ON THE RADICAL OF THE CENTER OF SMALL SYMMETRIC LOCAL ALGEBRAS

This article is motivated by some results from Chlebowitz and Külshammer on how the structure of a symmetric local algebra is influenced by its center. They have shown that a symmetric local algebra is almost always commutative if its center is at most 5-dimensional. In this article we are interested in how the ideal property of the radical of the center of a symmetric local algebra is influenced by the dimension of the algebra itself. Mathematics Subject Classification (2020): 16N40, 20C20


Introduction
When studying the modular representation theory of a block B of a group algebra If one tries to study the representation theory of a block B it is sometimes useful to replace B by its basic algebra A. If l(B) = 1 then A is a symmetric local algebra. Hence, studying properties of symmetric local algebras adds to the knowledge about blocks of group algebras. Some more recent examples of this principle can be found in the literature, e. g. in [4], the Morita equivalence class of a whole family of 3-blocks with elementary abelian defect groups of order 9 was determined by investigating certain 9-dimensional symmetric local algebras. In [7] the authors were able to determine the isomorphism type of the center of a class of 2-blocks with elementary abelian defect groups of order 16. A big part of the argument was, again, based on investigating symmetric local algebras of a certain type.
In 1984, Külshammer proved that symmetric local algebras with a center being of dimension at most 4 have to be commutative (see [5]). Some years later, Chlebowitz and Külshammer could show that symmetric local algebras with 5-dimensional center have to be either of dimension 5, and hence commutative, or of dimension 8 (see [1]). However, in order to obtain this result, a fair amount of computation was needed. These two articles suggest that there should be some more connections between a symmetric local algebra and its center, at least in low dimensions.
In this article we are interested in the question whether the radical of the center of a symmetric local algebra is an ideal of the whole algebra. Readers being familiar with the article [4] by Kessar will remember that showing this fact for a class of symmetric local algebras of dimension 9 was one of the crucial steps to get the computations going. In the present article we will, in fact, show that the radical of the center of a symmetric local algebra is an ideal of the whole algebra if the dimension of the algebra is at most 10. Hence the corresponding fact in the article [4] would be an easy corollary from our main result.
The present article is organized as follows: In Section 2 we will gather some well known properties of symmetric local algebras. In the third section we will first prove a lemma which characterizes when the radical of the center of a symmetric local algebra is an ideal of the whole algebra. Moreover, this section will contain the main result of the present paper. In the final section we will apply a modification of the main result in characteristic 2 to the setting in [7]. This will allow us to considerably shorten the computations and give a more elegant proof of the main result of that article.

Preliminaries
Throughout this section F will be an algebraically closed field (of arbitrary characteristic) and A will be a finite dimensional F -algebra. We will denote the For elements x, y ∈ A we will denote their commutator by [x, y] := xy − yx. For By "ideal" we will always mean "two-sided ideal".

ON THE RADICAL OF THE CENTER OF SMALL SYMMETRIC LOCAL ALGEBRAS 177
If A is symmetric, we fix a symmetrizing linear form s : A → F on A. For an The following lemma gathers some well-known facts about symmetric local algebras. Lemma 2.1. Let A be a symmetric local F -algebra. Then the following facts hold.   Proof. The first two items are well-known properties of non-degenerate symmetric bilinear forms. Note that λ : The definition of soc(A) implies soc(A) ⊆ J ⊥ . Moreover, from the first part of (i) we deduce dim F J ⊥ = 1. Since soc(A) = 0, one immediately obtains the first part of (iii), as well as (v). The second part of (iii) is proved in [5,Lemma C]. The proofs for (iv), (vi), and (viii) of can be found in [4]. Thus, only (vii) needs to be proved.
Let I be an ideal of A, and let x ∈ I ⊥ and a ∈ A be arbitrary. Then Thus, we have proved that I ⊥ is an ideal of A. We deduce further From this, we conclude I · I ⊥ ⊆ A ⊥ = 0. Similarly, we obtain I ⊥ · I = 0. This finishes the proof.
We need another lemma which gives a characterization of when a factor algebra of a symmetric algebra is again symmetric.

Lemma 2.2. Let A be a symmetric F -algebra and I be an ideal of A. Then the
From Lemma 2.2 we immediately get the following corollary.

Corollary 2.3. Let A be a symmetric local F -algebra and 0 = z ∈ Z(A). Then
A/(Az) ⊥ also is a symmetric local F -algebra.

Proof. By Lemma 2.2 we just need to show that
Next we will recap a result which gives a nice way of computing a generating set of a Loewy-layer of a given algebra if a generating set of the previous Loewy-layer is known.

Lemma 2.4. Let
A be an F -algebra, let I be an ideal of A and let n ∈ N. Suppose that there is some d ∈ N such that with elements x ik ∈ I. Then we have Proof. This is a slightly simplified version of [5, Lemma E].
As a last fact in this section we recap one of the results mentioned in the introduction of this article.

The main result
In this section F will again be an algebraically closed field and A will be a finite dimensional F -algebra. First we will point out a characterization of when the radical (resp. the socle) of the center of a symmetric local algebra A is an ideal of A. Although not hard to prove, this result seems to be new so we will give a proof here.
Switching to orthogonal spaces and using (ii), (iii), (iv), (vi) and (vii) of Lemma 2.1, we obtain that were arbitrary too, this implies J(Z) · A = A · J(Z) ⊆ J(Z). This shows the claim.
Since K(A) ⊆ (Az) ⊥ , we have K(A ) = 0. In particular, A is not commutative.
By Lemma 2.5 we immediately get We will now show that dim and hence A must be commutative, a contradiction.
Next, assume that dim F A = dim F Z(A ) + 2. Similarly to the previous case, But then, using (i) and (iii) of Lemma 2.1, we obtain a contradiction. This shows that dim F A ≥ dim F Z(A ) + 3 holds. Combining this result with (1) and (2), we obtain dim F (Az) ≥ 5 + 3 = 8. Since A is a local F -algebra, we have A = F 1 ⊕ J. Therefore, using z ∈ J(Z) we obtain . We distinguish these two cases.

