[Paper Review] Cosemisimple Hopf algebras with antipode of arbitrary finite order

Julien Bichon|ArXiv.org|Apr 18, 2002

Algebraic structures and combinatorial models13 references3 citations

TL;DR

This paper constructs, over an algebraically closed field of characteristic zero, cosemisimple Hopf algebras with antipode of arbitrary even finite order $2m$, using universal Hopf algebras associated to non-degenerate bilinear forms. The key contribution is proving the existence of such Hopf algebras—previously unknown—thereby providing counterexamples to Kaplansky's conjecture in the non-involutive case and extending the Frobenius-Schur indicator theory to cosemisimple Hopf algebras with finite antipode order.

ABSTRACT

We show that there exists cosemisimple Hopf algebras of arbitrary finite even order. We also discuss the Schur indicator for such Hopf algebras.

Motivation & Objective

• To construct cosemisimple Hopf algebras with antipode of arbitrary finite even order $2m$ over an algebraically closed field of characteristic zero.
• To provide counterexamples to Kaplansky's conjecture on the involutivity of the antipode in finite-dimensional cosemisimple Hopf algebras.
• To generalize the Frobenius-Schur indicator theory to cosemisimple Hopf algebras with finite antipode order.
• To analyze the Schur indicator for irreducible comodules in such Hopf algebras and demonstrate its possible values beyond $\pm1$.

Proposed method

• Constructs the universal Hopf algebra $\mathcal{B}(E)$ from a non-degenerate bilinear form via a matrix $E \in GL_n(k)$, satisfying $E^{-1}{}^t\!a E a = I = a E^{-1}{}^t\!a E$.
• Defines the comultiplication $\Delta(a_{ij}) = \sum_k a_{ik} \otimes a_{kj}$, counit $\varepsilon(a_{ij}) = \delta_{ij}$, and antipode $S(a) = E^{-1}{}^t\!a E$.
• Uses a category equivalence $\mathrm{Comod}(\mathcal{B}(E)) \cong^\otimes \mathrm{Comod}(\mathcal{O}(SL_q(2)))$ to prove cosemisimplicity when $q$ is not a root of unity or $q = \pm1$.
• Establishes that the antipode has order $2m$ by showing $S^{2m}(a) = a$ and $S^{2k}(a) = a$ only if $m \mid k$, using linear independence of matrix coefficients.
• Applies orthogonality relations for comodules to define the Schur indicator $\nu_2(V) = h(\chi_{V(1)}\chi_{V(2)})$ and derives its formula in terms of the matrix $E$ of a bilinear form.
• Uses the Hopf algebra $\mathcal{B}(E)$ with $E = \mathrm{AD}(\xi,1,1,1,1,1)$, $\xi$ a primitive $m$-th root of unity, to realize antipode order $2m$.

Experimental results

Research questions

• RQ1 Can cosemisimple Hopf algebras with antipode of arbitrary finite even order $2m$ exist over algebraically closed fields of characteristic zero?
• RQ2 Do examples exist of cosemisimple Hopf algebras with non-involutive antipode, thus refuting Kaplansky's conjecture in the absence of semisimplicity?
• RQ3 Can the Frobenius-Schur indicator theory be generalized to cosemisimple Hopf algebras with finite antipode order?
• RQ4 What values can the Schur indicator $\nu_2(V)$ take for irreducible comodules in such Hopf algebras, especially when $S^2 \neq \mathrm{id}$?
• RQ5 Is it possible to construct a cosemisimple Hopf algebra with antipode of order 4 and an irreducible comodule with $\nu_2(V) = n$ for arbitrary odd $n \geq 3$?

Key findings

• For any $m \geq 1$, there exists a cosemisimple Hopf algebra over an algebraically closed field of characteristic zero with antipode of order exactly $2m$, constructed via $\mathcal{B}(E)$ with $E = \mathrm{AD}(\xi,1,1,1,1,1)$, $\xi$ a primitive $m$-th root of unity.
• The antipode of $\mathcal{B}(E)$ has order $2m$ because $S^{2k}(a) = a$ only when $m \mid k$, as shown by the linear independence of the matrix coefficients $a_{ij}$ and the condition $\xi^k = 1$.
• The Hopf algebra $\mathcal{B}(E)$ is cosemisimple due to the equivalence $\mathrm{Comod}(\mathcal{B}(E)) \cong^\otimes \mathrm{Comod}(\mathcal{O}(SL_q(2)))$ and the fact that $\mathcal{O}(SL_q(2))$ is cosemisimple when $q = \pm1$ or $q$ is not a root of unity.
• The Schur indicator $\nu_2(V)$ for a self-dual irreducible comodule $V$ satisfies $\nu_2(V) = \frac{\dim(V)}{\mathrm{tr}(E{}^tE^{-1})}$, where $E$ is the matrix of an $A$-colinear bilinear form.
• When $S^2 = \mathrm{id}$, the Schur indicator takes values $\pm1$, corresponding to symmetric or skew-symmetric $A$-colinear forms.
• An example is constructed with antipode of order 4 and an irreducible comodule $V$ of dimension $2n$ ($n$ odd) such that $\nu_2(V) = n$, showing that the indicator can exceed $1$ in absolute value even for finite antipode order.