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Department of Pure Mathematics, Queen’s University Belfast, Belfast BT7 1NN, UK

- Corresponding Author:
- Natalia K Iyudu

Department of Pure Mathematics

Queen’s University Belfast, Belfast BT7 1NN, UK

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

**Received date:** July 21, 2015; **Accepted date:** August 03, 2015; **Published date:** August 31, 2015

**Citation:** Iyudu NK (2015) On the Representation Spaces of the Jordanian Plane. J Generalized Lie Theory Appl S1:e001. doi:10.4172/generalized-theory-applications.S1-e001

**Copyright:** © 2015 Iyudu NK. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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We announce here a number of results concerning representation
theory of the algebra , known as Jordanian
plane (or Jordan algebra). We consider the question on '**classification**'
of finite-dimensional modules over the Jordan algebra. Complete
description of irreducible components of the representation variety
mod(R, n) is given for any dimension n. It is obtained on the basis of
the stratification of this variety related to the Jordan normal form of Y.
Any irreducible component of the representation variety contains only
one stratum related to a certain partition of n and is the closure of this
stratum. The number of irreducible components therefore is equal to
the number of partitions of n.

As a preparation for the above result we describe the complete set
of pairwise non-isomorphic irreducible modules S_{a} over the Jordan
algebra, and the rule how they could be glued to indecomposables.
Namely, we show that Ext^{1}(S_{a}, S_{b}) = 0, if a ≠ b.

We study then properties of the image algebras in the endomorphism ring. Particularly, images of representations from the most important stratum, corresponding to the full Jordan block Y. This stratum turns out to be the only building block for the analogue of the Krull-Remark-Schmidt decomposition theorem on the level of irreducible components. Along this line we establish an analogue of the Gerstenhaber–Taussky–Motzkin theorem on the dimension of algebras generated by two commuting matrices. Another fact concerns with the tame-wild question for those image algebras. We show that all image algebras of n-dimensional repre-sentations are tame for n ≤ 4 and wild for n ≥ 5.

We consider here the quadratic algebra given by the presentation . This algebra appears in various different
contexts in mathematics and physics. First of all, it is a kind of quantum
plane: one of the two Auslander regular algebras of global dimension
two, as it is mentioned in the Artin and Shelter paper [1]. The other
one is the usual quantum plane . It served as one
of basic examples for the foundation of noncommutative geometry [2].
There were also studies of deformations of *GL*(2) analogous to GL_{q}(2)
with respect to R in 80-90th in Manin’s ’Quantum group’ [3,4], where
this algebra appeared under the name Jordan algebra.

This algebra is also a simplest element in the class of RIT (relativistic
internal time) algebras. The latter was defined and investigated [5-9].
The class of RIT algebras arises from a modification of the Poincare
algebra of the Lorenz group SO(3,1) by means of introducing the
additional generator corresponding to the relativistic internal time.
The algebra R above is a RIT algebra of type (1,1). Our studies of this
algebra are partially reported “Classification of **finite** dimensional
representations of one noncommutative quadratic algebra” [10].

Let us mention that R is a subalgebra of the first Weyl algebra A1. The latter has no finite dimensional representations, but R turns out to have quite a rich structure of them. Category of finite dimensional modules over R contains, for example, as a full subcategory mod GP (n, 2), where GP (n, 2) is the Gelfand–Ponomarev algebra [11] with the nilpotency degrees of variables x and y, n and 2 respectively. On the other hand we show that R is residually finite dimensional.

We are interested here in representations over an algebraically
closed field k of characteristic 0. Sometimes we just suppose k = , this
will be pointed out separately. We denote throughout the category of
all R-modules by Mod R, the category of finite dimensional R-modules
by mod R and ρ_{n} stands for an n-dimensional representation of R.

The most important question on finite dimensional representations
one can ask, is to classify them. This question may be solved directly by
parametrization of isoclasses of indecomposable modules, for example,
for tame **algebras**. But when the algebra is wild the problem turns to
a description of orbits under GL_{n} action by simultaneous conjugation
in the space *mod*(R,n) of n-dimensional representations. One can
think of the latter space also as of a variety of tuples of n × n matrices
(corresponding to generators) satisfying the defining relations of the
algebra R.

