Let us compute the cohomology equation We use the free resolutions of M1 and M2 as left D-modules given by

equation

and the definition of the differentials equation in the Yoneda complex, and obtain

equation

The base vector equation is represented by the 1-cocycle given by equation when equation for all i, j, it is clear that equation for equation

We conclude that equation is unobstructed. Hence, in the notation of Eriksen [1], the prorepresenting hull H of equation is given by

where equation is a basis of equation dual to the basis equation of equation for (i, j) = (1, 2) and (i, j) = (2, 1). We write equation and equation

In order to describe the versal family equation of left D-modules defined over H, we use Mfree resolutions in the notation of Eriksen [1]. In fact, the D-H bimodule equation has an M-free resolution of the form

equation

where equation This means that for any P,Q equation D, we have that equation and equation

In other words, S is an associative k-algebra of finite type such that the J-adic completion equation for the ideal equation The corresponding algebraization equation of the versal family equation is given by the M-free resolution

equation

with differential

equation

We shall determine the D-modules parameterized by the family equation over the noncommutative algebra S — this is much more complicated than in the commutative case. We consider the simple left S-modules as the points of the noncommutative algebra S, following Laudal [3], [4]. For any simple S-module T, we obtain a left D-module equation Therefore, we consider the problem of classifying simple S-modules of dimension n ≥ 1.

Any S-module of dimension n ≥ 1 is given by a ring homomorphism equation and we may identify equation by choosing a k-linear base equation for T. We see that S is generated by e1, s12, s21 as a k-algebra (since e2 = 1 - e1), and there are relations

equation

Any S-module of dimension n is therefore given by matrices equation satisfying the matric equations

equation

The S-modules represented by equation and equation are isomorphic if and only if there is an invertible matrix equation such that equation Using this characterization, it is a straight-forward but tedious task to classify all S-modules of dimension n up to isomorphism for a given integer n ≥ 1.

Let us first remark that for any S-module of dimension n = 1, p factorizes through the commutativization equation of S. It follows that there are exactly two non-isomorphic simple S-modules of dimension one, T1,1 and T1,2, and the corresponding deformations of equation are

equation

This reflects that M1 and M2 are rigid as left D-modules.

We obtain the following list of S-modules of dimension n = 2, up to isomorphism. We have used that, without loss of generality, we may assume that E1 has Jordan form:

equation   (2.1)
equation   (2.2)
equation   (2.3)
equation   (2.4)
equation   (2.5)
equation   (2.6)

We shall write equation for the corresponding S-modules of dimension two. Notice that equation is simple for all equation, while equation are extensions of simple S-modules of dimension one. In fact, equation are trivial extensions, while T2,4 is a non-trivial extension of T1,2 by T1,1 and T2,5 is a non-trivial extension of T1,1 by T1,2. The deformations of equation corresponding to the simple modules T2,6,α are given by M2,6,α equation for equation In fact, one may show that equation for any equation In particular, equation is a simple D-module if equation and in this case equation if and only if equation Furthermore, equation for equation for n = 2, 3, . . . .

We obtain the following list of S-modules of dimension n = 3, up to isomorphism. We have used that, without loss of generality, we may assume that E1 has Jordan form:

equation   (2.1)
equation   (2.2)
equation   (2.3)
equation   (2.4)
equation   (2.5)
equation   (2.6)
equation   (2.7)
equation   (2.8)
equation   (2.9)
equation   (2.10)
equation   (2.11)
equation   (2.12)

We shall write T3,1 – T3,6, T3,7,b, T3,8 – T3,11, and T3,12,c for the corresponding S-modules of dimension three. Notice that all S-modules of dimension three are extensions of simple S-modules of dimension one and two, so there are no simple S-modules of dimension n = 3.

In fact, equation equation and equation are trivial extensions, while T3,6 is a non-trivial extension of T1,2 by T2,5 and T3,11 is a non-trivial extension of T1,1 by T2,4.

We remark that there are no simple S-modules of finite dimension n ≥ 3. In fact, if T is a simple S-module, then equation is a surjective ring homomorphism. This implies that equation can be generated by equation as a k-algebra. To see that this is impossible, notice that we may choose a k-base of T such that

equation

where equation r)-matrix and Y is a (n − r) × r matrix. If r = 0 or r = n, then X = Y = 0, and this leads to a contradiction, because Mn(k) is not generated by diagonal matrices when n > 1. Moreover, r ≥ 2 leads to a contradiction, because equation is not generated by Ir and XY . Similarly equation leads to a contradiction, because equation is not generated by In−r and Y X. We conclude that n = r+(n−r) = 1+1 = 2, a contradiction.

Finally, we remark that the commutative deformation functor equation of the direct sum equation has pro-representing hull equation and an algebraization equation It is not difficult to find the family MS in this case. In fact, for any point equation the left D-module equation is given by

equation

We see that we obtain exactly the same isomorphism classes of left D-modules as commutative deformations of equation as we obtained as noncommutative deformations of the family equation However, the points of the algebraization S of the pro-representing hull of the noncommutative deformation functor equation give a much better geometric picture of the local structure of the moduli space of left D-modules. In fact, the family of left D-modules parametrized by the points of S contains few isomorphic D-modules, and the simple S-modules have algebraic properties – such as extensions – that reflect the algebraic properties of the corresponding D-modules.

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