Two-Parameter Quantum Groups and Drinfel'd Doubles
Georgia Benkart, Sarah Witherspoon
备注：
26 pages, AMS-TeX. Paper rewritten, new material added, some results appear in "Representations of Two-Parameter Quantum Groups and Schur-Weyl Duality"
We investigate two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl_n and sl_n. We show that these quantum groups can be realized as Drinfel'd doubles of certain Hopf subalgebras with respect to Hopf pairings. Using the Hopf pairing, we construct a corresponding R-matrix and a quantum Casimir element. We discuss isomorphisms among these quantum groups and connections with multiparameter quantum groups.
Based on a pairing of two regular multiplier Hopf algebras $A$ and $B$, Heisenberg double $\mathscr{H}$ is the smash product $A \# B$ with respect to the left regular action of $B$ on $A$. Let $\mathscr{D}=A\bowtie B$ be the Drinfel'd double, then Heisenberg double $\mathscr{H}$ is a Yetter-Drinfel'd $\mathscr{D}$-module algebra, and it is also braided commutative by the braiding of Yetter-Drinfel'd module, which generalizes the results in [10] to some infinite dimensional cases.
We study the Yetter--Drinfeld D(B)-module algebra structure on the Heisenberg double H(B^*) endowed with a "heterotic" action of the Drinfeld double D(B). This action can be interpreted in the spirit of Lu's description of H(B^*) as a twist of D(B). In terms of the braiding of Yetter--Drinfeld modules, H(B^*) is braided commutative. By the Brzezinski--Militaru theorem, H(B^*)#D(B) is then a Hopf algebroid over H(B^*). For B a particular Taft Hopf algebra at a 2p-th root of unity, the construction is adapted to yield Yetter--Drinfeld module algebras over the 2p^3-dimensional quantum group U_q(s…
The two isomorphic Borel subalgebras of gl(n), realized on upper and lower triangular matrices, allow us to consider the gl(n) \opus t_n algebra as a self-dual Drinfeld double. Compatibility conditions impose the choice of an orthonormal basis in the Cartan subalgebra and fix the basis of gl(n). A natural Lie bialgebra structure on gl(n) is obtained, that offers a new perspective for its standard quantum deformation.
We study infinite dimensional generalisations of the Heisenberg doubles of the Borel half of $U_q(sl(2))$ and of $U_q(osp(1|2))$ and find associated canonical elements which satisfy pentagon equation. The former reproduces the canonical element, expressed using the Faddeev's quantum dilogarithm, which has been found by Kashaev to be realised within quantised Teichmüller theory, while for the latter we show that it corresponds to an operator from quantised super Teichmüller theory. We study infinite dimensional representations of those two Heisenberg doubles and, using an algebra homomorphism b…
A description of all normal Hopf subalgebras of a semisimple Drinfeld double is given. This is obtained by considering an analogue of Goursat's lemma concerning fusion subcategories of Deligne products of two fusion categories. As an application we show that the Drinfeld double of any abelian extension is also an abelian extension.
In view of applications to conformal field theory or to other branches of theoretical physics and mathematics, new examples of character tables for Drinfeld doubles of finite groups (modular data) are made available on a website.
In the paper, we describe the Drinfel'd double structure of the $n$-rank Taft algebra and all of its simple modules, and then endow its $R$-matrices with some application to knot invariants. The knot invariants we get is a generalization of the Jones polynomial, in particular, it recovers the Jones polynomial in rank $1$ case, while in rank $2$ case, it is the one-parameter specialization of the two-parameter unframed Dubrovnik polynomial, and in higher rank case it is the composite ($n$-power) of the Jones polynomial.
The Hopf algebra $\mathcal{D}$ which is the subject of this paper can be viewed as a Drinfeld double of the bosonisation of the Jordan plane. Its prime and primitive spectra are completely determined. As a corollary of this analysis it is shown that $\mathcal{D}$ satisfies the Dixmier-Moeglin Equivalence, leading to the formulation of a conjecture on the validity of this equivalence for pointed Noetherian Hopf algebras.