It is commonly understood that the first step in the study of this variety should be the description of its irreducible components. This approach leads to such famous results of this kind as Kashiwara-Saito [12] description of the irreducible components of Lusztig’s nilpotent variety via the crystal basis [13].

We show that irreducible components of the representation space *mod(R, n)* of the **Jordan algebra** could be completely described for any
dimension n.

The first key point for this description is the choice of a stratification
of the Jordan variety. We choose a stratum corresponding to the
partition = (n_{1},…, n_{k}), n = n_{1} +…+ n_{k}, n_{1} ≥ … ≥ n_{k}, consisting of the
pairs of matrices (X, Y ), where Y has the Jordan normal form defined
by and X, Y satisfy the defining relation.

**Theorem 12.1** *Any irreducible component K _{j} of the representation
variety mod(R,n) of the Jordan algebra contains only one stratum from the stratification related to the Jordan normal form of Y , and is the
closure of this stratum.*

*The number of irreducible components of the variety mod(R, n) is
equal to the number of the partitions of n.*

*The variety mod (R, n) is equidimensional, the dimension of
components is n ^{2}.*

The importance of examples of algebras for which the irreducible
components of *mod* (R, n) could be described for each n was emphasized
[14] and it is mentioned there that known cases are restricted to
algebras of finite representation type (i.e., there are only finitely many isomorphism classes of indecomposable R modules) and one example
of infinite representation type [15]. There is some similarity between
the algebra generated by the pair of nilpotent matrices annihilating
each other considered [15] and the Jordan algebra, but while in the
case of [15] variables x and y act ’independently’, there is much more
interaction in the case of Jordan algebra, which makes the analysis in a
sense more difficult.

We also can answer the question, in which irreducible components, module in general position is indecomposable.

**Corollary:** *Only the irreducible component which is the
closure of the stratum corresponding to the trivial partition of n (the full
block Y) contains an open dense subset consisting of indecomposable
modules.*

First, we derive some **properties** of algebras which are images for
representations of R in the endomorphism ring. We show that they
are basic algebras, that is their semisimple parts are direct sums of r
copies of the field, where r is the number of different eigenvalues of X = *ρ _{n}*(x). This allows associating to any representation a quiver of its image
algebra, in a conventional way. This leads to a rough classification of
reps by these quivers. It turns out that indecomposable modules have
either a typical wild quiver with one vertex and two loops or the quiver
with one vertex and one loop. A simple, but important fact is that Y =

We show that all *irreducible* modules are one dimensional: S_{α} = (α,
0), and describe all finite dimensional modules, subject to the Jordan
normal form of Y. We study *indecomposable* modules and derive the
rule how one could glue irreducibles together: Ext^{1}(S_{α}, S_{β}) = 0, if α ≠
β. These provide us with enough information to suggest a stratification
of the variety mod (*R, n*) related to the Jordan normal form of Y and to
prove results on irreducible components.

Results on the structure of representation variety for R show an
exceptional role of the strata related to the full **Jordan block** Y, since
they turn out to be the only building blocks in the analogue of the Krull-
Remark-Schmidt decomposition theorem on the level of irreducible
components. Another evidence of the special role of this stratum is
the following. We prove an analogue of the Gerstenhaber-Taussky-
Motzkin theorem [16,17] on the dimension of algebras generated
by two commuting matrices. We show that the dimension of image
algebras of representations of R does not exceed n(n + 2)/4 for even
n and (n + 1)^{2}/4 for odd n and this estimate is attained in the stratum
related the full Jordan block Y.

Finally, we fulfill a more detailed study of tame-wild questions
for image algebras in the strata related to the full Jordan block Y.
From the above results we see that any algebra which is an image of indecomposable representation *ρ _{n}* is a

The work on this circle of questions was started in 2003-2004 during my visit at the Max-Planck-Institute for Mathematics in Bonn. Most of results apart from those on the irreducible components appeared in the MPI preprint. I am thankful to this institution for the support and hospitality and to many colleagues with whom I have been discussing ideas related to this work.

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