The Second Hochschild Cohomology Group for One-Parametric Self-Injective Algebras ()
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, ![](https://www.scirp.org/html/3-5300450\8c7b3511-ca5f-47e9-8ca7-27c8492b8191.jpg)
![](https://www.scirp.org/html/3-5300450\19115895-b823-454d-95bd-8727270a8d14.jpg)
![](https://www.scirp.org/html/3-5300450\20c3253c-0b9b-49b2-9ea4-de3ecb8bed1b.jpg)
![](https://www.scirp.org/html/3-5300450\93c4a792-0371-4f35-b3cb-b9c961fe9fff.jpg)
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, ![](https://www.scirp.org/html/3-5300450\aeed21a1-7afe-4f68-aaf0-3c3ed931cff7.jpg)
![](https://www.scirp.org/html/3-5300450\c633737f-aa15-4c41-86fc-da1f0216390f.jpg)
![](https://www.scirp.org/html/3-5300450\5c7cd4f4-22ca-421d-a3ec-5f39320cf89c.jpg)
and
. From [1, Section 5],
has quiver
:
![](https://www.scirp.org/html/3-5300450\dd926338-a73f-489b-b4b8-2187a886f680.jpg)
where, for any
,
denotes the path
![](https://www.scirp.org/html/3-5300450\daa2b3db-48bb-43ce-a79f-e9f63f05cfcb.jpg)
and
denotes the path
![](https://www.scirp.org/html/3-5300450\4d93cc08-52d3-4ef5-a8c7-cd3a6f1908cf.jpg)
Then
where
is the ideal generated by the relations
•
, for
•
, for
•
for
,
,
![](https://www.scirp.org/html/3-5300450\76d73cfa-506b-4d63-b1a3-5a1746b28af4.jpg)
for
,
,
![](https://www.scirp.org/html/3-5300450\c5d376f3-6e78-4cdc-9746-51996e2727d6.jpg)
for
, and
![](https://www.scirp.org/html/3-5300450\28f862cf-8bd9-433e-a9be-3d9c5cbf1940.jpg)
where
.
Next we describe the algebra
For
,
is given in [1, Section 6] by the quiver
:
![](https://www.scirp.org/html/3-5300450\10a1fc2d-14f7-4b09-abca-17c51b2ca0eb.jpg)
Then
where
is the ideal generated by the relations:
1) ![](https://www.scirp.org/html/3-5300450\78b5dc34-8369-42e5-9caf-ef101f2d5fb7.jpg)
2) ![](https://www.scirp.org/html/3-5300450\b2202415-7913-4e26-9ed3-33f674003f06.jpg)
![](https://www.scirp.org/html/3-5300450\c9b6b775-0d08-402b-8015-4ee3da42aac5.jpg)
![](https://www.scirp.org/html/3-5300450\074f309a-2650-4246-b47b-a49e153c49f1.jpg)
3) for all ![](https://www.scirp.org/html/3-5300450\222aed3c-029b-4a12-8c7b-fb8e59c3afc3.jpg)
![](https://www.scirp.org/html/3-5300450\ce7930e3-0af3-4ed5-80d3-98d7e5af5ff3.jpg)
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:
![](https://www.scirp.org/html/3-5300450\82eeae75-e18f-4e97-bf94-5fa3ebc2406d.jpg)
where the projective
-bimodules
are given by
![](https://www.scirp.org/html/3-5300450\b96ad071-1b69-43d4-bd1d-b6a541d80cd0.jpg)
![](https://www.scirp.org/html/3-5300450\a8d89906-aa67-4e13-bcff-b0bb83ecf03b.jpg)
![](https://www.scirp.org/html/3-5300450\2cc42ef6-62af-4629-a8d8-e76eb877cfdb.jpg)
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
![](https://www.scirp.org/html/3-5300450\1ed19925-ffc3-4dea-afd1-988931e84a81.jpg)
