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The locally complexified-gentle algebras

Jie Li, Chao Zhang

TL;DR

This work studies $\,\mathbb{R}$-algebras that become gentle over $\mathbb{C}$ after complexification, introducing locally complexified-gentle algebras and connecting them to modulated quivers and semilinear clannish algebras. It distinguishes two principal constructions—uniform type (where every vertex is ordinarily gentle with $\mathbb{R}$ or $\mathbb{H}$) and special type (more intricate configurations)—and establishes Morita-equivalence descriptions via complexified presentations $\mathbb{C}\Gamma/J$ and to $\mathbb{C}$-semilinear clannish algebras. Uniform-type algebras recover real or quaternionic gentle algebras and hence are semilinear clannish, while special-type algebras are Morita-equivalent to gentle-type semilinear clannish algebras, providing a complete Morita classification in this setting. Consequently, finite-dimensional representations admit a string-and-band description in the uniform case and reduce to gentle-type classifications in the special case, thereby unifying the real/gentle theory with semilinear clannish algebras and enabling systematic representation-theoretic analysis.

Abstract

We call an $\mathbb{R}$-algebra locally complexified-gentle if it becomes a locally gentle $\mathbb{C}$-algebra up to Morita equivalence after complexification. We use modulated quivers to introduce two types of locally complexified-gentle algebras and show that they are Morita equivalent to some semilinear clannish algebras.

The locally complexified-gentle algebras

TL;DR

This work studies -algebras that become gentle over after complexification, introducing locally complexified-gentle algebras and connecting them to modulated quivers and semilinear clannish algebras. It distinguishes two principal constructions—uniform type (where every vertex is ordinarily gentle with or ) and special type (more intricate configurations)—and establishes Morita-equivalence descriptions via complexified presentations and to -semilinear clannish algebras. Uniform-type algebras recover real or quaternionic gentle algebras and hence are semilinear clannish, while special-type algebras are Morita-equivalent to gentle-type semilinear clannish algebras, providing a complete Morita classification in this setting. Consequently, finite-dimensional representations admit a string-and-band description in the uniform case and reduce to gentle-type classifications in the special case, thereby unifying the real/gentle theory with semilinear clannish algebras and enabling systematic representation-theoretic analysis.

Abstract

We call an -algebra locally complexified-gentle if it becomes a locally gentle -algebra up to Morita equivalence after complexification. We use modulated quivers to introduce two types of locally complexified-gentle algebras and show that they are Morita equivalent to some semilinear clannish algebras.

Paper Structure

This paper contains 16 sections, 15 theorems, 98 equations, 3 tables.

Table of Contents

  1. Introduction
  2. The -algebras and their complexifications
  3. Tensor algebras of modulated quivers over
  4. Complexified quiver presentations
  5. The isomorphism
  6. The locally complexified-gentle algebras
  7. Locally complexified-gentle algebras
  8. Gentle vertices in
  9. Locally complexified-gentle algebras of uniform type and of special type
  10. Examples
  11. The -semilinear clannish algebras of gentle type
  12. The algebra
  13. The basic algebra of
  14. The -semilinear clannish algebras of gentle type
  15. The -semilinear clannish algebras of gentle type and the locally complexified-gentle algebras
  16. ...and 1 more sections

Key Result

Proposition 2.3

LJ2023 Let $\Gamma$ be the complexified quiver of a modulated quiver $(Q,\mathcal{M})$. Then there is an isomorphism of $\mathbb{C}$-algebras where $\mathbf{e}$ is a full idempotent of $T(Q, \mathcal{M})\otimes_{\mathbb{R}}\mathbb{C}$. Moreover, for each path $p$ in $Q$, where $q_1,\dots,q_n$ are all the paths in $\Gamma$ that are fibers of $p$.

Theorems & Definitions (38)

  • Proposition 2.3
  • Theorem 2.4
  • Example 2.5
  • Lemma 2.6
  • proof
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4
  • ...and 28 more