We introduce the Hom-analogue of the L-R-smash product and use it to define the Hom-analogue of the diagonal crossed product. When H is a finite dimensional Hom-Hopf algebra with bijective antipode and bijective structure map, we define the Drinfeld double of H; its algebra structure is a Hom-diagonal crossed product and it has all expected properties, namely it is quasitriangular and modules over it coincide with left-right Yetter-Drinfeld modules over H.
For a Hopf algebra B with bijective antipode, we show that the Heisenberg double H(B^*) is a braided commutative Yetter--Drinfeld module algebra over the Drinfeld double D(B). The braiding structure allows generalizing H(B^*) = B^{*cop}\braid B to "Heisenberg n-tuples" and "chains" ...\braid B^{*cop}\braid B \braid B^{*cop}\braid B\braid..., all of which are Yetter--Drinfeld D(B)-module algebras. For B a particular Taft Hopf algebra at a 2p-th root of unity, the construction is adapted to yield Yetter--Drinfeld module algebras over the 2p^3-dimensional quantum group U_qsl(2).
For a finite dimensional Hopf algebra we show that an associated natural inclusion of infinite crossed products is the crossed product by the Drinfeld double, and that this is a characterisation of the double.
We produce an explicit embedding of the planar algebra of the Drinfeld double of a finite-dimensional, semisimple and cosemisimple Hopf algebra $H$ into the two-cabling of the planar algebra of the dual Hopf algebra $H^*$ and characterise the image.
We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group G, focusing on surfaces without marked points or with one marked point. We obtain concrete descriptions of such representations in terms of finite group data. This allows us to establish various properties of these representations. In particular we show that they have finite images, and that for surfaces of genus at least 3 their restriction to the Torelli group is non-trivial iff G is non-abelian.
Let $q$ be an $n^{th}$ root of unity for $n > 2$ and let $T_n(q)$ be the Taft (Hopf) algebra of dimension $n^2$. In 2001, Susan Montgomery and Hans-Jürgen Schneider classified all non-trivial $T_n(q)$-module algebra structures on an $n$-dimensional associative algebra $A$. They further showed that each such module structure extends uniquely to make $A$ a module algebra over the Drinfel'd double of $T_n(q)$. We explore what it is about the Taft algebras that leads to this uniqueness, by examining actions of (the Drinfel'd double of) Hopf algebras $H$ "close" to the Taft algebras on finite-dimen…
Hopf algebra quantizations of 4-dimensional and 6-dimensional real classical Drinfel'd doubles are studied by following a direct "analytic" approach. The full quantization is explicitly obtained for most of the Drinfel'd doubles, except a small number of them for which the dual Lie algebra is either sl(2) or so(3). In the latter cases, the classical r-matrices underlying the Drinfel'd double quantizations contain known standard ones plus additional twists. Several new four and six-dimensional quantum algebras are presented and some general features of the method are emphasized.
We study representations of nilpotent type nontrivial liftings of quantum linear spaces and their Drinfel'd quantum doubles. We construct a family of Verma- type modules in both cases and prove a parametrization theorem for simple modules. We compute the Loewy and socle series of Verma modules under a mild restriction on the datum of a lifting. We find bases and dimensions of simple modules.
Drinfeld doubles of finite subgroups of SU(2) and SU(3) are investigated in detail. Their modular data - S, T and fusion matrices - are computed explicitly, and illustrated by means of fusion graphs. This allows us to reexamine certain identities on these tensor product or fusion multiplicities under conjugation of representations that had been discussed in our recent paper [J. Phys. A: Math. Theor. 44 (2011), 295208, 26 pages, arXiv:1103.2943], proved to hold for simple and affine Lie algebras, and found to be generally wrong for finite groups. It is shown here that these identities fail also…
The twisted Drinfeld double (or quasi-quantum double) of a finite group with a 3-cocycle is identified with a certain twisted groupoid algebra. The groupoid is the loop (or inertia) groupoid of the original group and the twisting is shown geometrically to be the loop transgression of the 3-cocycle. The twisted representation theory of finite groupoids is developed and used to derive properties of the Drinfeld double, such as representations being classified by their characters. This is all motivated by gerbes and 3-dimensional topological quantum field theory. In particular the representation …
We study irreducible spherical unitary representations of the Drinfeld double of a $q$-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In the case of $SU_q(3)$, we give a complete classification of such representations. As an application, we show the Drinfeld double of the quantum group $SU_q(2n+1)$ has property (T), which also implies central property (T) of the dual of $SU_q(2n+1)$.