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
![](https://www.scirp.org/html/3-5300450\58f1ab9d-51f7-4a03-bd39-0ecfe52a1308.jpg)
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
![](https://www.scirp.org/html/3-5300450\abc360c9-2326-4d77-8c77-ed528fc542d6.jpg)
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
![](https://www.scirp.org/html/3-5300450\acc67b90-6717-42df-aaa9-8055512ab475.jpg)
Applying
to this part of a minimal projective bimodule resolution of
gives us the complex
![](https://www.scirp.org/html/3-5300450\30650868-731e-4bf7-bcdd-25c4947dfd05.jpg)
where
is the map induced from
for
. Then ![](https://www.scirp.org/html/3-5300450\6d56e338-c852-4a56-8c14-d9e204a053bb.jpg)
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 ![](https://www.scirp.org/html/3-5300450\987e0ce3-90f9-4e12-b893-77850a72e109.jpg)
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
![](https://www.scirp.org/html/3-5300450\db5195f6-5d4d-4ae2-bcaa-ecd4f5c46493.jpg)
![](https://www.scirp.org/html/3-5300450\ef4733ef-878e-417e-9916-ebc2e1680a8e.jpg)
![](https://www.scirp.org/html/3-5300450\701cd2bb-b8a6-4fee-8c27-cc67cb950ff5.jpg)
![](https://www.scirp.org/html/3-5300450\a886848c-0b04-47b1-85ad-b5e5eb7e3424.jpg)
![](https://www.scirp.org/html/3-5300450\cdf2095e-17e0-40db-8412-5011d720e8bb.jpg)
![](https://www.scirp.org/html/3-5300450\2dd72052-2abe-4b86-ab6c-c419560ec6dc.jpg)
The remaining relations given in Section 2 are all linear combinations of the above relations. For example, the relation
can be written as
![](https://www.scirp.org/html/3-5300450\fc3ed79d-4c44-4ec0-a807-6f7c7be8e4ef.jpg)
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
![](https://www.scirp.org/html/3-5300450\0beceda4-3c27-4315-a2ca-af5d64f26e84.jpg)
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
![](https://www.scirp.org/html/3-5300450\0df5d38b-c699-4558-8515-f27fab9e9160.jpg)
is induced by
.
First we will find
. Since
let
so that
. We consider the cases
and
separately.
Let
and
![](https://www.scirp.org/html/3-5300450\1f452d99-d943-4c4a-93bf-8beba7f3249a.jpg)
![](https://www.scirp.org/html/3-5300450\35ca6cdc-12b4-49a4-8f46-826f513fc524.jpg)
![](https://www.scirp.org/html/3-5300450\00ab2467-2340-4c13-9ca7-344b50b8a210.jpg)
![](https://www.scirp.org/html/3-5300450\89f1751d-a28f-4907-ad9a-4a1484a5ebf0.jpg)
![](https://www.scirp.org/html/3-5300450\475bdd7d-ca2f-4951-acdc-f3e075ce4e7a.jpg)
where all coefficients
for
for
Now we find
.
First we have,
![](https://www.scirp.org/html/3-5300450\26f2ce5b-a088-4bdc-9a5c-375163fdeb60.jpg)
Similarly for
,
![](https://www.scirp.org/html/3-5300450\dc9f57aa-d11e-49e7-8c2e-4d2a7f2f363a.jpg)
For the remaining terms,
where
for all
,
and
.
