On the Lie algebra structure of 𝐻𝐻¹(𝐴) of a finite-dimensional algebra 𝐴

Abstract

Let A be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of A is a simple directed graph, then HH 1 (A) is a solvable Lie algebra. The second main result shows that if the Ext-quiver of A has no loops and at most two parallel arrows in any direction, and if HH 1 (A) is a simple Lie algebra, then char(k) = 2 and HH 1 (A) ∼ = sl 2 (k). The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.

The Lie algebra structure of the $HH^1$ of the blocks of the sporadic Mathieu groups

“…Let p = 3. The isomorphism of 3-dimensional simple Lie algebras HH 1 (B 1 ) ∼ = HH 1 (kC 3 ) agrees with results of Waki[37, Theorem 2.4] and Linckelmann and Rubio y Degrassi[30, Theorem 1.3]: the former shows that B 1 contains a unique isomorphism class of simple kGmodules with representative S, of dimension 231, whose Loewy structure gives dim k (Ext 1 kG (S, S)) = 1. The latter then tells us that if HH 1 (B 1 ) is simple, then it must be nilpotent.Theorem 2.10.…”

“…a block with a non-trivial cyclic defect group, giving an explicit formula for its dimension (though this is already folklore) and a characterisation of a basis for HH 1 (B) as a Lie algebra. This is a solvable Lie algebra in general (see [30,Example 5.7]), and we note that for some blocks B with a non-trivial cyclic defect group, we see that HH 1 (B) is 1-dimensional. In this case the we are also able to determine that there is p-toral basis of HH 1 (B).…”

“…Using Erdmann's classification of tame blocks [9, pp.296] one verifies that the decomposition and Cartan matrices of B 1 match those of D(3K). We note that by [30,Proposition 5.3] this implies that HH 1 (B 1 ) is a solvable Lie algebra. The isomorphism and dimension of HH 1 (B 1 ) is then given by Holm [16, 2.1(2)], and by Table 1 we recover dim k (HH 1 (B 0 )) = 13.…”

“…In the exceptional case, G = M 12 has a non-principal 2-block with Klein four defect group and so to calculate the dimensions and show solvability of the HH 1 of this block we use results of Holm [16], Erdmann [9] and Linckelmann and Rubio y Degrassi [30].…”

“…Theorem 2.1 ( [30,34]). Let A be a finite dimensional k-algebra such that dim k (Ext 1 A (S, S)) = 0 and dim k (Ext 1 A (S, T )) ≤ 1, for all non-isomorphic simple A-modules S and T .…”