Higher Frobenius–Schur indicators of small quantum groups: effective formulas for $u_q(\mathfrak{g})$ beyond $\mathfrak{sl}_2$ and the gcd-divisibility of $\nu_n(H)$

Shimizu (Israel J. Math. 207 (2015) 155–201) proved that the higher Frobenius–Schur indicators of the small quantum group $u_q(\mathfrak{sl}_2)$ at a root of unity $q$ of odd order $N$ satisfy $\nu_n(u_q(\mathfrak{sl}_2))=|\nu_n(T_{N^2}(q^2))|^2/\gcd(N,n)$, where $T_{N^2}(q^2)$ is a Taft algebra, and reduced the computation of $\nu_n(T_{N^2}(\omega))$ to Warnaar-type admissible-partition q-binomial sums for which no effective evaluation is known. The paper asks for analogous closed formulas for $\nu_n(u_q(\mathfrak{g}))$ for general semisimple Lie algebras $\mathfrak{g}$, and poses Question 6.11: is $\nu_n(H)/\gcd(n,\dim_k(H))$ an algebraic integer for every finite-dimensional Hopf algebra $H$? Both issues remain open in the published literature. The open problem recorded here is to determine the indicator sequence of small quantum groups for arbitrary semisimple $\mathfrak{g}$ and to settle the gcd-divisibility arithmetic, with small quantum groups as the central non-semisimple test family.

Background

A Hopf algebra over a field $k$ is an algebra $H$ equipped with a coproduct $\Delta:H\to H\otimes H$ (an algebra map that lets one tensor representations), a counit, and an antipode $S$ (the linear map implementing duals); group algebras $kG$ and universal enveloping algebras $U(\mathfrak{g})$ are the basic examples. Fix a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ and a root of unity $q$ of odd order $N>1$ (with $N\not\equiv 0 \pmod 3$ if $\mathfrak{g}$ has a $G_2$ component). The small quantum group $u_q(\mathfrak{g})$ — the finite-dimensional quotient of the quantum group of $\mathfrak{g}$ at $q$ obtained by imposing $K_i^N=1$ and $E_i^N=F_i^N=0$ — is the canonical non-semisimple finite-dimensional Hopf algebra; its finite tensor category of representations underlies logarithmic conformal field theory. It belongs to the pointed Hopf algebras — Hopf algebras whose simple subcoalgebras are all one-dimensional — of the form $u(\mathcal{D},\lambda,\mu)$, classified in finite Cartan type by Andruskiewitsch and Schneider.

For a finite-dimensional Hopf algebra $H$, Kashina, Montgomery and Ng introduced the $n$-th (higher) Frobenius–Schur indicator

$$\nu_n(H)=\mathrm{Tr}\big(S_H\circ P_H^{(n-1)}\big),$$

where $P_H^{(m)}$ is the $m$-th Sweedler power, the $m$-th convolution power of the identity map ($h\mapsto h_{(1)}\cdots h_{(m)}$ in Sweedler notation for iterated coproducts); equivalently $\nu_n(H)=\langle\lambda,\Lambda^{[n]}\rangle$ for normalized left integrals $\lambda\in H^*$ and $\Lambda\in H$. For $H=kG$ one recovers $\nu_n(kG)=\#\{g\in G: g^n=e\}$, and the classical theorem of Frobenius asserts that this number is divisible by $\gcd(n,|G|)$. The indicators are gauge invariants: they depend only on $\mathrm{Rep}(H)$ as a tensor category (they are unchanged under Drinfeld twists, the coproduct deformations realizing monoidal equivalences), which makes them a basic arithmetic fingerprint of the category. Shimizu proved that $\nu_n(H)$ is always a cyclotomic integer (it lies in $\mathbb{Z}[\zeta]$ for $\zeta$ a primitive $n\,\mathrm{ord}(S^2)$-th root of unity), that $\nu_n(D(H))=|\nu_n(H)|^2$ for the Drinfeld double $D(H)$ — the canonical Hopf algebra built on $H\otimes H^*$, whose representation category is the center of $\mathrm{Rep}(H)$ — (resolving a conjecture of Kashina–Montgomery–Ng), and that $\nu_n$ is unchanged by passing to the coradical-graded Hopf algebra $\mathrm{gr}\,H$ (the graded algebra attached to the coradical filtration, the coalgebra filtration whose degree-zero part, the coradical, is the maximal cosemisimple subcoalgebra). The last fact reduces indicators of $u(\mathcal{D},\lambda,\mu)$ to those of the bosonization $B(V_{\mathcal{D}})\#k\Gamma$ of a Nichols algebra $B(V_{\mathcal{D}})$ — the graded braided Hopf algebra generated by the skew-primitive root vectors — with the group algebra $k\Gamma$.

