Examples of polynomial identities distinguishing the Galois objects over finite-dimensional Hopf algebras

Christian Kassel

doi:10.5802/ambp.325

Abstract

Examples of polynomial identities distinguishing the Galois objects over finite-dimensional Hopf algebras

Highlights

By the celebrated Amitsur-Levitzki theorem [4], the standard polynomial of degree 2n S2n =sign(σ) Xσ(1)Xσ(2) · · · Xσ(2n) σ∈Sn is a polynomial identity for the algebra Mn(C) of n×n-matrices with complex entries, and Mn(C) has no non-zero polynomial identity of degree < 2n
By the celebrated Amitsur-Levitzki theorem [4], the standard polynomial of degree 2n sign(σ) Xσ(1)Xσ(2) · · · Xσ(2n) σ∈Sn is a polynomial identity for the algebra Mn(C) of n×n-matrices with complex entries, and Mn(C) has no non-zero polynomial identity of degree < 2n. It follows that the identities S2n distinguish the finite-dimensional simple associative algebras over C up to isomorphism
Consider the class of Hcomodule algebras. This class contains the G-graded k-algebras; such a algebra is nothing but a comodule algebra over the group algebra kG equipped with its standard Hopf algebra structure

Summary

Introduction

By the celebrated Amitsur-Levitzki theorem [4], the standard polynomial of degree 2n. Sign(σ) Xσ(1)Xσ(2) · · · Xσ(2n) σ∈Sn is a polynomial identity for the algebra Mn(C) of n×n-matrices with complex entries, and Mn(C) has no non-zero polynomial identity of degree < 2n It follows that the identities S2n distinguish the finite-dimensional simple associative algebras over C up to isomorphism. A comodule algebra over the Hopf algebra of k-valued functions on a finite group G is the same as a G-algebra, i.e., an associative k-algebra equipped with a left G-action by algebra automorphisms In this context we may wonder whether the following assertion holds: if H is a Hopf algebra over an algebraically closed field of characteristic zero, any finite-dimensional simple H-comodule algebra is determined up to H-comodule algebra isomorphism by its polynomial H-identities. We prove a similar result for the Hopf algebra E(n) in § 4, exhibiting a finite set of polynomial E(n)-identities which distinguishes the Galois objects over E(n)

Polynomial identities for comodule algebras

Taft algebras