We construct a generalized cluster structure compatible with the Poisson bracket on the Drinfeld double of the standard Poisson-Lie group $GL_n$ and derive from it a generalized cluster structure on $GL_n$ compatible with the push-forward of the Poisson bracket on the dual Poisson--Lie group.
We provide a deformation, $\mathfrak{f}_β$, of Lusztig algebra $\mathbf{f}$. Various quantum algebras in literatures, including half parts of two-parameter quantum algebras, quantum superalgebras, and multi-parameter quantum algebras/superalgebras, are all specializations of $\mathfrak{f}_β$. Moreover, $\mathfrak{f}_β$ is isomorphic to Lusztig algebra $\mathbf{f}$ up to a twist. As a consequence, half parts of those quantum algebras are isomorphic to Lusztig algebra $\mathbf{f}$ over a big enough ground field up to certain twists. We further construct the entire algebra $\mathbf U_{β,ξ}$ by Dr…
In this paper we prove a formula that relates Müger's centralizer in the category of representations of a factorizable Hopf algebra to the notion of Hopf kernel of a representation of the dual Hopf algebra. Using this relation we obtain a complete description for Müger's centralizer of some fusion subcategories of the fusion category of finite dimensional representations of a Drinfeld double of a semisimple Hopf algebra.
We find the defining structures of two-parameter quantum groups $U_{r,s}(\frak g)$ corresponding to the orthogonal and the symplectic Lie algebras, which are realized as Drinfel'd doubles. We further investigate the environment conditions upon which the Lusztig's symmetries exist between $(U_{r,s}(\frak g), < ,>)$ and its associated object $(U_{s^{-1}, r^{-1}}(\frak g), < | >)$.
We determine a substantial part of the unitary representation theory of the Drinfeld double of a $q$-deformation of a compact Lie group in terms of the complexification of the compact Lie group. Using this, we show that the dual of every $q$-deformation of a higher rank compact Lie group has central property ${\rm (T)}$. We also determine the unitary dual of $SL_q(n,\mathbb{C})$.
Consider a Frobenius kernel G in a split semisimple algebraic group, in very good characteristic. We provide an analysis of support for the Drinfeld center Z(rep(G)) of the representation category for G, or equivalently for the representation category of the Drinfeld double of kG. We show that thick ideals in the corresponding stable category are classified by cohomological support, and calculate the Balmer spectrum of the stable category of Z(rep(G)). We also construct a $π$-point style rank variety for the Drinfeld double, identify $π$-point support with cohomological support, and show that …
We investigate the Drinfel'd doubles $D(Λ_{n,d})$ of a certain family of Hopf algebras. We determine their simple modules and their indecomposable projective modules, and we obtain a presentation by quiver and relations of these Drinfel'd doubles, from which we deduce properties of their representations, including the Auslander-Reiten quivers of the $D(Λ_{n,d})$. We then determine decompositions of the tensor products of most of the representations described, and in particular give a complete description of the tensor product of two simple modules. This study also leads to explicit examples of…
Higher genus partition functions of two-dimensional conformal field theories have to be invariants under linear actions of mapping class groups. We illustrate recent results [4,6] on the construction of such invariants by concrete expressions obtained for the case of Drinfeld doubles of finite groups. The results for doubles are independent of the characteristic of the underlying field, and the general results do not require any assumptions of semisimplicity.
We report on two classes of autoequivalences of the category of Yetter-Drinfeld modules over a finite group, or, equivalently the Drinfeld center of the category of representations of a finite group. Both operations are related to the $r$-th power operation, with $r$ relatively prime to the exponent of the group. One is defined more generally for the group-theoretical fusion category defined by a finite group and an arbitrary subgroup, while the other seems particular to the case of Yetter-Drinfeld modules. Both autoequivalences preserve higher Frobenius-Schur indicators up to Galois conjugati…
The representation theory of the Drinfeld doubles of dihedral groups is used to solve the Yang-Baxter equation. Use of the 2-dimensional representations recovers the six-vertex model solution. Solutions in arbitrary dimensions, which are viewed as descendants of the six-vertex model case, are then obtained using tensor product graph methods which were originally formulated for quantum algebras. Connections with the Fateev-Zamolodchikov model are discussed.
