The Second Hochschild Cohomology Group for One-Parametric Self-Injective Algebras ()

Deena Al-Kadi

Department of Mathematics, Taif University, Taif, KSA.

**DOI: **10.4236/apm.2013.35065
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Department of Mathematics, Taif University, Taif, KSA.

In this paper, we determine the second Hochschild cohomology group for a class of self-injective algebras of tame representation type namely, which are standard one-parametric but not weakly symmetric. These were classified up to derived equivalence by Bocian, Holm and Skowroński in [1]. We connect this to the deformation of these algebras.

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D. Al-Kadi, "The Second Hochschild Cohomology Group for One-Parametric Self-Injective Algebras," *Advances in Pure Mathematics*, Vol. 3 No. 5, 2013, pp. 458-469. doi: 10.4236/apm.2013.35065.

1. Introduction

This paper determines the second Hochschild cohomology group for all standard one-parametric but not weakly symmetric self-injective algebras of tame representation type. Bocian, Holm and Skowroński give, in [1], a classification of these algebras by quiver and relations up to derived equivalence. The algebras in [1] are divided into two types, namely the algebra where are integers such that p, and and the algebra where. Thus the second Hochschild cohomology group will be known for all the classes of the algebras given in [1]. We remark that an algebra of the type is never isomorphic to an algebra of the type as their stable Auslander-Reiten quivers are not isomorphic. We refer the reader to [1] which gives precise conditions for two algebras of the same type or to be isomorphic.

We start, in Section 2, by introducing the algebras, for both types, by quiver and relations. Section 3 of this paper describes the projective resolution of [2] which we use to find. In the third section, we determine for the algebra, considering separately the cases and. The main result in this section is Theorem 4.9, which shows that has dimension 1 for. This group measures the infinitesimal deformations of the algebra; that is, if then has no non-trivial deformations, which is not the case here. We include, in Section 4, Theorem 4.10 where we find a non-trivial deformation of associated to our nonzero element in. This illustrates the connection between the second Hochschild cohomology group and deformation theory. In the final section, we determine for. The main result in Section 5 is Theorem 5.4 which shows that. The results we found in this paper are in contrast to the majority of self-injective algebras of finite representation type (see [3]). Since Hochschild cohomology is invariant under derived equivalence, the second Hochschild cohomology group is now known for the standard one-parametric but not weakly symmetric self-injective algebras of tame representation type which are derived equivalent to the algebra of the type or.

2. The One-Parametric Self-Injective Algebras

In this chapter we describe the algebras of [1]. We start with the algebra. Let K be an algebraically closed field and let be integers such that p, and. From [1, Section 5], has quiver:

where, for any, denotes the path

and denotes the path

Then where is the ideal generated by the relations

• , for• , for• for, ,

for, ,

for, and

where.

Next we describe the algebra For, is given in [1, Section 6] by the quiver:

Then where is the ideal generated by the relations:

1)

2)

3) for all

Note that we write our paths from left to right.

In order to compute, the next section gives the necessary background required to find the first terms of the projective resolution of as a -bimodule. Section 4 and Section 5 uses this part of a minimal projective bimodule resolution for our algebras to determine the second Hochschild cohomology group and provides the main results of this paper.

3. Projective Resolutions

To find the second Hochschild cohomology group, we could use the bar resolution given in [4]. This bar resolution is not a minimal projective resolution of as -bimodule. In practice, it is easier to compute the Hochschild cohomology group if we use a minimal projective resolution. So here we use the projective resolution of [2]. More generally, let be a finite dimensional algebra, where K is an algebraically closed field, is a quiver, and I is an admissible ideal of. Fix a minimal set of generators for the ideal I. Let. Then, that is, x is a linear combination of paths for and and there are unique vertices v and w such that each path starts at v and ends at w for all j. We write and Similarly is the origin of the arrow a and is the end of a.

