Polynomial invariants for low dimensional algebras

2025/03/07 by Alvarez, María Alejandra, Lopatin, Artem · 1 citation
#13A50 #15A72 #1630 #17A30 #17A36 #20F29 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2503.05337

Abstract

We classify all two-dimensional simple algebras (which may be non-associative) over an algebraically closed field. For each two-dimensional algebra A, we describe a minimal (with respect to inclusion) generating set for the algebra of invariants of the m-tuples of A in the case of characteristic zero. In particular, we establish that for any two-dimensional simple algebra A with a non-trivial automorphism group, the Artin--Procesi--Iltyakov Equality holds for A^m; that is, the algebra of polynomial invariants of m-tuples of A is generated by operator traces. As a consequence, we describe two-dimensional algebras that admit a symmetric or skew-symmetric invariant nondegenerate bilinear form.

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