In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by Lusztig led to the first general construction of PBW bases of these algebras. Also, they have important applications to the representation theory of these and related algebras. In the present paper the Drinfel'd double for a class of graded Hopf algebras is investigated. Various quantum algebras, including multiparameter quantizations of semisimple Lie algebras and of Lie superalgebras, are covered by the given definition. For these Drinfel'd doubles Lusztig maps are defined. It is shown that these…
We prove that the Drinfeld double of an arbitrary finite group scheme has finitely generated cohomology. That is to say, for G any finite group scheme, and D(G) the Drinfeld double of the group ring kG, we show that the self-extension algebra of the trivial representation for D(G) is a finitely generated algebra, and that for each D(G)-representation V the extensions from the trivial representation to V form a finitely generated module over the aforementioned algebra. As a corollary, we find that all categories rep(G)*_M dual to rep(G) are of also of finite type (i.e. have finitely generated c…
The Shapovalov determinant for a class of pointed Hopf algebras is calculated, including quantized enveloping algebras, Lusztig's small quantum groups, and quantized Lie superalgebras. Our main tools are root systems, Weyl groupoids, and Lusztig type isomorphisms. We elaborate powerful novel techniques for the algebras at roots of unity, and pass to the general case using a density argument. Key words: Hopf algebra, Nichols algebra, quantum group, representation
If G is a finite group and k is a field, there is a natural construction of a Hopf algebra over k associated to G, the Drinfel'd double D(G). We prove that if G is any finite real reflection group with Drinfel'd double D(G) over an algebraically closed field k of characteristic not 2, then every simple D(G)-module has Frobenius-Schur indicator +1. This generalizes the classical results for modules over the group itself. We also prove some new results about Weyl groups. In particular, we prove that any abelian subgroup is inverted by some involution. Also, if E is any elementary abelian 2-subgr…
Over many decades, the word "double" has appeared in various contexts, at times seemingly unrelated. Several have some relation to mathematical physics. Recently, this has become particularly strking in DFT (double field theory). Two 'doubles' that are particularly relevant are double vector bundles and Drinfel'd doubles. The original Drinfel'd double occurred in the contexts of quantum groups and of Lie bialgebras. Quoting T. Voronov: "Double Lie algebroids arose in the works on double Lie groupoids and in connection with an analog for Lie bialgebroids of the classical Drinfel'd double of Lie…
We present a new approach to calculating the higher Frobenius-Schur indicators for the simple modules over the Drinfeld double of a finite group. In contrast to the formula by Kashina-Sommerh{ä}user-Zhu that involves a sum over all group elements satisfying a certain condition, our formula operates on the level of conjugacy classes and character tables. It can be implemented in the computer algebra system GAP, efficiently enough to deal, on a laptop, with symmetric groups up to $S_{18}$ (providing further evidence that indicators are non-negative in this case) or simple groups of order up to $…
The Drinfled realization of quantum affine algebras has been tremendously useful since its discovery. Combining techniques of Beck and Nakajima with our previous approach, we give a complete and conceptual proof of the Drinfeld realization for the twisted quantum affine algebras using Lusztig's twisted braid group action.