ON THE RADICAL OF THE CENTER OF SMALL SYMMETRIC LOCAL ALGEBRAS 181
First, let dim F A = 9. Then we must have A = F 1⊕Az ⊆ F 1+F z+J 2 and J/J 2 is spanned by z + J 2 . Using Lemma 2.4, this implies that A is generated, as an F -algebra, by {1, z}. Since 1, z ∈ Z(A), this means that A must be commutative.
Hence A = Z(A) and J(Z) = J is an ideal of A, a contradiction to our assumption.
Finally, let dim F A = 10. Similarly to the previous case, we deduce that A = F 1 + F x + F z + J 2 for some x ∈ J. Thus, J/J 2 is spanned by a subset of x + J 2 , z + J 2 and A is generated, as an F -algebra, by a subset of {1, x, z}.
Since 1, z ∈ Z(A), all the generators of A commute and thus A is commutative. We get a contradiction with the same argument as before. This finishes the proof.

Remark 3.3.
Comparing the statement of Theorem 3.2 with that of Lemma 2.5 or that of the main theorem of [1], the reader will notice some analogy. While in Lemma 2.5 or [1] a bound on the dimension of the center of an algebra is used to obtain structural results on the whole algebra, Theorem 3.2 uses a bound on the dimension of the algebra itself to specify the relationship between an algebra and its center.

An application
In this section we are revisiting the article [7], especially its fourth section. The In fact, Charles Eaton has classified the 2-blocks with elementary abelian defect group of order 16, up to Morita equivalence (see [3]). The resulting list contains just one non-nilpotent block with exactly one isomorphism type of simple modules. His approach, however, relies on the classification of finite simple groups. The approach taken in the article [7] and the thesis [6] can be seen as a classification-free attempt to determine the structure of the unique non-nilpotent 2-block with elementary abelian defect group of order 16 and one isomorphism type of simple modules.
We will, in this section, prove a stronger result of our From now on, F will denote an algebraically closed field of characteristic 2 and B will be a non-nilpotent block with elementary abelian defect group of order 16 and one isomorphism type of simple modules. Moreover, A will denote a basic algebra of B. We will, again, use J := J(A) and J(Z) := J (Z(A)). First we obtain Lemma 4.1. Let F , B and A be as above. Then (iv) For every w ∈ J we have w 2 ∈ K(A).
As in the article [7], we will now assume that Z(A) = F {W 0 := 1, W 1 , . . . , W 7 } where the structure constants are given by the following We immediately obtain J(Z) = F {W 1 , . . . , W 7 }. this, in turn, implies the iden- In particular, we have dim F J(Z) = 7 and dim F J(Z) 2 = 3. In addition, from W 2 i = 0 for i = 1, . . . , 7 and char(F ) = 2 we conclude z 2 = 0 for any z ∈ J(Z). The aim of this section is to prove that the assumption of this isomorphism type for Z(A) leads to a contradiction. In order to do that, we first prove an analogue to Theorem 3.2 for the case we are considering in this section. In particular, we always have dim F (zJ(Z)) = 2. But this contradicts our computation above where we have shown that dim F (zJ(Z)) = 2 must hold. This shows that our assumption from the beginning must be wrong and finishes the proof.
Next, we will prove a lemma which will enable us to show that the ideal J(Z) is annihilated by J 3 . This will be the key ingredient for showing that the assumed isomorphism type of Z(A) leads to a contradiction. In the following, we will make extensive use of Lemma 2.4 for computing generating sets of Loewy-layers without explicitly quoting this lemma every single time. which, by Nakayama's lemma, implies (J ) 3 = 0, a contradiction.
Hence, we may assume dim F ((J ) 2 /(J ) 3 ) ≥ 2. Using Lemma 2.4, we conclude that also dim F (J /(J ) 2 ) ≥ 2 must hold. Assuming dim F (J /(J ) 2 ) = 2, we could write J = F x + F y + (J ) 2 with certain x , y ∈ J . However, using the notes from the start of the proof, this would imply (J ) 2 = F x y + (J ) 3 , contradict- A is symmetric and local, we conclude from Lemma 2.2 and z = 0 that A is also symmetric and local. Using dim F J(Z) 2 = 3. This shows that our assumption is wrong and we indeed have We immediately obtain the following.  Proof. This is [7, Lemma 4.2 (iv)]. Now we have everything we need to prove the desired theorem. First, let us assume dim F J(Z) + J 2 /J 2 = 0. Then J(Z) ⊆ J 2 . However, this implies J(Z) 2 ⊆ J 2 · J(Z) ⊆ soc(A) contradicting dim F J(Z) 2 = 3.
Finally, let dim F J(Z) + J 2 /J 2 = 2. Then we find z 1 , z 2 ∈ J(Z) with J(Z) = F z 1 +F z 2 +(J(Z)∩J 2 ). Keeping in mind that z 2 1 = z 2 2 = 0 and z 1 z 2 = z 2 z 1 , we obtain J(Z) 2 ⊆ F z 1 z 2 + (J 2 · J(Z)) ⊆ F z 1 z 2 + soc(A). Since the right hand side is at most 2-dimensional over F , this is again a contradiction. Thus our assumption from the beginning of the proof cannot hold and we have shown the claim.