On Contractions of Three-Dimensional Complex Associative Algebras

Contraction is one of the most important concepts that motivated by numerous applications in different fields of physics and mathematics. In this work, the contractions of complex associative algebras are considered. We focus on the variety A3() of all complex associative algebras of dimension three (including nonunital). Various contractions criteria are collected and new criteria are proposed to test the possible existence of contraction for each pair of associative algebras. One of the main tools is the use of the low-dimensional cohomology groups of these algebras. As a result, we prove that the variety A3() has seven irreducible components, two of dimension 5, four of dimension 7 and one of dimension 9.


Introduction
The notion of contractions was first introduced by Segal [1] and Inonu [2] for Lie algebras.According to references, the contractions can be divided into two major categories.The first one is more physical that deals with the applications of contractions.Another one is pure algebraic that is mainly oriented to the abstract algebraic structure and mathematical background.For associative algebras, Gabriel [3] studied the irreducible components of the algebraic variety of 4-dimensional unital associative algebras.Mazzola's paper [4] concerns unital associative algebras of dimension five.Classification of lowdimensional nilpotent rigid associative algebras and the description of the irreducible components have been treated by Makhlouf [5,6].
The main purpose of this work is to study the variety of all 3-dimensional complex associative algebras.In the paper, we deal with the algebraic point of view of the contractions.We study orbit closures of the variety of complex associative algebras of dimension three.The paper is organized as follows.Some notations on associative algebras, degeneration, rigidity and contractions of associative algebras are given in Section 2. In Section 3, we list some important invariance arguments for contractions.Calculation and collection of invariance arguments are adduced to conclude the possible existence of contractions for an arbitrary pair of associative algebras in Section 4.

Preliminaries
In this section, we recall some terminology that are used in the paper.Let A=(V,λ) be an algebra of dimension n with an underlying vector space V over a field  and product λ:V×V→V.Let g:(0,1]→GL(V) be a continuous function.More precisely for any t∈(0,1], a nonsingular linear operator g t on V is assigned.A parameterized family of new isomorphic to A=(V,λ) algebra structures on V is determined as follows: A contraction B of A to algebra B is called trivial if B is abelian and improper if B is isomorphic to A. Consider an n 3 -dimensional vector space Hom(V⊗V,V) formed by bilinear maps V×V→V, where V is an n-dimensional vector space over an algebraically closed field  (char=0) denoted by Alg n ().An algebra A=(V,λ) is given as an element λ(A) of Alg n () through the linear mapping λ:V⊗V→V.The linear reductive group GL n () acts on Alg n () by

Definition 2.2
Let A be an algebra over a field .We call A an associative algebra if its bilinear mapping λ satisfies the following condition ( ( , ), ) = ( , ( , )), , , x y z x y z x y z V λ λ λ λ ∈ .
Let A n () be the set of all associative algebra structures on n-dimensional space over a field .The set A n () is an algebraic subset of the affine variety Alg n ().For a fixed basis {e 1 ,e 2 ,…,e n } of the vector space V, the multiplication table of A on this basis is given as a point 3 ( ) Let A be an associative -algebra, D be an A-bimodule and Φ:A p →D be a multilinear mapping.The set of all multilinear mappings from A p to D is called p-dimensional cochain of A and denoted by C p (A,D).The coboundary homomorphism is a mapping δ (p) from C p (A,D) to C p+1 (A,D) given by ( , ) = ( , ) / ( , ) is called the cohomology space (group) of A of degree p.In this paper, we will consider a particular case that D=A as A-bimodule.

Definition 2.3
An associative algebra A is called geometrically rigid whenever its orbit is Zariski open in A n () and called algebraically rigid if the second cohomology group H 2 (A,A) is trivial [7].

Invariance Arguments
In this section, we list some invariance arguments which are helpful for studying the variety of a given class of algebras.Let A be an associative algebras over a field .We define: where k∈; -the right annihilator of A; -the left annihilator of A; -the group of automorphisms of A;

Der A d End A d x y d x y x d y x y A
-the algebra of derivations of A; • ( ) n r A -the nilpotency rank of an associative algebra A; is the space of (α,β,γ)-derivations of A, for fixed α,β,γ∈.
The following theorem is very useful to study the irreducible components of the subvariety A n () of Alg n () [8,9].

Theorem 3.1
The the following subsets of A n () are closed relative to the Zariski topology for any r,s∈:0.60tw0.60tw The proof of 1-4 is the same to that of Lie algebras [10,11].For the parts 5 and 6, the proof is obtained by the following significant fact: let N be a Zariski closed subset of A n () and A 1 ,A 2 in A n ().If A 1 lies in N and A 1 →A 2 then A 2 also lies in A. More precisely, the subset N is not GL n ()-setwise stabilizer.However, it is B-setwise stabilizer, where B is the Borel subgroup GL n ()-made up of upper triangular matrices.The statements 7,8 and 9 are equivalent because of the following relation between the dimensions of GL n ()--orbits, automorphism's groups and derivation algebras.
The proof of 10 is directly coming behind the following fact: let A be an associative algebra.We define the lower central series: If A is a nilpotent associative algebra then it has nilpotency rank denoted by r n (A), i.e., it is a minimal positive integer l such that A l =0.It is not hard to see that if A 1 degenerates to A 2 then dim ≥ The proof of 11 and 12 are the same of Lie algebras [12].
The next corollary is used to reject existence of degenerations for each pair of associative algebras A and B.

Corollary 3.1
If an algebra A degenerates to an algebra B. Then the following conditions are valid: for some r, • In the sequel, all algebras suppose to be over the field of complex numbers .

The Variety of Complex Associative Algebras of Dimension Three
In this section, we recall the complete list of non-isomorphic classes of three-dimensional complex associative algebras, which was obtained in Rikhsiboev et al. [13] to study the subvariety A 3 () of Alg 3 ().

Theorem 4.1
Any 3-dimensional complex associative algebra A is isomorphic to one of the following pairwise non-isomorphic algebras.  .

e e e e As e e e e e e e e e As e e e e e e As Abelian
In Table 1, we assemble some contraction invariants of threedimensional associative algebras that were obtained in refs.[14,15].
In Table 2 below we present the values of more contraction invariants.
Table 3 contains the values of contraction invariants applying the algorithm, which was stated in ref. [16] to describe the (α,β,γ)derivations of three-dimensional complex associative algebras.
By using all the criteria presented above, we give in the following table all possibilities of degenerations for 3-dimensional associative algebras.The checkmark denotes that there is a degeneration deg A B → .The other symbols stand for the reason why such a degeneration is impossible.Indeed, there is more than just one reason for a non-degeneration.However, we have written down only one in the table (Table 4).

According to
As As As As As As As As As As As As As As As As As 18 3 As The calculated contraction invariants of three-dimensional associative algebras. As

Table 3 :
Contraction invariants of three-dimensional associative algebras.Volume 11 • Issue 3 • 1000282 J Generalized Lie Theory Appl, an open access journal ISSN:

Theorem 4. 2
The rigid irreducible components of the variety A 3 () are generated by the algebras2

Table 4 :
All possibilities of degenerations of three-dimensional associative algebras.