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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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A Algebras Derived from Associative Algebras with a Non-Derivation Differential

Kaj Borjeson*

Department of Mathematics, Stockholm University, 10691 Stockholm, Sweden

*Corresponding Author:
Borjeson K
Department of Mathematics
Stockholm University, 10691 Stockholm, Sweden
Tel: 08-16 4531
E-mail: [email protected]

Received date: November 09, 2014; Accepted date: February 25, 2015; Published date: March 28, 2015

Citation: Borjeson K (2015) A∞-Algebras Derived from Associative Algebras with a Non-Derivation Differential. J Generalized Lie Theory Appl 9: 214. doi:10.4172/1736-4337.1000214

Copyright: © 2015 Borjeson K. 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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Abstract

Given an associative graded algebra equipped with a degree +1 differential delta we define an A∞-structure that measures the failure of delta to be a derivation. This can be seen as a non-commutative analog of generalized BV-algebras. In that spirit we introduce a notion of associative order for the operator delta and prove that it satisfies properties similar to the commutative case. In particular when it has associative order 2 the new product is a strictly associative product of degree +1 and there is compatibility between the products, similar to ordinary BV-algebras. We consider several examples of structures obtained in this way. In particular we obtain an A∞-structure on the bar complex of an A∞-algebra that is strictly associative if the original algebra is strictly associative. We also introduce strictly associative degree +1 products for any degree +1 action on a graded algebra. Moreover, an A∞-structure is constructed on the Hochschild cocomplex of an associative algebra with a non-degenerate inner product by using Connes’ B-operator.

Keywords

A-Algebras; Hochschild cocomplex; BV-algebra; coassociative coalgebra

Introduction

Consider a graded commutative algebra equipped with a degree +1 differential Δ. There is an L-structure measuring the failure of Δ to be a derivation. The L-operations are called the Koszul hierarchy [1,2], and are defined as image where the brackets are commutators of operators and La is the operator multiplying by a from the left. Unwrapping this compact definition we see that the first few operations are: image, imageand image

Disregarding signs from element degrees; the operations are a sum over all different ways of applying Δ to a subset of the inputs with minus signs if there is an odd number of an element outside Δ.The data of the commutative product and the operator Δ is called a generalized BV-algebra. Note that no compatibility is assumed and that there is no explicit mention of Lie brackets. If we require that b3=0 we obtain the definition of an ordinary BV-algebra. Put differently, that b3 =0 is equivalent to that Δ is a second order operator or that b2 is a strict degree +1 Lie bracket. Saying that bn+1 =0 is equivalent to Δ being of n:th order. However, if the algebra is not graded commutative the operations bn does not form an L-structure, [3] for one approach to repair this. In this note we define a set of operations mn for an arbitrary graded associative algebra with a degree +1 differential Δ. We prove that these operations form an A-structure measuring the failure of Δ to be a derivation. In analogy with the commutative case we define a notion of associative order of the operator Δ by saying that it has associative order n if mn+1=0. In case Δ has associative order 2 or equivalently that m3=0, the operation m2 is a strict degree +1 graded associative algebra and it turns out that there is extra compatibility between the products. We consider the combined structure of the two different degree products and Δ as a non-commutative analog of BValgebras. Forgetting the operator Δ yields a non-commutative analog of Gerstenhaber algebras. Note that it does not reduce to the usual notion of BV- algebra in the case the starting algebra is commutative. It should perhaps rather be seen as a BV-algebra in the “associative world” in the sense of [4].

Any A-algebra determines and is determined by the bar differential on the tensor (co)-algebra of the underlying complex. The bar differential is a coderivation of the coproduct but it is not a derivation with respect to the tensor product. We apply our construction in this case and obtain an A-structure on the tensor module that is strict in case the original A-structure is strict.

Consider an odd element and a degree 0 graded associative multiplications on a complex V. As an easy example we obtain a degree +1 graded associative algebra structure from our construction by letting Δ be the left multiplication of the odd element. The same construction yields a strict degree +1 associative product from any degree +1 left action on an associative algebra.

