On the Hochschild cohomology theory of -algebra
By
Alaa Hassan Noreldeen
Dept. of Mathematics, Faculty of Science ,Aswan University, Aswan, Egypt.
E-mail: ala2222000@yahoo.com
Abstract
We will study the simplicial (co)homology of Hochschild complex of –algebra with homotopical properties. We study some of the relation between the Hochschild cohomology of a commutative -algebra and the set of twisted cochain D(A,A) of this complex. The vanishing of Hochschild cohomology of special degree leads to vanish of D(A,A) . In the last part we get an extension of special case of –algebra.
2000 Mathematics Subject Classification: 55N35, 16E4
Key words : Hoschschild cohomology- twisted cochain – -algebra .
0- Introduction
The concepts of -modules, -algebra and its related with simplicial (Hochschild ) (co)homology has been studied in [2,4,6,7]. An burgee esteem between the habitual of throughout -algebra structures on unchanging differential graded algebra and the Hochschild cohomology of that algebra has been studied in [7]. Reliance of Hochschild (co)homology intricate for -modules go away from -algebras has been studied in [5]. The callousness of the Hochschild cohomology of measure (n,2-n) -algebra is rational in [4]. In our compounding we are careful by Hochschild cohomology of differential -algebra. For a inclined cochain Hochschild (simplicial) detailed for differential -algebra to leave a bound bulk of non unsatisfying contemptuous augment and differential -algebra A, we show that the Hochschild cohomology is trivial. Surely, we expansive the neutrality of the Hochschild cohomology in [4]. An important relation between the set of all -algebra structures on fixed differential graded algebra and the Hochschild cohomology of that algebra has been studied in [7]. Idea of Hochschild (co)homology complex for -modules over -algebras has been studied in [5]. The triviality of the Hochschild cohomology of dimension (n,2-n) -algebra is proved in [4]. In our article we are concerned by Hochschild cohomology of differential -algebra. For a given cochain Hochschild (simplicial) complex for differential –algebra with condition a finite number of non trivial high multiplication and differential –algebra A, we show that the Hochschild cohomology is trivial. Finally, we generalized the triviality of the Hochschild cohomology in [4].
1 – Hochschild complex for differential -algebra
We recall the requisites definitions and results relating to concepts of differential module and differential algebra. The main references are [3], [4], [5],[7] and [9], so all modules are defined on Z2.
Definition (1.1):
A differential module is a module and equipped with a morphism , called the differential of the module X, with degree (−1) such that .
A mapping of differential modules is a mapping of modules such that .
A differential homotopy between mappings of differential modules is a mapping of module Module X , where the space is called a space n-dimensional cycles and spaces are called spaces of n-dimensional boundaries. It is clear that . such that .
It is easy to see that the homotopy relation is an equivalence relation.
The modules X and Y are called homotopy equivalent (denoted by X Y), if there is a chain map , such that , .
The module is called contractible, if it is homotopically equivalent to the zero. The factor space is called the homology of X.
For a module X, denote by the dual module of X, , for which -conjugate to . The differentials induce the differentials . The homology of the dual complex is called cohomology of X and denoted by .
Definition (1.2):
A differential module is a an arbitrary Banach module with a family of homeomorphisms such that the following relations holds for each integer , .
If i=0 , and is an ordinary differential module, if i=1 we have , that is the mappings and are anti-commuting maps. This means that the composition is an endomorphism of the differential module . For k = 2, we obtain .This means that the mapping is a differential homotopy between zero map and map of differential modules. Therefore, the mapping is a differential within a homotopy.
Definition (1.3):
A differential algebra is a differential module over algebra with the multiplication such that the associate law holds.
Definition (1.4):
Let A be algebra. The triple is called –algebra , where is graded module over algebra such that:
(1-1)
The morphism between -algebras A, A∕ is a family of homeomorphism such that and
(1-2)
The summation in (1-1) and (1-2) are given in all possible place of mj and the right hand side of (1-2) we can put : .
The forms (1-1) and (1-2) are called stasheff relation for A∞ -algebra [7].
Definition(1.5):
A differential algebra -algebra is – module A together with a set operations , with the following identity :
.
For example if n=0, then , this is associated homotopy relation. If n=1 , then
,
this means that there is a homotopy relation between and .
Definition (1.6) :
The module A is called differential coalgebra, if there is a specified operation of dimensions , satisfying the relations :
.
From [4] the Hochschild complex C*(A, A) for algebras A is a A-module over Z2 with the multiplication with the associate law .
The cochain Hochschild complex is given by (C*(A, A), δ) such that C*(A, A) = ∑ Cn (An, A), Cn (A, A) =Hom (An, A) and The relation between operators and is given by:
The homology of (C*(A, A), δ) is Hochschild cohomology and defined by H*(A, A).
Definition (1.7):
For any differential -algebra A we can define the coalgebra BA which is called B-construction over A. Consider the tensor algebra such that
The tensor algebra TX with the following differential , such that
is called B-construction over A and denoted by .
Consider the differential -algebra A with finite integer nontrivial exterior multiplication , then there is -algebra such that for , .
Consider Hom(BA, A), then Homn(BA, A) = [f:BAi→Ai+n] .
Note that if f Homn (BA, A) , then there is .The identity map is Id1=d, Idk=0 for k>1 .
