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Burt-Butler algebras of the bocs associated to a finite partially ordered set

R. Bautista, J. A. Jimenez Gonzalez

Abstract

Given an algebra $A$ and an $A-A$-bimodule $U$ with co-algebra structure, a bocs, the algebras of endomorphisms of $A$ as left or right module of the bocs are known as Burt-Butler algebras (up to an appropriate opposite). Here we give a description of these algebras for the bocs associated to a finite partially ordered set in terms of incidence algebras and their balanced versions. We also exhibit their quasi-hereditary structure, provide bound quiver presentations for their Ringel duals, describe the embedding of $A$ as exact Borel subalgebra and characterize the corresponding subcategories of induced and co-induced modules.

Burt-Butler algebras of the bocs associated to a finite partially ordered set

Abstract

Given an algebra and an -bimodule with co-algebra structure, a bocs, the algebras of endomorphisms of as left or right module of the bocs are known as Burt-Butler algebras (up to an appropriate opposite). Here we give a description of these algebras for the bocs associated to a finite partially ordered set in terms of incidence algebras and their balanced versions. We also exhibit their quasi-hereditary structure, provide bound quiver presentations for their Ringel duals, describe the embedding of as exact Borel subalgebra and characterize the corresponding subcategories of induced and co-induced modules.
Paper Structure (20 sections, 10 theorems, 81 equations, 1 table)

This paper contains 20 sections, 10 theorems, 81 equations, 1 table.

Table of Contents

  1. Preliminary notions
  2. Bocses and their Burt-Butler algebras
  3. Grouplike elements and ditalgebras
  4. Incidence algebras
  5. Computation of the Burt-Butler algebras associated to a finite poset
  6. The bocs of a finite poset
  7. The right algebra
  8. The left algebra
  9. Simple and injective $\mathcal{L}$-modules
  10. A basic version for $\mathcal{L}$
  11. Quasi-hereditary structure of incidence algebras
  12. Regular and directed bocses
  13. Quasi-hereditary algebras and their exact Borel subalgebras
  14. Induced and co-induced modules
  15. Restrictions to the incidence algebra $K\mathcal{P}$
  16. ...and 5 more sections

Key Result

Lemma 2.1

The right $\omega$-action of $\mathcal{R}$ on $e_0A$ is faithful. Moreover, the structural constants of this action with respect to the $K$-basis $\mathbb{B}_0=\{\alpha_1,\ldots,\alpha_n,e_0\}$ of $e_0A$ satisfy that $f^{\alpha_j,b} \neq 0$ implies $b=\alpha_i$ for some $i \preceq j$, for any $f \in

Theorems & Definitions (28)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Lemma 3.1
  • ...and 18 more