Simple modules of small quantum groups at dihedral groups
Gastón Andrés García, Cristian Vay
备注：
The paper is mostly self-contained and includes figures to lighten the reading. v2: The major changes are in Sections 3.2, 3.3, 4.4 and the beginning of 5. It is the final version accepted in Documenta Mathematica
Based on previous results on the classification of finite-dimensional Nichols algebras over dihedral groups and the characterization of simple modules of Drinfeld doubles, we compute the irreducible characters of the Drinfeld doubles of bosonizations of finite-dimensional Nichols algebras over the dihedral groups $\mathbb{D}_{4t}$ with $t\geq 3$. To this end, we develop new techniques that can be applied to Nichols algebras over any Hopf algebra. Namely, we explain how to construct recursively irreducible representations when the Nichols algebra is generated by a decomposable module, and show that the highest-weight of minimum degree in a Verma module determines its socle. We also prove that tensoring a simple module by a rigid simple module gives a semisimple module.
Let $k$ be an algebraically closed field of odd characteristic $p$, and let $D_n$ be the dihedral group of order $2n$ such that $p\mid 2n$. Let $D(kD_n)$ denote the quantum double of the group algebra $kD_n$. In this paper, we describe the structures of all finite dimensional indecomposable left $D(kD_n)$-modules, equivalently, of all finite dimensional indecomposable Yetter-Drinfeld $kD_n$-modules, and classify them.
In this paper we describe the Jordan-Holder series of the standard modules over the rational Cherednik algebras associated with the dihedral group. In particular, we compute the characters of the irreducible representations from the category O and classify the finite dimensional representations.
We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GK-dimension for short, through the study of Nichols algebras over $\mathbb{D}_{\infty}$, the infinite dihedral group. We find all the irreducible Yetter-Drinfeld modules $V$ over $\mathbb{D}_{\infty}$, and determine which Nichols algebras $\mathcal{B}(V)$ of $V$ are finite GK-dimensional.
In this paper we complete the $\mathrm{ADE}$-like classification of simple transitive $2$-representations of Soergel bimodules in finite dihedral type, under the assumption of gradeability. In particular, we use bipartite graphs and zigzag algebras of $\mathrm{ADE}$ type to give an explicit construction of a graded (non-strict) version of all these $2$-representations. Moreover, we give simple combinatorial criteria for when two such $2$-representations are equivalent and for when their Grothendieck groups give rise to isomorphic representations. Finally, our construction also gives a large cl…
In this paper, we continue to investigate finite-dimensional Nichols algebras over simple Yetter-Drinfeld modules of the Suzuki algebras $A_{N\, n}^{μλ}$. It is finished for the case $A_{N\, 2n}^{μλ}$. As for the case $A_{N\, 2n+1}^{μλ}$, it boils down to the long-standing open problem: calculate dimensions of Nichols algebras of dihedral rack type $\Bbb D_{2n+1}$. It is interesting to see that the Suzuki algebras are of set-theoretical type. We pose some question or problems for our future research. In particular, we are curious about how to generalize the correspondence between braidings of …
The quantum Satake correspondence relates dihedral Soergel bimodules to the semisimple quotient of the quantum $\mathfrak{sl}_2$ representation category. It also establishes a precise relation between the simple transitive $2$-representations of both monoidal categories, which are indexed by bicolored $\mathsf{ADE}$ Dynkin diagrams. Using the quantum Satake correspondence between affine $\mathsf{A}_{2}$ Soergel bimodules and the semisimple quotient of the quantum $\mathfrak{sl}_3$ representation category, we introduce trihedral Hecke algebras and Soergel bimodules, generalizing dihedral Hecke …
We consider $AD$-type orbifolds of the triplet vertex algebras $\mathcal{W}(p)$ extending the well-known $c=1$ orbifolds of lattice vertex algebras. We study the structure of Zhu's algebras $A(\mathcal{W}(p)^{A_m})$ and $A(\mathcal{W}(p)^{D_m})$, where $A_m$ and $D_m$ are cyclic and dihedral groups, respectively. A combinatorial algorithm for classification of irreducible $\mathcal{W}(p)^Γ$-modules is developed, which relies on a family of constant term identities and properties of certain polynomials based on constant terms. All these properties can be checked for small values of $m$ and $p$ …
We are continuing our study of ADE-orbifold subalgebras of the triplet vertex algebra W(p). This part deals with the dihedral series. First, subject to a certain constant term identity, we classify all irreducible modules for the vertex algebra $\bar{M(1)} ^+$, the $\Z_2$--orbifold of the singlet vertex algebra $\bar{M(1)}$. Then we classify irreducible modules and determine Zhu's and $C_2$--algebra for the vertex algebra $\triplet ^{D_2}$. A general method for construction of twisted $\triplet$--modules is also introduced. We also discuss classification of twisted $\bar{M(1)}$--modules includ…
In the present paper, a class of new simple modules over the $N=1$ Ramond algebra are constructed, which are induced from simple modules over some finite dimensional solvable Lie superalgebras. These new modules are simple restricted modules over the $N=1$ Ramond algebra. Combined with the result in \cite{L1}, a classification of simple weak $ψ$-twisted $\bar W(0,c)$-modules under certain conditions is also given. At last, some examples of simple restricted $N=1$ Ramond modules as various versions of Whittaker modules are presented (classical Whittaker modules were studied in \cite{LPX}).