Consider the Hochschild cocomplex of an associative algebra with an invariant non-degenerate bilinear form. The form allows us to move Connes’ B-operator from the complex to the cocomplex. Applying our construction with this operator and the cup product yields an A- structure. It turns out that m2 is (up to a sign) the Gerstenhaber bracket.

Conventions

Definition 1. Let A be a graded associative (not necessarily unital) algebra over a field image of characteristic zero, such that the underlying image -graded vector space is a complex, that is, it has a degree +1 linear operator Δ such that Δ2 =0. We call this an algebra with differential.

Remark 1. Note that we do not require any compatibility between Δ and the multiplication. The symbol Δ is chosen to remind of the odd Laplacian operator in a BV-algebra structure.

Remark 2. We work in the symmetric monoidal category of complexes, thus we employ the Koszul sign rule.That is, when we permute two homogeneous odd elements we multiple the result by -1.

Definition 2. Let A be an algebra with differential Δ and product

image . Suppose that image. Then we call Δa derivation and A a differential graded algebra.

Definition 3. Suppose C is a graded vector space. Now let image

T(C) is an associative algebra with the product given by concatenation of tensor words. It is also a coassociative coalgebra given by the sum over all ways to split a tensor word in two without permuting any elements.

Definition 4. Suppose A is a image -graded vector space with a degree +1 differential Δ.An A-structure on A is a collection image of degree 2 − n maps

image

such that the following identity is satisfied for every n (where we put a1=Δ).

image

Equivalently, we can define the structure on the shifted space A[1] (where (A[1])i=Ai−1).

An A-structure on A [1] is a collection {mn}n≥2 of degree +1 maps

image

such that the following identity is satisfied for every n (where we put m1=Δ).

image

Remark 3. An A-structure on A where ak vanishes for k≠2 is an ordinary graded associative product on A. An A-structure on A where ak vanishes for k>2 is a differential graded algebra. An A-structure on A[1] with mk=0 for k≠2 is an associative algebra with product of degree +1 on A. Since the main construction of this note deals with the interplay of products of different degrees we cannot regrade to get rid of the products with odd degree. No matter how we choose it some product will be more complicated. We prefer to construct an A- structure on A[1] to avoid the presence of too complicated signs in the identities we have to check

A-Structure from Non-Derivation Differential

Theorem 1. Let A be a graded associative algebra with differential Δ. Denote multiplication of n ordered elements by the map image. There is an A-structure on A[1] given by

m1=Δ,

image

and

image

For n ≥ 3.

Actually the proof gives a bit more. We have the following a bit more elegant and general results. Looking at the associators of operations mn can be seen as taking a kind of square. The theorem says that this operation yields the same result as squaring the operator Δ first. In the case Δ2=0 it reduces to the previous theorem. This formulation is analogous to a result in the commutative case, see Theorem 2 [5].

Theorem 2. Let A be a graded associative algebra with a degree +1 operator Δ, not necessarily satisfying Δ2=0. Denote multiplication of n ordered elements by the mapγ : image. Define maps

image (1)

image (2)

and

image (3)

for n ≥ 3. Now let

Associator image

Then we have the identity

Associator image

Proof. For every n we have to check the identity

image

Every term is either of the form (Case 1)

image

Or of the form (Case 2)

image

We will prove the identity by checking that the coefficient in front of every type of term not containing Δ2 vanishes and that the coefficients of the terms with Δ2 agree.

Case 1

We look at the coefficient in front of

image

for fixed image. In the definition of the product there are no nonzero terms where there are more than one id in front or more than id one id behind Δ.From this we see that the coefficient in front of

image

Therefore it remains to check the following terms: image

image

The term image has contributions from image and image .They contribute +1 and −1 respectively; they have different signs because the Δ:s pass each other when calculating one of the terms.

The term image has contributions from image and image. They contribute with opposite signs; again using the Koszul sign rule.

The term image vanishes similarly.

