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Group gradings on finite dimensional incidence algebras

Ednei A. Santulo, Jonathan P. Souza, Felipe Y. Yasumura

Abstract

In this work, we classify the group gradings on finite-dimensional incidence algebras over a field, where the field has characteristic zero, or the characteristic is greater than the dimension of the algebra, or the grading group is abelian. Moreover, we investigate the structure of $G$-graded $(D_1,D_2)$-bimodules, where $G$ is an abelian group, and $D_1$ and $D_2$ are the group algebra of finite subgroups of $G$. As a consequence, we can provide a more profound structure result concerning the group gradings on the incidence algebras, and we can classify their isomorphism classes of group gradings.

Group gradings on finite dimensional incidence algebras

Abstract

In this work, we classify the group gradings on finite-dimensional incidence algebras over a field, where the field has characteristic zero, or the characteristic is greater than the dimension of the algebra, or the grading group is abelian. Moreover, we investigate the structure of -graded -bimodules, where is an abelian group, and and are the group algebra of finite subgroups of . As a consequence, we can provide a more profound structure result concerning the group gradings on the incidence algebras, and we can classify their isomorphism classes of group gradings.

Paper Structure

This paper contains 14 sections, 32 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Incidence algebras
  4. Graded Jacobson radical
  5. Group algebras over fields
  6. Group gradings on incidence algebras
  7. On the structure of graded bi-vector spaces
  8. The isomorphism problem
  9. Abelian grading group
  10. Examples
  11. It is not always possible to reduce to cyclic groups.
  12. Non-existence of homogeneous multiplicative basis.
  13. Non-abelian grading.
  14. Reconstruction of algebras

Key Result

Theorem 1

Let $\mathbb{F}$ be a field, $X$ a finite poset, and let $I(X)$ be endowed with a $G$-grading. Assume at least one of the following conditions: $\mathrm{char}\,\mathbb{F}=0$, $\mathrm{char}\,\mathbb{F}>\dim I(X)$, or $G$ is abelian. Then, up to a graded isomorphism, there exist finite abelian subgro where each $M_{i,j}$ is a graded $(\mathbb{F}H_i,\mathbb{F}H_j)$-bimodule.

Theorems & Definitions (62)

  • Example 1
  • Theorem 1
  • Theorem 2
  • Example 2
  • Lemma 3: Corollary 3.3 of Gord
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • proof
  • ...and 52 more