In [2, Theorem 2.9], it is shown that there is a minimal projective resolution of as a -bimodule which begins:

where the projective -bimodules are given by

and the maps, and are -bimodule homomorphisms, defined as follows. The map is the multiplication map so is given by. The map is given by

for each arrow. With the notation for given above, the map is given by

where.

In order to describe the projective bimodule and the map in the -bimodule resolution of in [2], we need to introduce some notation from [5]. Recall that an element is uniform if there are vertices such that We write and. In [5], Green, Solberg and Zacharia show that there are sets in, for, consisting of uniform elements such that

for unique elements such that. These sets have special properties related to a minimal projective -resolution of, where is the Jacobson radical of. Specifically the n-th projective in the minimal projective -resolution of is

In particular, to determine the set, we follow explicitly the construction given in [5, §1]. Let denote the set of arrows of. Consider the intersection

. Set this intersection equal to some. We then discard all elements of the form that are in; the remaining ones form precisely the set.

Thus, for we have that

. So we may write

with, such that are in the ideal generated by the arrows of, and unique. Then [2] gives that

and, for in the notation above, the component of in the summand of is

Applying to this part of a minimal projective bimodule resolution of gives us the complex

where is the map induced from for. Then

Throughout, all tensor products are tensor products over, and we write for. When considering an element of the projective -bimodule

it is important to keep track of the individual summands of. So to avoid confusion we usually denote an element in the summand by using the subscript “a” to remind us in which summand this element lies. Similarly, an element lies in the summand

of and an element lies in the summand of. We keep this notation for the rest of the paper.

4. for

We have given by quiver and relations in Section 2. However, these relations are not minimal. So next we will find a minimal set of relations for this algebra.

Let

The remaining relations given in Section 2 are all linear combinations of the above relations. For example, the relation can be written as

So this relation is in I and is not in.

Proposition 4.1 For and with the above notation, the minimal set of relations is

In contrast to the majority of self-injective algebras of finite representation type, we will show that the algebra has non-zero second Hochschild cohomology group (see [3, Theorem 6.5]). Recall that, where

is induced by.

First we will find. Since

let so that. We consider the cases and separately.

Let and

where all coefficients for for Now we find.

First we have,

Similarly for,

For the remaining terms, where for all,

and.

Let

for and

for

Thus for and, fA_{2} is given by

where with. So

For, we let

where for all the coefficients for for are in

Then we can find for in the same way as the previous case to see that it is given by

where with. Note that there is no dependency between the So

Proposition 4.2 If, we have If, we have

Next we find and again consider the two cases separately. Let and . Then is defined by

where.

Therefore Hence,

For and, is given by

where are in K for Thus

Proposition 4.3 If, we have

If,

Corollary 4.4 If, we have. If,

In order to find Kerd_{3} and hence determine we start by giving a non-zero element in for all s.

Proposition 4.5 Define by

Then is in.

Proof. We note that so is a non-zero map. To show that we show that. First, observe that and Hence. Similarly we have

Recall that where

and are in the ideal generated by the arrows. For the component of

in is

Then

Thus

As is in the arrow ideal of, So we have Similarly

as Therefore

for all so. Thus as required.

Theorem 4.6 For where are positive integers, , with

and, we have.

Proof. Consider the element of

where is given as in Proposition 4.5 by

Suppose for contradiction that Then. So and so

. Also where Then where But this contradicts having. Therefore, that is,. So is a nonzero element in□

Note that we can also define maps by

for. However, all represent the same element of.

As we have found a non-zero element in we know that. In the case

we have the following result, the proof of which is immediate from Proposition 4.2, Corollary 4.4 and Theorem 4.6.

Proposition 4.7 For where, we have and

For the case, we need more details to find. Following [5] we may choose the set to consist of the following elements:

where

Thus the projective bimodule is

Now we determine in the case. Let, so and. Recall that for, is given by

where are in.

Then for, we have

In a similar way we can show that.

For, we have

As we have for.

Similarly it can be shown that

so that.

We also have for

and Finally, putting

does not give any new information for,.

