Arithmetic nature of the basic invariants of semisimple Hopf algebras: algebraic integrality and cyclotomicity

For a finite-dimensional semisimple Hopf algebra $H$ over $\mathbb{C}$, Ehud Meir's geometric-invariant-theoretic study (Adv. Math. 2017) attaches a system of scalar "basic invariants", built from the integrals of $H$ and $H^*$ via multiplication and comultiplication, that determine $H$ up to isomorphism and include higher Frobenius–Schur indicators, braid-group traces on the Drinfeld double, and Reshetikhin–Turaev invariants of 3-manifolds. The paper proves that if all basic invariants are algebraic integers then $H$ is of Frobenius type, i.e. Kaplansky's sixth conjecture holds for $H$, yet leaves wide open the arithmetic of these invariants in general — even for the concrete invariant $\lambda(\ell_1\ell_3\ell_2\ell_4\ell_5)$ it is unknown whether it is an algebraic integer or lies in a cyclotomic field. This problem asks to determine the arithmetic nature of the basic invariants.

Background

A Hopf algebra over $\mathbb{C}$ is an algebra that is simultaneously a coalgebra (a vector space with a comultiplication $\Delta$ and a counit $\epsilon$ compatible with multiplication), together with an antipode $S$; group algebras $\mathbb{C}G$ and their duals are the basic examples, and a finite-dimensional Hopf algebra that is semisimple as an algebra behaves representation-theoretically much like a finite group (in characteristic zero, semisimplicity of $H$ is equivalent to semisimplicity of the dual $H^*$ by a theorem of Larson and Radford). A left integral in $H$ is an element $\ell$ with $h\ell=\epsilon(h)\ell$ for all $h\in H$; dually, a left integral $\lambda\in H^*$ satisfies $\mathrm{id}\ast\lambda=\lambda(1)\,\mathrm{id}$ for the convolution product. For semisimple $H$ these are unique up to scalar and play the role of the Haar measure: $\lambda$ is the character of the regular representation of $H$ and $\ell=\dim(H)\,e_1$ where $e_1$ is the idempotent projecting onto the trivial representation.

Meir (2017) studied the moduli of Hopf algebra structures using geometric invariant theory (the theory of quotients of algebraic-group actions on varieties): Hopf structures on a fixed vector space form an affine variety cut out by polynomial equations, quotiented by change of basis, and classical matrix invariant theory describes the resulting ring of invariants by trace-type generators. Meir showed that for a semisimple $H$ the corresponding scalar invariants — the basic invariants, spanned by numbers obtained from tensor powers of the integrals $\lambda$ and $\ell$ combined through the multiplication and comultiplication of $H$, concretely including scalars of the form $\lambda(\ell_{i_1}\ell_{i_2}\cdots\ell_{i_n})$ in which the subscripts denote Sweedler legs produced by iterated comultiplication of copies of $\ell$ inside $H$ and the product is taken in $H$ — span the ring of invariants up to localization, determine $H$ up to isomorphism, and realize the invariant spaces as fixed points of the (finite) group of Hopf automorphisms of $H$. These invariants subsume familiar objects: the higher Frobenius–Schur indicators $\nu_n(\chi)$ (generalizations of the classical indicator that detects the self-duality symmetry type of an irreducible representation), traces of braid generators on tensor powers of the Drinfeld double $D(H)$ (the quasitriangular Hopf algebra of dimension $\dim(H)^2$ — quasitriangular meaning it carries a universal $R$-matrix, which equips its representations with braided isomorphisms $V\otimes W\cong W\otimes V$ satisfying the braid relations, hence actions of the braid groups $B_n$), and, by a theorem of Shimizu, the Reshetikhin–Turaev invariants of closed 3-manifolds associated with $D(H)$.

The arithmetic of these scalars is well understood in special cases. For $H=\mathbb{C}G$ every basic invariant is an integer, of the form $|G|^a\#\mathrm{Hom}(P,G)$ for a finitely presented group $P$. Indicator-type invariants of a semisimple $H$ lie in cyclotomic rings $\mathbb{Z}[\zeta_n]$ (Kashina–Sommerhäuser–Zhu). If $H$ is group-theoretical — meaning its representation category is categorically Morita equivalent to a pointed category of $G$-graded vector spaces — Etingof–Rowell–Witherspoon proved that the braid group action on $D(H)^{\otimes n}$ factors over a finite quotient, which forces the braid-derived basic invariants into some $\mathbb{Z}[\zeta_m]$. For a general semisimple Hopf algebra nothing of the sort is known: Meir records explicitly that he does not know whether the single concrete invariant $\lambda(\ell_1\ell_3\ell_2\ell_4\ell_5)$ can be expressed through Frobenius–Schur indicators, whether it lies in a cyclotomic extension of $\mathbb{Q}$ (an algebraic integer being a root of a monic polynomial with integer coefficients, and a cyclotomic field being $\mathbb{Q}(\zeta_m)$ for a root of unity $\zeta_m$), or whether it is an algebraic integer at all.