By finite quantum groups we mean Lusztig's finite-dimensional pointed Hopf algebras called quantum Frobenius Kernels [9, 10], and their natural generalizations due to Andruskiewitsch and Schneider [2, 3]. For a Hopf algebra $H$ in a special class of the latter generalizations, which arises from a pair of quantum linear spaces, Krop and Radford [8] described the simple modules over $H$ and over the Drinfel'd double $D(H)$, showing that they fall into a simple pattern of parametrization. We extend the description to a wider class of Hopf algebras which includes the quantum Frobenius Kernels, ren…
On the Drinfeld doubles of Laistrygonian algebras in positive characteristic
Vrije Universiteit Brussel（布鲁塞尔自由大学）；Shenzhen International Center for Mathematics, Southern University of Science and Technology（南方科技大学深圳数学中心）；Departamento de Matemática, Universidade Federal de Santa Catarina（圣卡塔琳娜联邦大学数学系）；Departamento de Matemática, Universidade Federal de Santa Maria（圣玛丽亚联邦大学数学系）
We provide an analog of the Joyal-Street center construction and of the Kassel-Turaev categorical quantum double in the context of the crossed categories introduced by Turaev. Then, we focus or attention to the case of categories of representation. In particular, we introduce the notion of a Yetter-Drinfeld module over a crossed group coalgebra H and we prove that both the category of Yetter-Drinfeld modules over H and the center of the category of representations of H as well as the category of representations of the quantum double of H are isomorphic as braided crossed categories.
Let $H$ be a $\times$-bialgebra in the sense of Takeuchi. We show that if $H$ is $\times$-Hopf, and if $H$ fulfills the finiteness condition necessary to define its skew dual $H^\vee$, then the coopposite of the latter is $\times$-Hopf as well. If in addition the coopposite $\times$-bialgebra of $H$ is $\times$-Hopf, then the coopposite of the Drinfeld double of $H$ is $\times$-Hopf, as is the Drinfeld double itself, under an additional finiteness condition.
Let $G$ be any group and let $K(G)$ denote the multiplier Hopf algebra of complex functions with finite support in $G$. The product in $K(G)$ is pointwise. The comultiplication on $K(G)$ is defined with values in the multiplier algebra $M(K(G) \otimes K(G))$ by the formula $(Δ(f)) (p,q) = f(pq)$ for all $f \in K(G)$ and $p, q \in G$. In this paper we consider multiplier Hopf algebras $B$ (over $\Bbb C$) such that there is an embedding $I: K(G) \to M(B)$. This embedding is a non-degenerate algebra homomorphism which respects the comultiplication and maps $K(G)$ into the center of $M(B)$. These …
Let $G$ be a compact connected Lie group and $k \in H^4(BG,\mathbb{Z})$ a cohomology class. The String 2-group $G_k$ is the central extension of $G$ by the 2-group $[\ast/U(1)]$ classified by $k$. It has a close relationship to the level $k$ extension of the loop group $LG$. We compute the Drinfel'd centre of $G_k$ as a smooth 2-group. When $G$ is semisimple, we prove that the Drinfel'd centre is equal to the invertible part of the category of positive energy representations of $LG$ at level $k$ (as long as we exclude factors of $E_8$ at level 2).
Consider a field k of characteristic p > 0, G_r the r-th Frobenius kernel of a smooth algebraic group G, DG_r the Drinfeld double of G_r, and M a finite dimensional DG_r-module. We prove that the cohomology algebra H*(DG_r,k) is finitely generated and that H*(DG_r,M) is a finitely generated module over this cohomology algebra. We exhibit a finite map of algebras θ_r:H*(G_r,k) \otimes S(g) \to H*(DG_r,k) which offers an approach to support varieties for DG_r-modules. For many examples of interest, θ_r is injective and induces an isomorphism of associated reduced schemes. Additionally, for M an …
This paper exposes the fundamental role that the Drinfel'd double $\dkg$ of the group ring of a finite group $G$ and its twists $\dbkg$, $β\in Z^3(G,\uk)$ as defined by Dijkgraaf--Pasquier--Roche play in stringy orbifold theories and their twistings. The results pertain to three different aspects of the theory. First, we show that $G$--Frobenius algebras arising in global orbifold cohomology or K-theory are most naturally defined as elements in the braided category of $\dkg$--modules. Secondly, we obtain a geometric realization of the Drinfel'd double as the global orbifold $K$--theory of glob…
The zx-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of dagger-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of this structure, Hopf-Frobenius algebras, starting from a single Hopf algebra which is not necessarily commutative or cocommutative. We provide a few necessary and sufficient conditions for a Hopf algebra to be a Hopf-Frobenius algebra, and show that every Hopf algebra in the category of finite dimensional vector spaces is a Hopf-Frobenius algebra. In additi…
