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Two-sided crossed products

Florin Panaite

Abstract

Given two associative algebras A, C and a linear space V together with some linear maps R_1, R_2, R_3, E satisfying some conditions, we define an associative algebra structure on A\otimes V\otimes C called a two-sided crossed product. Particular cases of this construction are the iterated twisted tensor product of algebras and the two-sided crossed product over a quasi-bialgebra.

Two-sided crossed products

Abstract

Given two associative algebras A, C and a linear space V together with some linear maps R_1, R_2, R_3, E satisfying some conditions, we define an associative algebra structure on A\otimes V\otimes C called a two-sided crossed product. Particular cases of this construction are the iterated twisted tensor product of algebras and the two-sided crossed product over a quasi-bialgebra.

Paper Structure

This paper contains 3 sections, 5 theorems, 36 equations.

Table of Contents

  1. Preliminaries
  2. The definition of the two-sided crossed product
  3. Examples

Key Result

Proposition 1.1

(brz) Let $(A, \mu , 1_A)$ be an (associative unital) algebra and $V$ a vector space equipped with a distinguished element $1_V\in V$. Then the vector space $A\otimes V$ is an associative algebra with unit $1_A\otimes 1_V$ and whose multiplication has the property that $(a\otimes 1_V)(b\otimes v)= a If this is the case, the multiplication of $A\otimes V$ is given explicitly by where $\mu _2=\mu \

Theorems & Definitions (10)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Example 3.1
  • Example 3.2
  • Example 3.3