The term image has contributions from image, again canceling.

The term image has contributions by image and imageThese cancel by the Koszul sign rule.

The term image vanishes similarly.

The term image has contributions from image and image, also vanishing. The term image has contributions from image and image thus also vanishes by the Koszul rule.

Case 2

We now look at the coefficient in front of

image

Where either j or l is non-zero. Changing I or m only multiplies it with1, −1 or 0 so it is enough to check the vanishing of coefficient for cases of the form image

The term image has contributions from image and from image which vanishes by equation1and 2.

The term image vanishes similarly.

For j, l ≥1 the term imagehas contributions from image, image, image and from imageThe sum of the contributing coefficients vanishes by equation1,2and3.

When j=l=0 it does not necessarily vanish. We want to prove that the coefficient from

image

Is the same as the coefficient from image But in this case there is only one contributing term on both sides, the coefficient comes from the definition of m∗,I which is the same in both cases.

Failure of Being A Derivation and Associative Order of Operators

Lemma 1. Suppose A is a graded associative algebra with a differential Δ and let mn be a sin Theorem1.If mn =0 then mi=0 for i >n.

Proof. Suppose m1=Δ=0, then it is clear that all mn vanishes since all terms use Δ.Suppose instead that image

We want to show that

image

By writing image and using that m2 =0, we see that m3 vanishes. Now suppose n≥4 and that mn−1 vanishes. We want to show that image

Similarly to the previous case we rewrite image and use that mn−1 =0 to see that mn also vanishes. Now the lemma follows by induction.

This lemma motivates the following definition inspired by the commutative case.

Definition 5. Suppose A is a graded associative algebra with a differential Δ. We say that Δ has associative ordern if mn+1 vanishes.

Remark 4. In the case of a unital algebra A, Definition 5 has to be tweaked in some way to get the right notion. Suppose that mn=0 for some n>3, then we see that image

Thus if mn vanishes for some n, then we obtain a dg algebra. Note that we still obtain an on-trivial A-structure in the unital case, however the notion of order has to be modified.

Remark 5. An operator of associative order1 is the same thing as a derivation, thus we can look at the A-structure as measuring the failure of Δ to be a derivation.

The next theorem shows that there is compatibility in the case when the operator has associative order2. This is analogous to an ordinary BV-algebra and the Gerstenhaber part of it.

Theorem 3. Suppose A is a graded associative algebra with multiplication γ2 and differential Δ of associative order 2. Then the identities

image

and

image

hold.

Proof. That Δ has associative order 2 is equivalent to the identity

image

Now we have

image

The other identity is proved in the same way.

Triviality on Δ-Cohomology

Given an A-algebra one has an induced structure on cohomology. The structure from Theorem 1 measures the incompatibility of Δ with an associative product. Since passing to cohomology kills Δ one can guess that the induced structure is trivial. This is indeed the case and is analogous to the commutative case.

Theorem 4. Let A and Δ be as in Theorem 1 .The operations mk are trivial on Δ-cohomology.

Proof. On cohomology every element in the image of Δ is zero. But every term in the definition of mk contains images of Δ.

Triangular Matrices and Odd Actions

As a first very concrete example we consider the algebra of upper triangular matrices.

Example 1. Let A be the algebra of upper triangular 2×2-matrices.

This has a grading where we consider matrices of the form image as degree 0 and matrices of the form image as degree 1. Now let us consider the differential given by image. It is easy to check

that the multiplication and differential respect the grading. We will determine the structure given by Theorem 1 in this case. By definition m1 is exactly Δ. Note that the mk : s all raise the degree by one, because of this all multiplications involving elements of degree 1 will vanish since case.

we have nothing in degree 2. Thus we only need to compute the mk : s on diagonal matrices. The defining formula gives

image

image

Similarly to determine m3 we apply the definition to obtain

image

image

image

By Lemma 1 we can now see that mk vanishes for all higher k.

Remark 6. Note that there is nothing really special about this example except that is small and easily computable. We could have chosen any graded associative algebra with any differential.