It is proved that uniformly bounded simple modules over higher rank super-Virasoro algebras are modules of the intermediate series, and that simple modules with finite dimensional weight spaces are either modules of the intermediate series or generalized highest weight modules.
We study a class of representations over the degenerate double affine Hecke algebra of gl_n by an algebraic method. As fundamental objects in this class, we introduce certain induced modules and study some of their properties. In particular, it is shown that these induced modules have unique simple quotient modules under certain conditions. Moreover, we show that any simple module in this class is obtained as such a simple quotient, and give a classification of all the simple modules.
In the present paper, using the technique of localization, we determine the center of the quantum Schrödinger algebra $§_q$ and classify simple modules with finite-dimensional weight spaces over $§_q$, when $q$ is not a root of unity. It turns out that there are four classes of such modules: dense $U_q(\mathfrak{sl}_2)$-modules, highest weight modules, lowest weight modules, and twisted modules of highest weight modules.
We present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. When a certain commutative subalgebra is finitely generated over an algebraically closed field we obtain a classification of a class of locally finite simple weight modules as those induced from simple modules over a subalgebra isomorphic to a tensor product of noncommutative tori. As an application we describe simple weight modules over the quantized Weyl algebra.
The Suzuki algebra $A_{Nn}^{μλ}$ was introduced by Suzuki Satoshi in 1998, which is a class of cosemisimple Hopf algebras. It is not categorically Morita-equivalent to a group algebra in general. In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over the Suzuki algebra $A_{N\,2n}^{μλ}$ and investigates the Nichols algebras over those simple Yetter-Drinfeld modules. The involved finite dimensional Nichols algebras of diagonal type are of Cartan type $A_1$, $A_1\times A_1$, $A_2$, $A_2\times A_2$, Super type ${\bf A}_{2}(q;I_2)$ and the Nichols algebra ufo(8). Ther…
We show that certain twisting deformations of a family of supersolvable groups are simple as Hopf algebras. These groups are direct products of two generalized dihedral groups. Examples of this construction arise in dimensions 60 and p^2q^2, for prime numbers p, q with q dividing p-1. We also show that certain twisting deformation of the symmetric group is simple as a Hopf algebra. On the other hand, we prove that every twisting deformation of a nilpotent group is semisolvable. We conclude that the notions of simplicity and (semi)solvability of a semisimple Hopf algebra are not determined by i…
For the algebra $A$ in the title, its prime, primitive and maximal spectra are classified. The group of automorphisms of $A$ is determined. The simple unfaithful $A$-modules and the simple weight $A$-modules are classified.
We compute the finite-dimensional Nichols algebras over the sum of two simple Yetter-Drinfeld modules V and W over non-abelian quotients of a certain central extension of the dihedral group of eight elements or SL(2,3), and such that the Weyl groupoid of the pair (V,W) is finite. These central extensions appear in the classification of non-elementary finite-dimensional Nichols algebras with finite Weyl groupoid of rank two. We deduce new information on the structure of primitive elements of finite-dimensional Nichols algebras over groups.
In this paper we use Block's classification of simple modules over the first Weyl algebra to obtain a complete classification of simple weight modules, in particular, of Harish-Chandra modules, over the 1-spatial ageing algebra age(1). Most of these modules have infinite dimensional weight spaces and so far the algebra age(1) is the only Lie algebra having simple weight modules with infinite dimensional weight spaces for which such a classification exists. As an application we classify all simple weight modules over the (1+1)-dimensional space-time Schrodinger algebra S that have a simple age(…