In rank one this program terminates at the Taft algebra $T_{N^2}(\omega)$ (generated by $g,x$ with $g^N=1$, $x^N=0$, $gxg^{-1}=\omega x$, $\omega$ a primitive $N$-th root of unity), because the positive Borel part $u_q^{\geq 0}(\mathfrak{sl}_2)$ (the Hopf subalgebra generated by the Cartan and raising operators) is isomorphic to $T_{N^2}(q^2)$. Shimizu's Theorem 6.4 evaluates $\nu_n(T_{N^2}(\omega))$ as $d\sum_{i\geq 0}\{\begin{smallmatrix}N-1\\ di,\, n-1\end{smallmatrix}\}_\omega\,\omega^{-\check{n}di^2}$, where $d=\gcd(N,n)$ and $\check{n}$ is any integer with $\check{n}n\equiv d \pmod N$, in terms of the q-partition sums

$$\left\{\begin{matrix}L\\ a,m\end{matrix}\right\}_q=\sum_{\substack{j_1+\cdots+j_m=a\\ L\geq j_1\geq\cdots\geq j_m\geq 0}}q^{j_1^2+\cdots+j_m^2}\binom{L}{j_1}_q\binom{j_1}{j_2}_q\cdots\binom{j_{m-1}}{j_m}_q,$$

the generating functions of $(m,m+1;L,a)$-admissible partitions in the sense of Warnaar ($\binom{m}{a}_q$ denotes the Gaussian q-binomial coefficient). Theorem 6.5 then gives the closed formula for the small quantum group:

$$\nu_n(u_q(\mathfrak{sl}_2))=\frac{1}{\gcd(N,n)}\bigl|\nu_n(T_{N^2}(q^2))\bigr|^2.$$

Two issues remain open at exactly this point. First, no effective way to compute the admissible-partition sums at roots of unity is known: the paper evaluates them by hand for $N=2,3$ and by computer for $N\leq 7$, where the values exhibit a quasi-periodic pattern $\nu_n(T_{N^2}(\omega))=c_\omega(n)\,n$ with $c_\omega(n+N)=c_\omega(n)$ that is machine-observed but unproved in general; the author states that 'a combinatorial approach would be helpful for this problem'. Second, by analogy with the Frobenius theorem for group algebras, Question 6.11 asks whether $\nu_n(H)/\gcd(n,\dim_k(H))$ is always an algebraic integer — known positively only for several semisimple families (Tambara–Yamagami categories and other computed classes, by Shimizu), and open even in the semisimple case. The rank-one formula makes the small quantum group the test family: $\nu_n(u_q(\mathfrak{sl}_2))$ appears to be divisible by $n^2$ (proved for $\mathrm{ord}(q)=3$, where $\nu_n=n^2$; derived from the machine tables for $\mathrm{ord}(q)=5,7$). For $\mathfrak{g}$ of higher rank only the second indicator is known — it factors over the simple components through the Taft-type factors $u_q^{\geq 0}(\mathfrak{g}_i)$ — and the paper's Remark 6.7 asks for the analogue of Theorem 6.5 in general.

Problem Statement

Part (A) — the general-rank formula (Remark 6.7 of the source). Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ and let $q$ be a root of unity of odd order $N>1$ (with $N\not\equiv 0 \pmod 3$ if $\mathfrak{g}$ has a $G_2$ component). Find a closed formula for $\nu_n(u_q(\mathfrak{g}))$, valid for all $n\geq 1$, that expresses it through simpler building blocks — Taft-type Borel factors $u_q^{\geq 0}(\mathfrak{g}_i)$, Nichols-algebra bosonizations $B(V)\#k\Gamma$, or analogous rank-one objects — in the way that the rank-one formula $\nu_n(u_q(\mathfrak{sl}_2))=|\nu_n(T_{N^2}(q^2))|^2/\gcd(N,n)$ does for $\mathfrak{sl}_2$. As context for the target form of such a formula: in the rank-one case the reduction terminates at the admissible-partition q-binomial sums, and the source notes (§6.5) that no effective way to compute these sums at roots of unity is known — hand computations reach $N\leq 3$, machine verification $N\leq 7$ — while expecting that a combinatorial approach would be helpful for that evaluation problem.

Part (B) — the arithmetic divisibility (Question 6.11 of the source). For a finite-dimensional Hopf algebra $H$ over an algebraically closed field of characteristic zero, is $\nu_n(H)/\gcd(n,\dim_k(H))$ always an algebraic integer? Decide this in general — it is open even when $H$ is semisimple — and in particular for $H=u_q(\mathfrak{g})$, where the unexplained phenomenon $\nu_n(u_q(\mathfrak{sl}_2))\equiv 0 \pmod{n^2}$ (proved for $\mathrm{ord}(q)=3$, where $\nu_n=n^2$; derived from machine-verified tables for $\mathrm{ord}(q)=5,7$) awaits explanation.

Parts (A) and (B) are the structural and arithmetic faces of one objective — determining the indicator sequence of the small quantum groups, the canonical non-semisimple family — and are linked in the source itself: the $n^2$-divisibility evidence for (B) is a corollary of the rank-one formula that (A) asks to generalize.

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