Automorphism Groups of Finite-Dimensional Algebras Acting on Subalgebra Varieties

2018/09/25 by Alexander Sistko, Sistko, Alexander H.
Mathematics · #14L30 #14N99 #16S99 #16U60 #16W20 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1809.09760

openalex publication_date 2018/09/25 · openalex created_date 2018/10/05 · openalex updated_date 2026/07/28

Abstract

Let k be an algebraically-closed field, and B a unital, associative k-algebra with n := dim_kB < ∞. For each 1 ≤ m ≤ n, the collection of all m-dimensional subalgebras of B carries the structure of a projective variety, which we call AlgGr_m(B). The group Aut_k(B) of all k-algebra automorphisms of B acts regularly on AlgGr_m(B). In this paper, we study the problem of explicitly describing AlgGr_m(B), and classifying its Aut_k(B)-orbits. Inspired by recent results on maximal subalgebras of finite-dimensional algebras, we compute the homogeneous vanishing ideal of AlgGr_n-1(B) when B is basic, and explictly describe its irreducible components. We show that in this case, AlgGr_n-1(B) is a finite union of Aut_k(B)-orbits if B is monomial or its Ext quiver is Schur, but construct a class of examples to show that these conditions are not necessary.

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