On Isoclasses of Maximal Subalgebras Determined by Automorphisms

2018/10/01 by Sistko, Alexander H.
#05C60 #05E15 #16S99 #16W20 #16Z99 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1810.00806

Abstract

Let k be an algebraically-closed field, and let B = kQ/I be a basic, finite-dimensional associative k-algebra with n := dim_kB < ∞. Previous work shows that the collection of maximal subalgebras of B carries the structure of a projective variety, denoted by msa (Q), which only depends on the underlying quiver Q of B. The automorphism group Aut_k(B) acts regularly on msa (Q). Since msa (Q) does not depend on the admissible ideal I, it is not necessarily easy to tell when two points of msa (Q) actually correspond to isomorphic subalgebras of B. One way to gain insight into this problem is to study Aut_k(B)-orbits of msa (Q), and attempt to understand how isoclasses of maximal subalgebras decompose as unions of Aut_k(B)-orbits. This paper investigates the problem for B = kQ, where Q is a type \mathbbA Dynkin quiver. We show that for such B, two maximal subalgebras with connected Ext quivers are isomorphic if and only if they lie in the same Aut_k(B)-orbit of msa (Q).

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