The Loewy structure of certain fixpoint algebras, Part I

Abstract

In Part I of this paper, we introduced a class of certain algebras of finite dimension over a field. All these algebras are split, symmetric and local. Here we continue to investigate their Loewy structure. We show that in many cases their Loewy length is equal to an upper bound established in Part I, but we also construct examples where we have a strict inequality.

The Loewy Structure of Certain Fixpoint Algebras, Part Ii

“…Let q, e ∈ N such that gcd(q, e) = 1. In [3,Section 6], we defined m(q, e) as the smallest positive integer t with the property that there exists a sum of t powers of q which is divisible by e. Then 1 ≤ m(q, e) ≤ e; moreover, m(q, e) = e if and only if q ≡ 1 (mod e), and m(q, e) = 1 if and only if e = 1 (cf. [3,Example 6.1]).…”

“…+ x n q n . In [3,Proposition 6.1] we proved that m(q, e) = min{s q (ke) : k ∈ N} = min{s q (ke) : k = 1, . .…”

“…Then m := m(q, e) ≥ e 3 if and only if one of the following holds: Proof. For the "if" direction, observe that the claimed values of m for given q and e are equal to both e 1 = gcd(e, q − 1) and s q (e) in the cases (i)-(iii), so that we can apply [3, Lemma 6.2 (i)] and [3,Proposition 6.1]. In case (iv), the values for m are given in Table 1.…”

“…In [3,Corollary 5.1] we proved that A(q, n, e) is uniserial if and only if e is divisible by q n −1 q−1 . For an arbitrary finite-dimensional algebra A, the Loewy length LL(A) measures how far A is away from being semisimple.…”

“…The main purpose of this paper is to show that, in many cases, the inequality in (1) is in fact an equality. For example, we will show that LL(A(q, n, e)) = n q − 1 m + 1 (2) whenever one of the following conditions is satisfied: • m(q, e) = 2, see [3,Lemma 6.3].…”