[Paper Review] The quasi-Hopf analogue of $u_q(sl_2)$

Gongxiang Liu|arXiv (Cornell University)|Feb 8, 2012

Algebraic structures and combinatorial models8 references3 citations

TL;DR

This paper constructs the quasi-Hopf analogue of Lusztig's small quantum group $\mathbf{u}_q(\mathfrak{sl}_2)$, denoted $\operatorname{Q}_s\mathbf{u}_q(\mathfrak{sl}_2)$, as the Drinfeld double of a quasi-Hopf algebra $A(n,s,q)$, which generalizes the Taft algebra. The key result is that when $n$ is even and $s$ is odd, this double is not twist equivalent to any Hopf algebra, establishing a genuine non-semisimple quasitriangular quasi-Hopf algebra distinct from $\mathbf{u}_q(\mathfrak{sl}_2)$.

ABSTRACT

In [4], some quasi-Hopf algebras of dimension $n^{3}$, which can be understood as the quasi-Hopf analogues of Taft algebras, are constructed. Moreover, the quasi-Hopf analogues of generalized Taft algebras are considered in [7], where the language of the dual of a quasi-Hopf algebra is used. The Drinfeld doubles of such quasi-Hopf algebras are computed in this paper. The authors in [5] shew that the Drinfeld double of a quasi-Hopf algebra of dimension $n^{3}$ constructed in [4] is always twist equivalent to Lusztig's small quantum group $u_q(sl_2)$ if $n$ is odd. Based on computations and analysis, we show that this is \emph{not} the case if $n$ is even. That is, the quasi-Hopf analogue $Qu_q(sl_2)$ of $u_q(sl_2)$ is gotten.

Motivation & Objective

• To construct a quasi-Hopf analogue of Lusztig’s small quantum group $\mathbf{u}_q(\mathfrak{sl}_2)$ using the Drinfeld double of a quasi-Hopf algebra.
• To determine whether the Drinfeld double of the quasi-Hopf analogue of a Taft algebra is twist equivalent to a Hopf algebra.
• To identify conditions under which the resulting quasi-Hopf algebra is genuinely non-Hopf, i.e., not twist equivalent to any Hopf algebra.
• To provide a new class of finite-dimensional non-semisimple quasitriangular quasi-Hopf algebras as non-trivial generalizations of $\mathbf{u}_q(\mathfrak{sl}_2)$.

Proposed method

• Construct the Drinfeld double $D(A(n,s,q))$ of the quasi-Hopf algebra $A(n,s,q)$, which is the quasi-Hopf analogue of a generalized Taft algebra.
• Define the new quasi-Hopf algebra $\operatorname{Q}_s\mathbf{u}_q(\mathfrak{sl}_2)$ via generators and relations in Section 2, using the Drinfeld double construction.
• Use a 1-dimensional representation $X$ of $\operatorname{Q}_s\mathbf{u}_q(\mathfrak{sl}_2)$ to generate a subtensor category $\langle X \rangle$ isomorphic to the representation category of $\mathbb{Z}_{2s}$.
• Analyze the restriction of the reassociator $\phi_s$ to $\langle X \rangle$ using group cohomology, showing it is a nontrivial 3-cocycle on $\mathbb{Z}_{2s}$.
• Prove that the 3-cocycle $f(g_{2s}^i, g_{2s}^j, g_{2s}^k) = (-1)^{i[\frac{j+k}{2s}]}$ is not a 3-coboundary, implying non-triviality of the associator.
• Apply Tannaka-Krein duality to show that if $\operatorname{Q}_s\mathbf{u}_q(\mathfrak{sl}_2)$ were twist equivalent to a Hopf algebra, the reassociator would be a 3-coboundary, leading to a contradiction.

Experimental results

Research questions

• RQ1 Is the Drinfeld double of the quasi-Hopf analogue of a generalized Taft algebra twist equivalent to a Hopf algebra?
• RQ2 What conditions on $n$ and $s$ ensure that the resulting quasi-Hopf algebra is not twist equivalent to any Hopf algebra?
• RQ3 Can a genuine quasi-Hopf analogue of $\mathbf{u}_q(\mathfrak{sl}_2)$ be constructed that is not twist equivalent to the original Hopf algebra?
• RQ4 How does the representation theory of the Drinfeld double reflect the non-triviality of the associator in the quasi-Hopf setting?
• RQ5 What role does group cohomology play in detecting non-triviality of the reassociator in the context of tensor categories?

Key findings

• The Drinfeld double $D(A(n,s,q))$ is isomorphic to the quasi-Hopf algebra $\operatorname{Q}_s\mathbf{u}_q(\mathfrak{sl}_2)$ as a quasi-Hopf algebra.
• When $n = 2^m l$ and $s = 2^{m'} l'$ with $(l,2) = (l',2) = 1$ and $m' < m$, the double $D(A(n,s,q))$ is not twist equivalent to any Hopf algebra.
• In particular, if $n$ is even and $s$ is odd, then $D(A(n,s,q))$ is not twist equivalent to a Hopf algebra, establishing a non-trivial quasi-Hopf analogue.
• The 3-cocycle $f(g_{2s}^i, g_{2s}^j, g_{2s}^k) = (-1)^{i[\frac{j+k}{2s}]}$ on $\mathbb{Z}_{2s}$ is not a 3-coboundary, proving the non-triviality of the reassociator in the subtensor category $\langle X \rangle$.
• The representation $X$ defined by $g_1 \mapsto -1$, $g_2 \mapsto (-1)^{1/s}$, $x,y \mapsto 0$ generates a subtensor category whose associator obstruction proves non-twist-equivalence.
• The result confirms that the quasi-Hopf analogue $\operatorname{Q}_1\mathbf{u}_q(\mathfrak{sl}_2)$ is not twist equivalent to a Hopf algebra when $n$ is even, providing a new example of a non-semisimple quasitriangular quasi-Hopf algebra.