Let
![](https://www.scirp.org/html/3-5300450\a4e65e68-a95b-4981-832f-cfa018cfb9d0.jpg)
for
and
![](https://www.scirp.org/html/3-5300450\8c1bae7b-5596-414b-8aff-ceaea9f3a226.jpg)
for ![](https://www.scirp.org/html/3-5300450\19ef03e3-3e59-44c3-8c16-a34bcc1c6b89.jpg)
Thus for
and
, fA2 is given by
![](https://www.scirp.org/html/3-5300450\114f0ffa-b554-4da0-a9cc-2fb6195c0b20.jpg)
![](https://www.scirp.org/html/3-5300450\ffd6c4ad-5dc7-4264-9b1f-d9a52fc3d568.jpg)
![](https://www.scirp.org/html/3-5300450\d166c92f-8f02-40e4-9d28-7d6e6890233f.jpg)
![](https://www.scirp.org/html/3-5300450\7c032bf4-5796-4b9e-9de9-f8b2c72cfee7.jpg)
![](https://www.scirp.org/html/3-5300450\86c63616-7fb2-4acf-99d2-93c2597631b7.jpg)
where
with
. So
![](https://www.scirp.org/html/3-5300450\dd830072-4066-44fd-9671-8bd2772a27d6.jpg)
For
, we let
![](https://www.scirp.org/html/3-5300450\e8ec0139-3f45-4ef9-a048-19db98250335.jpg)
![](https://www.scirp.org/html/3-5300450\e3eae03a-929c-4b4e-9752-98a82c9bebce.jpg)
![](https://www.scirp.org/html/3-5300450\37aa155a-8768-4d45-8ad7-447f21063676.jpg)
![](https://www.scirp.org/html/3-5300450\516c25e8-d1e7-4bf0-9f26-5e534cbb93b4.jpg)
![](https://www.scirp.org/html/3-5300450\2b0a629f-1f79-480f-bfc7-4cc28599b54f.jpg)
![](https://www.scirp.org/html/3-5300450\10464798-8d23-45af-88aa-f78c2be79b38.jpg)
where for all
the coefficients
for
for
are in ![](https://www.scirp.org/html/3-5300450\5a3b39ca-24af-4fba-908d-898087f5b821.jpg)
Then we can find
for
in the same way as the previous case to see that it is given by
![](https://www.scirp.org/html/3-5300450\5aafd9fe-1f0e-44e1-9be5-b0ee78d752c0.jpg)
![](https://www.scirp.org/html/3-5300450\0f7d2ff5-c14f-402a-9545-b9396ee815f5.jpg)
![](https://www.scirp.org/html/3-5300450\ec87e1bc-ffb6-4ea7-a49d-fb73f00b2962.jpg)
![](https://www.scirp.org/html/3-5300450\5cdfc717-c468-4716-9b5b-29b15e759516.jpg)
![](https://www.scirp.org/html/3-5300450\44cbf1af-f110-45a1-a0d8-85c3e6b66466.jpg)
where
with
. Note that there is no dependency between the
So ![](https://www.scirp.org/html/3-5300450\77298c73-cb28-45d0-aa02-1a6ff59fea9a.jpg)
Proposition 4.2 If
, we have
If
, we have ![](https://www.scirp.org/html/3-5300450\d19e5934-6518-47db-9592-8867c89e2087.jpg)
Next we find
and again consider the two cases separately. Let
and
. Then
is defined by
![](https://www.scirp.org/html/3-5300450\2bb41112-13a1-4302-8aac-6f218508cd26.jpg)
where
.
Therefore
Hence, ![](https://www.scirp.org/html/3-5300450\fe5da5fa-3f79-4849-9553-25bd82643e57.jpg)
For
and
,
is given by
![](https://www.scirp.org/html/3-5300450\dc62e6ad-7b4d-4cd0-8087-061d15c0afc3.jpg)
where
are in K for
Thus ![](https://www.scirp.org/html/3-5300450\ae6e39bf-52a3-4c08-a99d-61a1bfaa69dd.jpg)
Proposition 4.3 If
, we have
If
,
![](https://www.scirp.org/html/3-5300450\99f4e05b-b4af-4f2d-b9e9-fd2248f0d180.jpg)
Corollary 4.4 If
, we have
. If
, ![](https://www.scirp.org/html/3-5300450\27531af0-8e07-4077-afa0-5b4b690b4586.jpg)
In order to find Kerd3 and hence determine
we start by giving a non-zero element in
for all s.
