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The derived-discrete algebras over the real numbers

Jie Li

TL;DR

This work extends the classification of derived-discrete algebras to the real setting by leveraging complexification. By explicitly constructing the complexified quiver $\Gamma$ and its relations $J$ from a modulated real quiver $(Q,\mathcal{M})$, the authors reduce the problem to the known complex classification: a real algebra is derived-discrete iff it is piecewise hereditary of Dynkin type or Morita equivalent to a gentle one-cycle without clock condition. The main contributions are the detailed complexification framework, the explicit quiver presentations $\mathbb{C}\Gamma/J$, and the two-direction proof (necessary and sufficient conditions) yielding a complete real-case classification. Practically, this provides concrete criteria to decide derived-discreteness for real algebras via their complexifications and modulated-quiver data. The results unify real and complex theories and supply a blueprint for analyzing derived-discrete structures over non-algebraically closed fields.

Abstract

We classify derived-discrete algebras over the real numbers up to Morita equivalence, using the classification of complex derived-discrete algebras in [{\sc D. Vossieck}, {\em The algebras with discrete derived category}, J. Algebra {\bf 243} (2001), 168--176]. To this end, we investigate the quiver presentation of the complexified algebra of a real algebra given by a modulated quiver and an admissible ideal.

The derived-discrete algebras over the real numbers

TL;DR

This work extends the classification of derived-discrete algebras to the real setting by leveraging complexification. By explicitly constructing the complexified quiver and its relations from a modulated real quiver , the authors reduce the problem to the known complex classification: a real algebra is derived-discrete iff it is piecewise hereditary of Dynkin type or Morita equivalent to a gentle one-cycle without clock condition. The main contributions are the detailed complexification framework, the explicit quiver presentations , and the two-direction proof (necessary and sufficient conditions) yielding a complete real-case classification. Practically, this provides concrete criteria to decide derived-discreteness for real algebras via their complexifications and modulated-quiver data. The results unify real and complex theories and supply a blueprint for analyzing derived-discrete structures over non-algebraically closed fields.

Abstract

We classify derived-discrete algebras over the real numbers up to Morita equivalence, using the classification of complex derived-discrete algebras in [{\sc D. Vossieck}, {\em The algebras with discrete derived category}, J. Algebra {\bf 243} (2001), 168--176]. To this end, we investigate the quiver presentation of the complexified algebra of a real algebra given by a modulated quiver and an admissible ideal.

Paper Structure

This paper contains 26 sections, 15 theorems, 88 equations.

Table of Contents

  1. introduction
  2. Real algebras and complexifications
  3. The division ring $\mathbb{R}$
  4. The division ring $\mathbb{H}$
  5. The division ring $\mathbb{C}$
  6. Simple $\mathbb{R}$-$\mathbb{R}$-bimodules
  7. Simple $\mathbb{R}$-$\mathbb{C}$-bimodules
  8. Simple $\mathbb{C}$-$\mathbb{R}$-bimodules
  9. Simple $\mathbb{R}$-$\mathbb{H}$-bimodules
  10. Simple $\mathbb{H}$-$\mathbb{R}$-bimodules
  11. Simple $\mathbb{H}$-$\mathbb{H}$-bimodules
  12. Simple $\mathbb{H}$-$\mathbb{C}$-bimodules
  13. Simple $\mathbb{C}$-$\mathbb{H}$-bimodules
  14. Simple $\mathbb{C}$-$\mathbb{C}$-bimodules
  15. The complexified quiver presentation
  16. ...and 11 more sections

Key Result

Lemma 3.2

We have the following statements for a modulated quiver $(Q,\mathcal{M})$. 1) If $\mathcal{M}$ is v-uniform with $\mathbb{R}$, then $T(Q,\mathcal{M})\simeq \mathbb{R} Q$. 2) If $\mathcal{M}$ is v-uniform with $\mathbb{H}$, then $T(Q,\mathcal{M})\simeq \mathbb{H} Q$. 3) If $\mathcal{M}$ is v-uniform

Theorems & Definitions (34)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Example 3.4
  • Proposition 3.5
  • proof
  • Theorem 3.7
  • Example 3.9
  • Lemma 3.10
  • ...and 24 more