A Generalized Double Crossproduct and Drinfeld Double

2003 ◽  
Vol 26 (1) ◽  
pp. 159-180
Author(s):  
Shuan-hong Wang
Keyword(s):  
Drinfeld Double

scholarly journals Unitary spherical representations of Drinfeld doubles

2018 ◽  
Vol 2018 (742) ◽  
pp. 157-186 ◽  
Author(s):  
Yuki Arano
Keyword(s):  
Quantum Group ◽  
Lie Group ◽  
Complete Classification ◽  
Unitary Representations ◽  
Compact Lie Group ◽  
Drinfeld Double ◽  
Quantum Analogue ◽  
Property T ◽  
Simply Connected

Abstract We study irreducible spherical unitary representations of the Drinfeld double of the 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 \mathrm{SU}_{q}(3) , we give a complete classification of such representations. As an application, we show the Drinfeld double of the quantum group \mathrm{SU}_{q}(2n+1) has property (T), which also implies central property (T) of the dual of \mathrm{SU}_{q}(2n+1) .

scholarly journals Generalized cluster structure on the Drinfeld double of GL

2016 ◽  
Vol 354 (4) ◽  
pp. 345-349 ◽  
Author(s):  
Michael Gekhtman ◽  
Michael Shapiro ◽  
Alek Vainshtein
Keyword(s):  
Cluster Structure ◽  
Drinfeld Double

scholarly journals Periodic Staircase Matrices and Generalized Cluster Structures

2020 ◽  
Author(s):  
Misha Gekhtman ◽  
Michael Shapiro ◽  
Alek Vainshtein
Keyword(s):  
Difference Operators ◽  
Drinfeld Double ◽  
Similar Role ◽  
Geometric Type ◽  
Plücker Relations ◽  
Lie Brackets

Abstract As is well known, cluster transformations in cluster structures of geometric type are often modeled on determinant identities, such as short Plücker relations, Desnanot–Jacobi identities, and their generalizations. We present a construction that plays a similar role in a description of generalized cluster transformations and discuss its applications to generalized cluster structures in $GL_n$ compatible with a certain subclass of Belavin–Drinfeld Poisson–Lie brackets, in the Drinfeld double of $GL_n$, and in spaces of periodic difference operators.

Two-Dimensional R-Matrices — Descendants of Three-Dimensional R-Matrices

1997 ◽  
Vol 12 (19) ◽  
pp. 1393-1410 ◽  
Author(s):  
S. M. Sergeev
Keyword(s):  
Three Dimensional ◽  
Baxter Equation ◽  
Two Dimensional ◽  
Affine Algebra ◽  
Tetrahedron Equation ◽  
Drinfeld Double ◽  
R Matrix ◽  
Operator Solution ◽  
Dimensional Models ◽  
Three Dimensional Models

Finite layers of three-dimensional models can be regarded as two-dimensional with complicated multi-stated weights. The tetrahedron equation in 3D provides the Yang–Baxter equation for this composite weights in 2D. Such solutions of the Yang–Baxter equation are constructed for the simplest operator solution of the tetrahedron equation. These R-matrices can be regarded as a special projection of universal R-matrix for some Drinfeld double [Formula: see text], associated with the affine algebra [Formula: see text]. Usual R-matrix for [Formula: see text] is another projection of [Formula: see text].

scholarly journals Comodule algebras and 2-cocycles over the (Braided) Drinfeld double

2019 ◽  
Vol 21 (04) ◽  
pp. 1850045 ◽  
Author(s):  
Robert Laugwitz
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Monoidal Category ◽  
Crossed Product ◽  
Braided Monoidal Category ◽  
Drinfeld Double ◽  
Comodule Algebra ◽  
Category Of Modules ◽  
Cohomology Spaces ◽  
Quasitriangular Hopf Algebra

We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces a 2-cocycle over their Drinfeld double. This construction also has an analogue for braided Drinfeld doubles.

scholarly journals C⁎-Basic Construction from the Conditional Expectation on the Drinfeld Double

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Qiaoling Xin ◽  
Lining Jiang ◽  
Tianqing Cao
Keyword(s):  
Finite Group ◽  
Conditional Expectation ◽  
Crossed Product ◽  
Drinfeld Double ◽  
Basic Construction ◽  
A Finite Group

Let D(G) be the Drinfeld double of a finite group G and D(G;H) be the crossed product of C(G) and CH, where H is a subgroup of G. Then the sets D(G) and D(G;H) can be made C⁎-algebras naturally. Considering the C⁎-basic construction C⁎〈D(G),e〉 from the conditional expectation E of D(G) onto D(G;H), one can construct a crossed product C⁎-algebra C(G/H×G)⋊CG, such that the C⁎-basic construction C⁎〈D(G),e〉 is C⁎-algebra isomorphic to C(G/H×G)⋊CG.

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