Proposition 4.5 Define
by
![](https://www.scirp.org/html/3-5300450\35709f53-f19d-4afa-8831-7195de2ef5a7.jpg)
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 ![](https://www.scirp.org/html/3-5300450\3e541199-83c6-4ff0-a697-24e2e672744b.jpg)
Recall that
where
and
are in the ideal generated by the arrows. For
the component of
in
is
![](https://www.scirp.org/html/3-5300450\0f95033a-9512-4b37-9ded-0e16f08c3edd.jpg)
Then
![](https://www.scirp.org/html/3-5300450\9b54105f-2fd1-480f-baa8-bf231d657514.jpg)
Thus
![](https://www.scirp.org/html/3-5300450\2d6eee22-5ea8-4068-9ad0-446fbf5bdf88.jpg)
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 ![](https://www.scirp.org/html/3-5300450\37b45c7a-1be9-40cd-b6a4-4df376359df4.jpg)
where
is given as in Proposition 4.5 by
![](https://www.scirp.org/html/3-5300450\d286e10c-cb46-408d-9008-940b2b9500f0.jpg)
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
![](https://www.scirp.org/html/3-5300450\f7184b1d-8792-4b44-8237-43000284005d.jpg)
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
![](https://www.scirp.org/html/3-5300450\b561336f-7338-4b82-94ca-275cd17d904a.jpg)
For the case
, we need more details to find
. Following [5] we may choose the set
to consist of the following elements:
where
![](https://www.scirp.org/html/3-5300450\8eee7c09-6887-43da-9508-a52f81896d81.jpg)
![](https://www.scirp.org/html/3-5300450\3acedc50-ff8b-4126-a321-23e8e5e49990.jpg)
![](https://www.scirp.org/html/3-5300450\ad34b523-75f8-4618-a92f-3735e2920300.jpg)
![](https://www.scirp.org/html/3-5300450\7f3d3ac3-6d1b-4293-9215-d5de9aceeafe.jpg)
![](https://www.scirp.org/html/3-5300450\9d5b01e2-6396-40b0-b9ea-6debafb01724.jpg)
![](https://www.scirp.org/html/3-5300450\647f50a6-8011-4f6c-b4db-60a67a7e3026.jpg)
![](https://www.scirp.org/html/3-5300450\8e416fc9-f787-42f9-baec-052ad34472cf.jpg)
![](https://www.scirp.org/html/3-5300450\6c1e2034-8591-4b8c-aa20-0aab4ce32fcc.jpg)
![](https://www.scirp.org/html/3-5300450\3b445050-6968-4439-b4c0-44a71b63c7cb.jpg)
![](https://www.scirp.org/html/3-5300450\b5ec093c-e211-4de1-ae2d-4144466b97ee.jpg)
![](https://www.scirp.org/html/3-5300450\3e8fb4b1-d4c8-43df-8c89-20496a8193bc.jpg)
![](https://www.scirp.org/html/3-5300450\481e1699-dc76-4c75-8a7f-605d945ec1f2.jpg)
![](https://www.scirp.org/html/3-5300450\1b7a1c81-e0f5-4dd0-b090-b6a2df5b19a2.jpg)
![](https://www.scirp.org/html/3-5300450\57485ae4-9499-4244-b020-3ea42538845e.jpg)
Thus the projective bimodule
is ![](https://www.scirp.org/html/3-5300450\1120dfd7-0208-48ac-bb16-533768bca8de.jpg)
![](https://www.scirp.org/html/3-5300450\642d0bac-d779-420c-8747-48a2b8c77b26.jpg)
Now we determine
in the case
. Let
, so
and
. Recall that for
,
is given by
![](https://www.scirp.org/html/3-5300450\9a2808e9-b6d0-41ef-825b-fdcaa534a3ae.jpg)
where
are in
.
Then for
, we have ![](https://www.scirp.org/html/3-5300450\ec7f6759-be66-40b9-99e2-06df7271b7d1.jpg)
![](https://www.scirp.org/html/3-5300450\4b74fa38-026b-42fe-a95e-bfcf2a897fad.jpg)
In a similar way we can show that
.