Thus h is given by

where for are in K. It is clear that there is no dependency between, and therefore.

Proposition 4.8 For and, we have

Using Propositions 4.2, 4.7, 4.8 and Theorem 4.6 we get the main result of this section.

Theorem 4.9 For where p, q, s, k are integers such that p, and, we have

We conclude this section by giving a deformation of which arises from the non-zero element in.

Let. Recall that

. We introduce a new parameter and define the algebra to be the algebra where is the ideal generated by the following elements:

1) where

2) for all, where

3) for all arrows a with4) for all arrows a with

We now need to show that to verify that is indeed a deformation of. First of all, it is clear that for all t and for all vertices e_{i} with. Now we consider and with, and with. These projective modules are described as follows:

In each case we see that

for all t. Hence. Moreover, when the algebras and are not isomorphic since, in this case, is not self-injective. Thus we have found a non-trivial deformation of.

Theorem 4.10 With and as defined above, then is a non-trivial deformation of. Moreover, the algebras and are socle equivalent.

5. for

We have given the algebra by quiver and relations in Section 2. Note that these relations are not minimal. So we will find a minimal set of relations for this algebra.

Let

The remaining relation can be written as. So this relation is in I and is not in.

Proposition 5.1 For and with the above notation, the minimal set of relations is

Recall that the projective. Thus we have

(We note that the projective is also described in [4] although Happel gives no description of the maps in the -projective resolution of.) Following [2], and with the notation introduced in Section 3, we may choose the set to consist of the following elements:

with where

We know that. First we will find. Let and so write

where

Now we find. We have

Also

We can show by direct calculation that

for all.

Thus is given by

So.

Proposition 5.2 For, we have

Now we determine. Let, so and. Then is given by

for some for

Then

As we have and

As we have and. So and

Next,

So we have and hence

Therefore as

Thus again we have

As above, we have as we already know.

Also

So we have and

Finally, for, we have

Therefore we have and. Hence and for as we have above and

Thus is given by

for some

Proposition 5.3 For, we have

Therefore

and a basis is given by the maps and where is given by

is given by

From Proposition 5.2 and Proposition 5.3 we get the main result of this section.

Theorem 5.4 For with we have

To connect this with deformations we use a similar discussion as Section 4. We introduce the parameter and define the algebra to be the algebra where is the ideal generated by the following elements:

1)

2)

3)

4)

We can show that. Hence this algebra has no non-trivial deformation.

From Theorem 4.9 and Theorem 5.4 we have now found for all standard one-parametric but not weakly symmetric self-injective algebras of tame representation type.

6. Acknowledgements

I thank Prof. Nicole Snashall for her encouragement and helpful comments.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | R. Bocian, T. Holm and A. Skowroński, “Derived Equivalence Classification of One-Parametric Self-Injective Algebras,” Journal of Pure and Applied Algebra, Vol. 207, No. 3, 2006, pp. 491-536. doi:10.1016/j.jpaa.2005.10.015 |

[2] | E. L. Green and N. Snashall, “Projective Bimodule Resolutions of an Algebra and Vanishing of the Second Hochschild Cohomology Group,” Forum Mathematicum, Vol. 16, No. 1, 2004, pp. 17-36. doi:10.1515/form.2004.003 |

[3] | D. Al-Kadi, “Self-Injective Algebras and the Second Hochschild Cohomology Group,” Journal of Algebra, Vol. 321, No. 4, 2009, pp. 1049-1078. doi:10.1016/j.jalgebra.2008.11.019 |

[4] | D. Happel, “Hochschild Cohomology of Finite-Dimensional Algebras,” Lecture Notes in Mathematics, Spring-Verlag, Berlin, 1989. doi:10.1090/S0002-9947-01-02687-3 |

[5] | E. L. Green, Ø. Solberg and D. Zacharia, “Minimal Projective Resolutions,” Transactions of the American Mathematical Society, Vol. 353, No. 7, 2001, pp. 2915-2939. doi:10.1007/BFb0084073 |

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