Medical, Pharma, Engineering, Science, Technology and Business

**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.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

Given an associative graded algebra equipped with a degree +1 differential Δ we define an A_{∞}-structure that measures the failure of Δ to be a derivation. This can be seen as a non-commutative analog of generalized BValgebras. In that spirit we introduce a notion of associative order for the operator Δ 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.

A_{∞}-Algebras; Hochschild cocomplex; BV-algebra; coassociative coalgebra

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 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: , and

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 b_{3}=0 we obtain the definition of an ordinary BV-algebra. Put differently, that b_{3} =0 is equivalent to that Δ is a second order operator or that b_{2} is a strict degree +1 Lie bracket. Saying that b_{n+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 m_{n+1}=0. In case Δ has associative order 2 or equivalently that m_{3}=0, the operation m_{2 }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 m_{2} is (up to a sign) the Gerstenhaber bracket.

Definition 1. Let A be a graded associative (not necessarily unital) algebra over a field of characteristic zero, such that the underlying -graded vector space is a complex, that is, it has a degree +1 linear operator Δ such that Δ^{2} =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

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

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

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 -graded vector space with a degree +1 differential Δ.An A_{∞}-structure on A is a collection of degree 2 − n maps

such that the following identity is satisfied for every n (where we put a_{1}=Δ).

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

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

such that the following identity is satisfied for every n (where we put m_{1}=Δ).

**Remark 3.** An A_{∞}-structure on A where a_{k} 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 m_{k}=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

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

m_{1}=Δ,

and

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 Δ^{2}=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 Δ ^{2}=0. Denote multiplication of n ordered elements by the mapγ : . Define maps *

(1)

(2)

and

(3)

for n ≥ 3. Now let

Associator

*Then we have the identity*

Associator

Proof. For every n we have to check the identity

Every term is either of the form (Case 1)

Or of the form (Case 2)

We will prove the identity by checking that the coefficient in front of every type of term not containing Δ^{2} vanishes and that the coefficients of the terms with Δ^{2} agree.

**Case 1**

We look at the coefficient in front of

for fixed . 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

Therefore it remains to check the following terms:

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

The term vanishes similarly.

The term has contributions from , again canceling.

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

The term vanishes similarly.

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

**Case 2**

We now look at the coefficient in front of

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

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

The term vanishes similarly.

For j, l ≥1 the term has contributions from , , and from The 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

Is the same as the coefficient from 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.

**Lemma 1.** Suppose A is a graded associative algebra with a differential Δ and let mn be a sin Theorem1.If m_{n} =0 then m_{i}=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

We want to show that

By writing and using that m_{2} =0, we see that m_{3} vanishes. Now suppose n≥4 and that m_{n−1} vanishes. We want to show that

Similarly to the previous case we rewrite and use that m_{n−1} =0 to see that m_{n} 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 m_{n+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

Thus if m_{n} 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 γ _{2} and differential Δ of associative order 2. Then the identities*

and

hold.

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

Now we have

The other identity is proved in the same way.

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 m _{k} are trivial on Δ-cohomology.*

**Proof. **On cohomology every element in the image of Δ is zero. But every term in the definition of m_{k} contains images of Δ.

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 as degree 0 and matrices of the form as degree 1. Now let us consider the differential given by . 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 *m _{1}* is exactly Δ. Note that the

we have nothing in degree 2. Thus we only need to compute the *m _{k}* : s on diagonal matrices. The defining formula gives

Similarly to determine m_{3} we apply the definition to obtain

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 L_{ξ} . It is a degree +1 endomorphism of A satisfying Consider the construction in Theorem 2 applied to . We see that and that

Now consider an arbitrary degree +1 endomorphism satisfying This also induces a strict degree +1 associative algebra structure since m_{3} vanishes by applying

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 {m_{n}} correspond to the coderivation*

**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 {m_{n}} on A[1] there is an A_{∞}- structure {t_{i}}on the shifted reduced tensor algebra T(A)[1]. We have

And for k ≥ 3 we have

**Proof. **By definition t1 is just application of Δ , the signs originate from the Koszul sign rule. The defnition of t_{2} is

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

The definition of t_{k} for k ≥ 3 is

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

**Remark 7.** *Note that t _{1} is the differential of the bar construction. If m_{i}=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*

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

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 Δ ^{2} = 0. Furthermore, on Hochschild cohomology the following identity holds*

*Where [,] is the Grestenhaber bracket and 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 {m_{n}}n>1 on the cohomology HH• (A, A)[ 1] such that m_{1} = Δ and m_{2} is the Gerstenhaber bracket up to a sign.

Proof. The A_{∞}-structure is built from the operator Δand the cup product .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 m_{2} we see that Theorem 7 shows that m_{2} is the Gerstenhaber bracket up to a sign.

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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