For
, we have ![](https://www.scirp.org/html/3-5300450\abdca72c-5d36-45a1-adb5-a794c3c2f59f.jpg)
![](https://www.scirp.org/html/3-5300450\7d1a5324-25b3-4d58-99b7-6ab87e06593a.jpg)
As
we have
for
.
Similarly it can be shown that
![](https://www.scirp.org/html/3-5300450\a1588071-803e-4976-899b-6d6fd32fe243.jpg)
so that
.
We also have
for
and
Finally, putting
![](https://www.scirp.org/html/3-5300450\55eefead-65c8-40f6-8e2d-c62e6c9a20ee.jpg)
does not give any new information for
,
.
Thus h is given by
![](https://www.scirp.org/html/3-5300450\2f3a6120-09c1-4408-a8bc-a7f6f50d8f53.jpg)
where
for
are in K. It is clear that there is no dependency between
, and therefore
.
Proposition 4.8 For
and
, we have ![](https://www.scirp.org/html/3-5300450\6fc33675-63a2-42e0-88d3-7bbf736b3870.jpg)
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, ![](https://www.scirp.org/html/3-5300450\e0c1f49f-4c29-42f0-a04b-f6f76ac2a5e9.jpg)
![](https://www.scirp.org/html/3-5300450\1652b18a-ca64-42e3-a8e0-28ffe84510b1.jpg)
![](https://www.scirp.org/html/3-5300450\d7b701c1-43b9-4af0-93b4-2c92d4518063.jpg)
![](https://www.scirp.org/html/3-5300450\e1a79445-778d-4d1b-bd04-5574e2a3f6c6.jpg)
and
, we have ![](https://www.scirp.org/html/3-5300450\26c5cdd9-28c8-4412-88b6-480870b673fc.jpg)
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 ![](https://www.scirp.org/html/3-5300450\7230d863-0621-42d9-907e-81b43d957a79.jpg)
2) for all
,
where
![](https://www.scirp.org/html/3-5300450\960b42ec-d624-422a-8ca9-fe5d740f80bb.jpg)
3)
for all arrows a with
4)
for all arrows a with ![](https://www.scirp.org/html/3-5300450\14231931-68a5-4547-9d3a-b17ed1fbbd30.jpg)
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 ei with
. Now we consider
and
with
, and
with
. These projective modules are described as follows:
![](https://www.scirp.org/html/3-5300450\88f3ff63-ab3a-4b94-a9b8-fe3ab759695b.jpg)
In each case we see that
![](https://www.scirp.org/html/3-5300450\5d537fba-88ba-4f31-9bbc-9e41a583a5bc.jpg)
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 ![](https://www.scirp.org/html/3-5300450\61ebb80c-757a-4c20-860b-f35bd9f17328.jpg)
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
![](https://www.scirp.org/html/3-5300450\c4156179-2fbb-42ac-80e9-ea999cfa0b24.jpg)
![](https://www.scirp.org/html/3-5300450\f63262d4-1cb8-4b64-ad57-447d1f8bb00e.jpg)
![](https://www.scirp.org/html/3-5300450\7eab2d6e-b2cb-457e-a442-f81973ac2924.jpg)
![](https://www.scirp.org/html/3-5300450\4fad5e65-2844-4f18-bc82-1760b48af057.jpg)
![](https://www.scirp.org/html/3-5300450\15ab877e-75f0-4829-ba85-81c5728a0bbb.jpg)
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
![](https://www.scirp.org/html/3-5300450\ece80e09-058f-43ab-825f-e55fb783ddb1.jpg)
Recall that the projective
. Thus we have
![](https://www.scirp.org/html/3-5300450\52cf5154-91b5-4272-8eb3-135c3bd018fa.jpg)
(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:
![](https://www.scirp.org/html/3-5300450\c5c1f0e8-47fb-4f82-b3a2-80c87aa51e3d.jpg)
with
where
![](https://www.scirp.org/html/3-5300450\a127a1dd-987b-4530-8e9c-84f32bb1bf78.jpg)
![](https://www.scirp.org/html/3-5300450\ede35d2d-4d26-40eb-a759-f1d24f7b89ec.jpg)
