[Paper Review] On the structure of weak Hopf algebras

Dmitri Nikshych|ArXiv.org|Jun 1, 2001

Algebraic structures and combinatorial models11 references4 citations

TL;DR

This paper establishes weak Hopf algebra analogues of classical Hopf algebra results, proving that the antipode of a finite-dimensional Frobenius weak Hopf algebra has finite order modulo trivial automorphisms, and derives a trace formula for the square of the antipode to give a sufficient condition for semisimplicity and cosemisimplicity. It further shows that dynamical twisting of a semisimple Hopf algebra yields a cosemisimple weak Hopf algebra.

ABSTRACT

We study the group of group-like elements of a weak Hopf algebra and derive an analogue of Radford's formula for the fourth power of the antipode S, which implies that the antipode has a finite order modulo a trivial automorphism. We find a sufficient condition in terms of Tr(S^2) for a weak Hopf algebra to be semisimple, discuss relation between semisimplicity and cosemisimplicity, and apply our results to show that a dynamical twisting deformation of a semisimple Hopf algebra is cosemisimple.

Motivation & Objective

• . To establish a weak Hopf algebra analogue of Radford's formula for the fourth power of the antipode.
• . To derive a condition for semisimplicity and cosemisimplicity using the trace of the square of the antipode.
• . To prove that dynamical twisting of a semisimple Hopf algebra yields a cosemisimple weak Hopf algebra.
• . To classify minimal weak Hopf algebras and define the group of group-like elements and its quotient by trivial group-like elements.
• . To show that semisimplicity and cosemisimplicity are equivalent when the bases coincide in a weak Hopf algebra.

Proposed method

• . Introduce and analyze the group of group-like elements G(H) and its normal subgroup G₀(H) of trivial group-like elements in the minimal weak Hopf subalgebra.
• . Define the distinguished coset of group-like elements to measure the difference between left and right integrals in the dual algebra.
• . Extend Radford's formula to weak Hopf algebras, proving that S⁴ is conjugate to the identity modulo trivial automorphisms.
• . Use the Larson-Radford trace formula for Tr(S²) to derive a sufficient condition for semisimplicity and cosemisimplicity.
• . Construct a twist (Θ, Θ̄) from a dynamical twist J in a semisimple Hopf algebra U, and define the twisted weak Hopf algebra HΘ.
• . Compute the antipode SΘ and trace Tr(S²Θ) using the twist parameters, showing Tr(S²Θ) = dim(HΘ) ≠ 0 to prove cosemisimplicity of HΘ.

Experimental results

Research questions

• RQ1 . Does the antipode of a finite-dimensional weak Hopf algebra have finite order modulo trivial automorphisms, analogous to Radford's result for ordinary Hopf algebras?
• RQ2 . Can the trace of the square of the antipode, Tr(S²), be used to determine semisimplicity and cosemisimplicity in weak Hopf algebras?
• RQ3 . Is a weak Hopf algebra obtained by dynamical twisting of a semisimple Hopf algebra necessarily cosemisimple?
• RQ4 . What is the structure of the group of group-like elements in a weak Hopf algebra, and how does its quotient by trivial elements relate to the classical group of group-like elements?
• RQ5 . Under what conditions are semisimplicity and cosemisimplicity equivalent in weak Hopf algebras?

Key findings

• . The fourth power of the antipode S⁴ in a finite-dimensional Frobenius weak Hopf algebra is conjugate to the identity modulo trivial automorphisms, generalizing Radford's result.
• . A sufficient condition for a weak Hopf algebra to be semisimple and cosemisimple is that Tr(S²) ≠ 0.
• . Semisimplicity and cosemisimplicity are equivalent when the left and right bases of the weak Hopf algebra coincide.
• . The dual of a weak Hopf algebra obtained by dynamical twisting of a semisimple Hopf algebra is semisimple, so the twisted algebra is cosemisimple.
• . For a dynamical twist J on a semisimple Hopf algebra U, the twisted weak Hopf algebra HΘ has Tr(S²Θ) = dim(HΘ) ≠ 0, implying HΘ is cosemisimple.
• . The trace Tr(S²|pHp) is strictly positive for all primitive idempotents p in the minimal weak Hopf subalgebra, which supports the non-degeneracy of integrals and semisimplicity.