On Cohomology Groups of Four-Dimensional Nilpotent Associative Algebras

: The study of cohomology groups is one of the most intensive and exciting researches that arises from algebraic topology. Particularly, the dimension of cohomology groups is a highly useful invariant which plays a rigorous role in the geometric classification of associative algebras. This work focuses on the applications of low dimensional cohomology groups. In this regards, the cohomology groups of degree zero and degree one of nilpotent associative algebras in dimension four are described in matrix form.


Introduction:
Algebraic topology is one of the main areas in mathematics. This area uses fundamental ingredients from abstract algebra in the study of topological spaces. One of the essential techniques of algebraic topology is cohomology groups. It is a general term of a sequence of groups associated with a topological space which is defined from a cochain. On the other hand, one of the ancient areas of the modern algebra is the theory of finite-dimensional associative algebras, and it has been studied by many investigators like Pierce 1 , Mazzola 2 and Basri 3 . In this regard, this study concentrates on cohomology groups for associative algebras.
The cohomology theory of groups has numerous applications in many sorts of mathematical and physical studies (see for instance 4,5 and 6 ). The cohomology groups H p (A, D), with an action associative algebra A on A-bimodule D, were presented by Hochschild 7 . One of the important invariant in studying cohomological problems is derivations. The derivations are linear transformations on algebras and exactly the elements of 1 ( , ) 1-cocycles, while the socalled inner derivations are 1-coboundaries denoted by 1 ( , ), 8,9 .
The present study concentrates on the description of low dimensional cohomology groupsH i (A, A); i = 0, 1, because the dimension of the cohomology groups is considered as one of the important invariants to study the properties of algebras. Particularly, this invariant plays a rigorous role in the geometric classification of associative algebras. Using the classification result of nilpotent associative algebras, the description of cohomology groups of degree zero and degree one of nilpotent associative algebras in dimension four are given in matrix form.

Preliminaries
This section provides some terminologies that are needed in this work. Definition 1 10 : An associative algebra over a field K is a K -vector space A equipped with bilinear map λ: A × A ⟶ A satisfying the associative law: λ(λ(x, y), z) = λ(x, (λ(y, z)), ∀x, y, z ∈ A Definition 2 10 : Let A be an associative algebra over K and define as: If there exists an integer s ∈ N,, such that 1 ⊇ 2 ⊇ 3 ⊇ ⋯ = {0} then this algebra is called nilpotent. Definition 3 10 : Let 1 and 2 be two associative algebras over . A homomorphism between 1 and 2 is an -linear mapping ∶ 1 ⟶ 2 such that ( ) = ( ) ( ) ∀ , ∈ 1 Definition 4 10 : A linear transformation of associative algebra is called a derivation if ( · ) = ( ) + ( ) ∀ , ∈ The set of all derivations is denoted by ( ). An important special case of derivation mapping so-called inner derivation mapping is defined as follows: Definition 5 10 : A linear transformation of the associative algebra A is called an inner derivation if ( ) = − ∀ ∈ , and ∈ The set of all inner derivations is represented by ( ).

Definition 6 10 :
Let be associative algebra on a field . An -dimensional vector space over the same field is called a − bimodule if is a right and a left -module, such that ( ) = ( ) ∀ , ∈ and ∈ ; = ∀ ∈ a n d ∈ . To simplify terminology, we will use the expression -bimodule instead of − bimodule.
Let Φ be a multilinear mapping from an associative algebra to an -bimodule . The set of all these multilinear mappings is called cochain of in dimensional and represented by ( , ). It is convenient to identify 0 ( , )with and ( , ) with {0} for < 0.

4-The quotient space
( , ) = ( , )/ ( , ) is known as the cohomology group of in degree . Following, a particular case is considered that = as -bimodule and all algebras considered are over a complex field ℂ.

Main Results:
This section is devoted to computing the zerocohomology groups 0 ( , ) and firstcohomology groups 1 ( , ) of fourdimensional nilpotent associative algebras. The algebraic classification of all nilpotent associative algebras in dimensional four is constructed by 3 and is provided with the following theorem. Note that denotes ℎ isomorphism class of associative algebra in dimension .

Theorem 1.
Any complex nilpotent associative algebra structure on four dimensional is isomorphic to one of the following classes of algebras: Following, the classification results will be used which is already known from Theorem 1 to compute cohomology groups ( , ) for = 0, 1.

The Cohomology Groups of Degree Zero
Based on the expression of cohomology group of A 0 ( , ) Thus, 0 ( , ) is the center of . Next the zero-cohomology group of complex nilpotent associative algebras in dimension four is described. Theorem 2 provides the following results: ∈ ∶ 0 ( )( ) = − = 0 , ∀ ∈ } = ( ) Then, the structure constants of 4 1 are given as follows: The remaining parts of zero cohomology groups in dimension four can be done in a similar manner as shown above.
The solutions to the system give the description of 1−coboundaries in matrix form. Following, our procedure is applied to obtain the group 1 ( , ) of nilpotent associative algebras in dimension four where IC is represented isomorphism classes of algebras.  The remaining parts of 1-coboundary algebras in dimension four can be done in a similar manner as shown above. Based on [9] and Theorem 3, the span bases and the dimensions of first cohomology groups will be found as shown in Corollary 1.
Corollary 1: Let 1 ( n q , n q ) be the cohomology group in degree one where n q denotes as ℎ isomorphism class of associative algebra in dimension . If = 4 and = 1, 2, . . .16 then, the span bases and the dimensions cohomology group in degree one of complex nilpotent associative algebra has the following form:   The span bases of cohomology group in degree one of the remaining parts can be done in a similar manner as shown above.

Conclusion:
The present work focuses on the applications of low dimensional cohomology groups ( , ), = 0, 1. The dimensions of zero cohomology groups and first cohomology groups for four-dimensional complex nilpotent associative algebras range between zero and four and between zero and six, respectively.