Define the differential such that
The complex Hom (BA, A) with differential δ (defined in relation (1-3)) is called the Hochschild complex for –algebra and denoted by . Consider the following operations in Hochschild complex C(A, A) from [4] :
(1-4)
where
We can rewrite the operations and on the Hochschild complex as follows :
(1-5)
where .
For some we can generalize the operation in relation (1-5) to be
(1-6)
and the summation will be in all place of elements . The relation between operators , and is given by:
.
From the relation (1-3), (1-4), (1-5), (1-6) we have:
(1-7)
The relation (1-7), when k=1, can be written in the form:
if we put and in (1-7) and in (1-5).
2- Twisted cochain Hochschild complex for A∞-algebra and related cohomology.
In this part we are concerned with the commutative -algebra and triviality of the Hochschild cohomology in [4]. Define a new concept of twisted cochain on Hochschild complex for -algebra and proof theorems (2.4) and (2.5) analog to theorems of kadishfili in [4].
Firstly we recall the definition of commutative -algebra and its related cohomology, we also define the twisted cochain and its properties on Hochschild complex from [1], [4] and [7].
Definition (2.1):
The twisted cochain is an element where , such that , since is multiplication in the Hochschild complex for algebra A. The set of twisted cochains is denoted by .
Definition 2.2.
Two twisted cochain a and are equivalent ( ) if there exist an element , such that:
The set , where ~ is an equivalent relation, is denoted by .
In the following we define the algebra commutative case and its related cohomology.
Definition(2.3):
algebra A is commutative algebra if , where the summation is got on the perturbation .
Definition(2.4):
If A is commutative algebra, then it’s Hochschild complex is called Harresona complex.
Definition(2.5):
The cohomology of the complex is called Harresona cohomology of commutative algebra A, then it’s Hochschild complex is called Harresona complex.
In the following we define a new concept of twisted cochain on Hochschild complex for algebra.
Definition (2.6):
Any element is called twisted cochain if the following hold:
(2-1)
(2-2)
All twisted cochain in Hochschild complex is denoted by the set .
Definition (2.7):
Two twisted cochains and are equivalent and denoted by g ~ g’ if there is , such that:
(2-3)
(2-4)
Where and are defined by formula (1-5), (1-6).
Suppose that ) where ~ is an equivalent relation, then the following holds.
Theorem (2.8):
Let be an arbitrary twisted cochain and , such that , then there exist Twisted cochain such that:
1.
2.
3. ~ .
Proof. By using the method of constructing element ~ . Note that , to use the condition of the theorem (2.8) we have the relation . For every element in definition (2.5), which make the equivalent relation ~ , we consider it as an element satisfies condition of theorem (2.8). Define from condition 1 of theorem (2.8). For elements g and , the first nontrivial elements in right hand side of relation (2-4) is given in (k+1)-dimension, such that , this relation is true if ( all remain ).
Theorem (2.9):
For , we get .
Proof: We must prove that the arbitrary twisted cochain, given condition, is equal zero. The formula (2-2), for element g, in (n+1)-dimension has the form , that is g is acyclic. By considering the condition there exist such that or . Following theorem (2.8) we can get a twisted cochain such that . Hence the formula (2-2) in (n+2)-dimension, for element , is given by , that is is acyclic. since , then there is such that or and so on. Repeating this process we get a sequence of twisted cochain such that The extension of this process to infinity get trivial twisted cochain with the element f with components and )
3- Extension of algebra and cohomology of Hochschild of for
algebra
Let A be algebra with nontrivial finite number of the multiplication i.e. ( . The extension of algebra is an algebra Ā such that A and Ā coincided and the high multiplication for .
In [4] is proved that there is a bijection between the set of structure algebra on fixed graded algebra, such that , , π is multiplication in algebra, denoted by and the set of twisted cochains Hochschild complex factored by the equivalent relation ~.
We give an important extension of this fact between the set of all extension of a fixed , denoted by , where is the structure on a fixed A, and the set of twisted cochains Hochschild complex factored by the equivalent relation ~(D(A,A)).
The following theorem is the main result in this part
Theorem (3.1):
There is a bijection map between the sets and .
Proof: For consider the stasheff relation (1-1) as follow:
(3-1)
Clearly that the first term of (3-1) is equal zero, following stasheff relation for fixed algebra A. The second and third terms of (3-1) , following (1-5), (1-6) can be written in the form . The fourth term in form where and .
Therefore the stasheff relation (1-1) takes the form , and hence g is twisted cochain.
Thus every -structure from defines a twisted cochain for Hochscihld complex C∞ (A, A). The inverse is true that is every twisted cochain defines – structure.
To complete the proof we must show that any two extension of -algebra are equivalent if and only if every equivalent result coincide with its twisted cochain. From theorem (3.1) and definition (2.3) we get the following assertion
Theorem (3.2):
If , then any structure of extension of a fixed -algebra is trivial.
Proof of this theorem is trivial.
References:
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[9] Stasheff J.D., “Homotopy associatively of H-space”, 1, 2 // transfer. Math. Soc. 1963, V.108, N.2, P.275-313.
Alaa Hassan Noreldeen
Dept. of Mathematics, Faculty of Science ,
Aswan University, Aswan, Egypt.