![](https://www.scirp.org/html/3-5300450\2b79f2c4-3219-4e80-9930-8fe593eed098.jpg)
![](https://www.scirp.org/html/3-5300450\898c6511-866e-4832-a8b5-7f27ab344d8c.jpg)
![](https://www.scirp.org/html/3-5300450\ff0f1d52-d51c-42fe-a0a6-b54ec319f632.jpg)
![](https://www.scirp.org/html/3-5300450\98605815-e03b-4a3e-8c61-e3dbb102d2f1.jpg)
![](https://www.scirp.org/html/3-5300450\e55dcfed-f781-4a05-a3bb-a4b390a4980f.jpg)
![](https://www.scirp.org/html/3-5300450\b2194603-ff7e-4500-959f-d2488d5715bf.jpg)
We know that
. First we will find
. Let
and so write
![](https://www.scirp.org/html/3-5300450\d7e3c6f6-93e2-4b14-a727-a6a355afcd50.jpg)
![](https://www.scirp.org/html/3-5300450\3baf10e4-e04c-454d-8a2a-11839fe321cd.jpg)
![](https://www.scirp.org/html/3-5300450\63467a6d-d59f-4f0c-89f9-a6549b847448.jpg)
where ![](https://www.scirp.org/html/3-5300450\cb1924d6-9cc2-4a91-941b-f6b3674be671.jpg)
Now we find
. We have
![](https://www.scirp.org/html/3-5300450\3661bfec-3f44-4033-8207-28e759f01f64.jpg)
Also
![](https://www.scirp.org/html/3-5300450\3b1b1824-e0d3-4475-8ace-c02fcb6b3ada.jpg)
![](https://www.scirp.org/html/3-5300450\7e27ed58-9b3d-4be1-8117-04052ebc3180.jpg)
We can show by direct calculation that
for all
.
Thus
is given by
![](https://www.scirp.org/html/3-5300450\3d92132d-8a0b-4cbd-9cca-0048013193d0.jpg)
![](https://www.scirp.org/html/3-5300450\c2ada732-346b-4ccd-8271-e2672a430554.jpg)
So
.
Proposition 5.2 For
, we have ![](https://www.scirp.org/html/3-5300450\982c9a2a-89b0-48c6-a25f-236430e79640.jpg)
Now we determine
. Let
, so
and
. Then
is given by
![](https://www.scirp.org/html/3-5300450\6bd1ee90-fedb-43c8-9c5a-7edce74f32fe.jpg)
![](https://www.scirp.org/html/3-5300450\64391b8a-6027-4f43-a504-ada87eec7297.jpg)
![](https://www.scirp.org/html/3-5300450\55884337-f33b-4518-aad6-aa0659bee113.jpg)
![](https://www.scirp.org/html/3-5300450\f335d1ca-2f8e-4009-9fb3-1d105533ce94.jpg)
![](https://www.scirp.org/html/3-5300450\8dce43ec-2519-41b2-aaaf-6681df56ca2b.jpg)
![](https://www.scirp.org/html/3-5300450\461aa025-a7ae-497a-90e5-9ddd1c7ab952.jpg)
for some
for ![](https://www.scirp.org/html/3-5300450\8f70b7a0-2461-4af5-bdd2-96482a280c24.jpg)
Then
![](https://www.scirp.org/html/3-5300450\744c0972-b6ce-48fe-b7fc-8364e1dfaf31.jpg)
As
we have
and ![](https://www.scirp.org/html/3-5300450\0d36dd51-89fa-44e3-a611-9d35dbd5f48e.jpg)
![](https://www.scirp.org/html/3-5300450\56579857-35a1-4ff8-8cdb-1472c9975558.jpg)
As
we have
and
. So
and ![](https://www.scirp.org/html/3-5300450\88de830d-ef6d-4040-a906-d6fd35d99716.jpg)
Next,
![](https://www.scirp.org/html/3-5300450\d27d3210-f585-485c-8e03-666b1baaf569.jpg)
So we have
and hence ![](https://www.scirp.org/html/3-5300450\4e2b1205-3b61-4b35-9e37-bbe7188c7d03.jpg)
![](https://www.scirp.org/html/3-5300450\740a0762-9ac0-4c0d-92e1-16e75e591f21.jpg)
Therefore
as ![](https://www.scirp.org/html/3-5300450\657ea8f6-6c4b-4f7c-8cef-928b66612f61.jpg)
![](https://www.scirp.org/html/3-5300450\13c0effe-8e77-4b12-83e8-ec5489438fa9.jpg)
Thus again we have ![](https://www.scirp.org/html/3-5300450\eaadbfd8-cb8c-46d8-af4e-1620435afdc3.jpg)
![](https://www.scirp.org/html/3-5300450\b2e27b2a-fca2-4ef8-984c-e66f6759876d.jpg)
As
above, we have
as we already know.
