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Primitive elements in Ringel-Hall algebras of tame hereditary algebras

Bangming Deng, Weihao Li

Abstract

We study primitive elements in the Ringel-Hall algebra H(A) of an algebra A over a finite field associated with a quiver with automorphism. When A is a tame hereditary algebra, we give a description of primitive elements in H(A) which generalizes and improves a result of Hennecart (IMRN 2021) for tame quivers. Moreover, we obtain an identity concerning primitive elements in the subalgebra of H(A) generated by regular A-modules which enables us to construct an explicit basis for the space of primitive elements in H(A).

Primitive elements in Ringel-Hall algebras of tame hereditary algebras

Abstract

We study primitive elements in the Ringel-Hall algebra H(A) of an algebra A over a finite field associated with a quiver with automorphism. When A is a tame hereditary algebra, we give a description of primitive elements in H(A) which generalizes and improves a result of Hennecart (IMRN 2021) for tame quivers. Moreover, we obtain an identity concerning primitive elements in the subalgebra of H(A) generated by regular A-modules which enables us to construct an explicit basis for the space of primitive elements in H(A).
Paper Structure (7 sections, 18 theorems, 191 equations)

This paper contains 7 sections, 18 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Ringel--Hall algebras and their primitive elements
  3. Primitive elements in subalgebras of ${\mathcal{H}}_v(Q,\sigma)$
  4. Primitive elements in the cyclic quiver case
  5. Regular primitive elements for tame hereditary algebras
  6. The proof of Theorem
  7. The proof of Lemma

Key Result

Theorem 1.1

Let $Q$ be an acyclic tame quiver with automorphism $\sigma$. Then for each $n\geqslant 1$,

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • proof
  • ...and 17 more