On Combined Standard-Nonstandard or Hybrid (q,h)-Deformations

A q-analogue of the Wigner-Eckart theorem for the nonstandard q-deformed algebra U'_q(so(n))

The tensor product of vector and arbitrary representations of the nonstandard q-deformation U'_q(so(n)) of the universal enveloping algebra U(so(n)) of Lie algebra so(n) is defined. The Clebsch-Gordan coefficients of tensor product of vector and arbitrary classical or nonclassical type representations of q-algebra U'_q(so(n)) are found in an explicit form. The Wigner-Eckart theorem for vector operators is proved.

General Construction of Nonstandard $R_h$-matrices as Contraction Limits of $R_{q}$-matrices

A class of transformations of $R_q$-matrices is introduced such that the $q\to 1$ limit gives explicit nonstandard $R_{h}$-matrices. The transformation matrix is singular itself at $q\to 1$ limit. For the transformed matrix, the singularities, however, cancel yielding a well-defined construction. Our method can be implemented systematically for R-matrices of all dimensions and not only for $sl(2)$ but also for algebras of higher dimensions. Explicit constructions are presented starting with ${\cal U}_q(sl(2))$ and ${\cal U}_q(sl(3))$, while choosing $R_q$ for (fund. rep.)$\otimes$(arbitrary ir…

Quantum Jordanian twist

The quantum deformation of the Jordanian twist F_qJ for the standard quantum Borel algebra U_q(B) is constructed. It gives the family U_qJ(B) of quantum algebras depending on parameters x and h. In a generic point these algebras represent the hybrid (standard-nonstandard) quantization. The quantum Jordanian twist can be applied to the standard quantization of any Kac-Moody algebra. The corresponding classical r-matrix is a linear combination of the Drinfeld- Jimbo and the Jordanian ones. The obtained two-parametric families of Hopf algebras are smooth and for the limit values of the parameters…

Nonclassical type representations of the q-deformed algebra $U'_q({\rm so}_n)$

The nonstandard q-deformation $U'_q({\rm so}_n)$ of the universal enveloping algebra $U({\rm so}_n)$ has irreducible finite dimensional representations which are a q-deformation of the well-known irreducible finite dimensional representations of $U({\rm so}_n)$. But $U'_q({\rm so}_n)$ also has irreducible finite dimensional representations which have no classical analogue. The aim of this paper is to give these representations which are called nonclassical type representations. They are given by explicit formulas for operators of the representations corresponding to the generators of $U'_q({\r…

Jordanian Quantum (Super)Algebras $U_{h}(g)$ via Contraction Method and Mapping:Review

Recently, a class of transformations of $R_q$-matrices was introduced such that the $q \to 1$ limit gives explicit nonstandard $R_h$-matrices. The transformation matrix is singular as $q \to 1$. For the transformed matrix, the singularities, however, cancel yielding a well-defined construction. We have shown that our method can be implemented systematically on $R_q$ matrices of all dimensions of $U_q(sl(N)), U_Q(osp(1|2))$ and $U_q(sl(2|1))$ algebras. Explicit constructions are presented for $U_q(sl(2)), U_q(sl(3)), U_q(osp(1|2))$ and $U_q(sl(2|1))$ algebras, while choosing $R_q$ matrix for (f…

关于混合量子群的范畴 $\u003cspan style="font-style: italic;"\u003eO\u003c/span\u003e$

On the category $\mathcal{O}$ of a hybrid quantum group

该研究聚焦Gaitsgory提出的单位根处混合量子群范畴O，探讨其基本性质与形变，构造次generic形变下大投射对象的自同态代数并计算，证明形变后范畴O中心与Langlands对偶群仿射Grassmann流形上ζ-不动点轨迹的等变上同调存在代数同构。