Also
![](https://www.scirp.org/html/3-5300450\52bc8593-6287-45d2-a643-cddd68d991bc.jpg)
So we have
and ![](https://www.scirp.org/html/3-5300450\61e5a86b-9400-4328-9b4a-e7feb4cf7488.jpg)
Finally, for
, we have
![](https://www.scirp.org/html/3-5300450\f2d679bf-54e8-446c-89bf-9d49eab158b5.jpg)
Therefore we have
and
. Hence
and
for
as we have above
and ![](https://www.scirp.org/html/3-5300450\631c42f1-8bd8-489f-8304-fdb2fe146e3c.jpg)
Thus
is given by
![](https://www.scirp.org/html/3-5300450\46ffc6cd-15aa-438f-9268-07bbac7a437d.jpg)
![](https://www.scirp.org/html/3-5300450\3aee9a62-6d14-44e4-9d43-6c2b5fa98f37.jpg)
![](https://www.scirp.org/html/3-5300450\53c886e3-a8e2-4425-8999-e7ac72601f89.jpg)
![](https://www.scirp.org/html/3-5300450\4ff6789f-253d-4265-8c68-e63551c8141f.jpg)
![](https://www.scirp.org/html/3-5300450\ea84ce2c-afee-4f16-93d8-90fc5620ee20.jpg)
![](https://www.scirp.org/html/3-5300450\ecec913b-3365-4a45-9014-3a9884dbce0d.jpg)
for some ![](https://www.scirp.org/html/3-5300450\72446231-6514-4aff-9624-4ac16123f9de.jpg)
Proposition 5.3 For
, we have ![](https://www.scirp.org/html/3-5300450\16b36bfe-cd80-4066-8c5c-a1ab6c5fb325.jpg)
Therefore
![](https://www.scirp.org/html/3-5300450\562b38b2-c8dc-4262-b30f-e9823379675a.jpg)
and a basis is given by the maps
and
where
is given by
![](https://www.scirp.org/html/3-5300450\74b41b47-0108-417b-87b4-3dc44115665d.jpg)
is given by
![](https://www.scirp.org/html/3-5300450\30c02902-8dfa-45bc-aff0-1726113e5702.jpg)
From Proposition 5.2 and Proposition 5.3 we get the main result of this section.
Theorem 5.4 For
with
we have ![](https://www.scirp.org/html/3-5300450\fd91cabf-cc8f-4d32-96d7-3d36316060b0.jpg)
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) ![](https://www.scirp.org/html/3-5300450\6e1efefb-79e8-4641-8b0d-7389dba67c7d.jpg)
2) ![](https://www.scirp.org/html/3-5300450\f3b4f04f-fb5e-421f-a11c-2ed2766d1e4a.jpg)
3) ![](https://www.scirp.org/html/3-5300450\cf3734c4-0666-44c8-9002-cb26cc228826.jpg)
4) ![](https://www.scirp.org/html/3-5300450\2d87e6b8-beb1-4104-8b55-0e9c647f9a86.jpg)
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.