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A negative answer to a Bahturin-Regev conjecture about regular algebras in positive characteristic

Lucio Centrone, Plamen Koshlukov, Kauê Pereira

Abstract

Let $A=A_1\oplus\cdots\oplus A_r$ be a decomposition of the algebra $A$ as a direct sum of vector subspaces. If for every choice of the indices $1\le i_j\le r$ there exist $a_{i_j}\in A_{i_j}$ such that the product $a_{i_1}\cdots a_{i_n}\ne 0$, and for every $1\le i,j\le r$ there is a constant $β(i,j)\ne 0$ with $a_ia_j=β(i,j) a_ja_i$ for $a_i\in A_i$, $a_j\in A_j$, the above decomposition is regular. Bahturin and Regev raised the following conjecture: suppose the regular decomposition comes from a group grading on $A$, and form the $r\times r$ matrix whose $(i,j)$th entry equals $β(i,j)$. Then this matrix is invertible if and only if the decomposition is minimal (that is one cannot get a regular decomposition of $A$ by coarsening the decomposition). Aljadeff and David proved that the conjecture is true in the case the base field is of characteristic 0. We prove that the conjecture does not hold for algebras over fields of positive characteristic, by constructing algebras with minimal regular decompositions such that the associated matrix is singular.

A negative answer to a Bahturin-Regev conjecture about regular algebras in positive characteristic

Abstract

Let be a decomposition of the algebra as a direct sum of vector subspaces. If for every choice of the indices there exist such that the product , and for every there is a constant with for , , the above decomposition is regular. Bahturin and Regev raised the following conjecture: suppose the regular decomposition comes from a group grading on , and form the matrix whose th entry equals . Then this matrix is invertible if and only if the decomposition is minimal (that is one cannot get a regular decomposition of by coarsening the decomposition). Aljadeff and David proved that the conjecture is true in the case the base field is of characteristic 0. We prove that the conjecture does not hold for algebras over fields of positive characteristic, by constructing algebras with minimal regular decompositions such that the associated matrix is singular.

Paper Structure

This paper contains 15 sections, 20 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Regular algebras
  4. Minimal decomposition of a regular algebra
  5. [(Zn x Zn)-graded regular algebras
  6. Technical results on bicharacters
  7. The determinant of the decomposition matrices
  8. An explicit counterexample
  9. The relatively free graded algebra
  10. Constructing the counterexample
  11. The counterexample for the conjecture in the general case
  12. The twisted group algebra
  13. The counterexample for infinite fields
  14. The counterexample for finite fields
  15. A construction of a regular algebra that verifies the conjecture

Key Result

Theorem 10

Let $A$, $B$ be algebras with regular decompositions $A=\oplus_{i=1}^{r} A_{i}$ and $B=\oplus_{i=1}^{r} B_{i}$ with corresponding decomposition matrices $M^{A}$ and $M^{B}$. Assume that $M^{B}=PM^{A}P^{-1}$ where $P$ is a permutation matrix. Then $A$ and $B$ satisfy the same multilinear identities (

Theorems & Definitions (49)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • Example 9
  • Theorem 10: regevz2, Theorem 3.1
  • ...and 39 more