The next example creates a degree +1 associative algebra from any graded associative algebra with a choice of degree +1 left action (for example left multiplication with a degree +1 element).

Example 2. Consider a dg algebra A with multiplication γ and a degree +1 element ξ . Denote left multiplication with ξ by Lξ . It is a degree +1 endomorphism of A satisfying image Consider the construction in Theorem 2 applied to image . We see that image and that

image

Now consider an arbitrary degree +1 endomorphism image satisfying image This also induces a strict degree +1 associative algebra structure since m3 vanishes by applying image

A-structure on the Bar Complex

An alternative characterization of A-structure is the following.

Theorem 5. An A-structure on A is equivalent to a degree +1 square-zero coderivation of the reduced tensor coalgebra on A [1]. An A-structure {mn} correspond to the coderivation

image

Proof. See for example, Section 9.2.1 in [6].

This coderivation is however not necessarily a derivation with respect to the tensor product, enabling us to apply Theorem 1.

Theorem 6. Given an A-structure {mn} on A[1] there is an A- structure {ti}on the shifted reduced tensor algebra T(A)[1]. We have image

image

image

image

And for k ≥ 3 we have

image

image

Proof. By definition t1 is just application of Δ , the signs originate from the Koszul sign rule. The defnition of t2 is

image

image

By the definition of Δ we see that the first term consists of all the way we can contract using the multiplication and the other terms are contractions using only elements from the first respectively the second tensor word. The remaining terms are thus the contractions involving elements from both words. This can be written as

image

image

The definition of tk for k ≥ 3 is image

image

Similarly to the previous case we see that the first term corresponds to any contractions, the second term corresponds to contractions not involving the last tens or word, the third corresponds to contractions not involving the first word and the fourth to contractions not involving the first or last word. Thus we see that the remaining terms are only the contractions involving elements both from the first and the last tensor word. Alternatively this can be written as

image

image

Remark 7. Note that t1 is the differential of the bar construction. If mi=0 for i ≥ 3 we see that ti vanishes for i ≥ 3 making the bar resolution complex into a dg algebra (with odd degree). In this case the product is easily described by

imageimage

and Theorem 3 says that there is a compatibility with the tensor product (which is also easy to prove directly).

Hochschild Cocomplex and the Dualized Connes’ B-Operator

For and introduction to Hoschild cohomology, see for example [7]. Consider a finite- dimensional associative unital algebra A with a symmetric, invariant non-degenerate inner product <,>. In [8], a degree -1 differential Δ is considered on Hochschild cochains C• (A, A) defined by

image

This operator is Connes' B-operator transferred from chains to cochains by using the inner product. The following is shown in Theorem 1 of [8].

Theorem 7. Δ is a chain map with respect to the Hochschild differential and Δ2 = 0. Furthermore, on Hochschild cohomology the following identity holds

image

Where [,] is the Grestenhaber bracket and image is the cup product as defined in [9].

Proof. [8].

It is clear that the construction in Theorem 1 works if we reverse gradings and in that case we obtain a homologically graded A- structure. We can therefore apply that machinery to C• (A, A) equipped with the cup product and the differentialΔ. Doing this we obtain an A- structure on the chain level.

Theorem 8. There is a homologically graded A1-structure on C• (A, A)[−1] such that it induces an A-structure given by maps {mn}n>1 on the cohomology HH• (A, A)[ 1] such that m1 = Δ and m2 is the Gerstenhaber bracket up to a sign.

Proof. The A-structure is built from the operator Δand the cup product image .Since both are compatible with the Hochschild coboundary δ , the A-structure is well defined on the cohomology. By definition m1 is given by Δ and by unwinding the definition of m2 we see that Theorem 7 shows that m2 is the Gerstenhaber bracket up to a sign.

Acknowledgement

I would like to thank J. Alm, A. Berglund, T. Backman and S. Merkulov for interesting discussions. A special thanks to S. Merkulov for many useful suggestions and comments during the preparation of this paper.

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