The stakes are high: Meir's Theorem 1.9 states that if all basic invariants of $H$ are algebraic integers, then every irreducible representation $W$ of $H$ satisfies $\dim(W)\mid\dim(H)$. This divisibility property (Frobenius type) is Kaplansky's sixth conjecture, posed in the mid-1970s and still open in general (Dai–Dong 2016 survey); it is one of the central structural questions about semisimple Hopf algebras. The braid-group finiteness statement is itself a named open problem: Naidu and Rowell conjectured that the braid action on tensor powers of $D(H)$ factors over a finite quotient for all semisimple $H$, indeed for all braided weakly integral fusion categories (categories whose categorical dimensions are algebraic integers of which a power is an integer — Rep$(D(H))$ is such a category), a statement closely related to the "property F" conjecture for modular categories, verified only for group-theoretical, metaplectic, and various low-rank classes; an affirmative resolution of it would force all braid-derived basic invariants, including the Reshetikhin–Turaev invariants $RT_{D(H)}(M)$, to be cyclotomic integers, and since it concerns only the braid-derived part of the invariant system it neither settles the integrality and cyclotomicity of the complete basic-invariant system nor is settled by it. Determining the arithmetic nature of the basic invariants is thus the concrete missing bridge between the invariant-theoretic/moduli description of semisimple Hopf algebras and the deepest open divisibility questions about them.

Problem Statement

Let $H$ be a finite-dimensional semisimple Hopf algebra over $\mathbb{C}$, with left integrals $\ell\in H$ and $\lambda\in H^*$ normalized by $\epsilon(\ell)=\lambda(1)=\dim(H)$ (equivalently, $\lambda$ is the character of the regular representation of $H$, and $\ell=\dim(H)e_1$ where $e_1$ is the central idempotent projecting onto the trivial representation), and let the basic invariants of $H$ be the scalars obtained by starting from tensor powers $\ell^{\otimes a}\otimes\lambda^{\otimes b}$, applying the comultiplication of $H$ and of $H^*$ to some of the tensor factors, permuting tensor factors, and pairing copies of $H$ with copies of $H^*$ through the evaluation map — concretely including all values $\lambda(\ell_{i_1}\ell_{i_2}\cdots\ell_{i_n})$ in which $\ell_1\otimes\ell_2\otimes\cdots$ denotes Sweedler legs produced by iterated comultiplication of copies of $\ell$ inside $H$ (a word may mix legs of several comultiplied copies of $\ell$ and of $\lambda$), the product $\ell_{i_1}\ell_{i_2}\cdots\ell_{i_n}$ is taken in the algebra $H$, and $\lambda$ is evaluated on the result (the finitely many such scalars span the invariant ring studied by Meir, up to localization). Determine the arithmetic nature of these invariants, with the following specific targets, which together establish the arithmetic (integrality and cyclotomicity) of the complete invariant system:

1. Is every basic invariant of every finite-dimensional semisimple Hopf algebra $H$ over $\mathbb{C}$ an algebraic integer? In particular, is $\lambda(\ell_1\ell_3\ell_2\ell_4\ell_5)$ an algebraic integer?
2. Is every basic invariant contained in some cyclotomic field $\mathbb{Q}(\zeta_m)$?

An affirmative answer to (1) for all semisimple $H$ proves Kaplansky's sixth conjecture (every irreducible representation of $H$, and likewise of $H^*$, has dimension dividing $\dim H$) via Meir's Theorem 1.9; the problem is to settle (1) and (2) — either affirmatively for all semisimple $H$ or by producing a semisimple Hopf algebra and a basic invariant of it that is demonstrably not an algebraic integer (respectively not cyclotomic).

References

1. Ehud Meir, Semisimple Hopf algebras via geometric invariant theory, Advances in Mathematics 311 (2017) 61–90, doi:10.1016/j.aim.2017.02.020, https://doi.org/10.1016/j.aim.2017.02.020 (preprint arXiv:1506.00314, https://arxiv.org/abs/1506.00314)
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