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To facilitate the parallel development of the structures and properties of two-parameter quantum groups with those of one-parameter quantum groups, this paper primarily elucidates the interrelations and distinctions between the derivation algebras of these two types of quantum groups. Additionally, we also proposed a method for deriving the derivation algebra of one-parameter quantum groups from known two-parameter quantum group derivations, and vice versa.

We construct finite$R$ -matrices for the first fundamental representation$V$of two-parameter quantum groups$U_{r,s}(\\mathfrak{g})$for classical$\\mathfrak{g}$ , both through the decomposition of$V\\otimes V$into irreducibles$U_{r,s}(\\mathfrak{g})$ -submodules as well as by evaluating the universal$R$ -matrix. The latter is crucially based on the construction of dual PBW-type bases of$U^{\\pm}_{r,s}(\\mathfrak{g})$consisting of the ordered products of quantum root vectors defined via$(r,s)$ -bracketings and combinatorics of standard Lyndon words. We further derive explicit formulas for affine$R$ -matrices, both through the Yang-Baxterization technique of [Internat. J. Modern Phys. A 6 (1991), 3735-3779] and as the unique intertwiner between the tensor product of$V(u)$and$V(v)$ , viewed as modules over two-parameter quantum affine algebras$U_{r,s}(\\widehat{\\mathfrak{g}})$for classical$\\mathfrak{g}$ . The latter generalizes the formulas of [Comm. Math. Phys. 102 (1986), 537-547] for one-parametric quantum affine algebras.

In this paper, the simple modules for the quantum superalgebra Ur,s(osp(1,2)) with two parameters at the root of unity (i.e., q=(rs−1)1/2 is a root of unity) are completely determined up to isomorphism. Furthermore, the classification of finite-dimensional simple modules over the restricted quantum superalgebra ¯Ur,s(osp(1,2))is given.

To facilitate the parallel development of the structures and properties of two-parameter quantum groups with those of one-parameter quantum groups, this paper primarily elucidates the interrelations and distinctions between the derivation algebras of these two types of quantum groups. Additionally, we also proposed a method for deriving the derivation algebra of one-parameter quantum groups from known two-parameter quantum group derivations, and vice versa.

Let U be the two-parameter quantized enveloping algebra U r , s ( s l 2 ) and F ( U ) the locally finite subalgebra of U under the adjoint action. The aim of this paper is to determine some ring-theoretical properties of F ( U ) in the case when rs −1 is not a root of unity. Then we describe the annihilator ideals of finite dimensional simple modules of U by generators.

We compute the derivations of the positive part of the two-parameter quantum group U r , s ( B 3 ) and show that the Hochschild cohomology group of degree 1 of this algebra is a three-dimensional vector space over the base field C. We also compute the groups of (Hopf) algebra automorphisms of the augmented two-parameter quantized enveloping algebra Ǔ r , s ≥0 ( B 3 ).

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) We study the representations of the restricted two-parameter quantum groups of types ... and ... . For these restricted two-parameter quantum groups, we give some explicit conditions which guarantee that a simple module can be factored as the tensor product of a one-dimensional module with a module that is naturally a module for the quotient by central group-like elements. That is, given ... a primitive th root of unity, the factorization of simple ... -modules is possible, if and only if ... . (ProQuest: ... denotes formulae and non-USASCII text omitted)

We provide a group-theoretic realization of two-parameter quantum toroidal algebras using finite subgroups of SL2(C)SL_2(\\mathbb C) via McKay correspondence. In particular our construction contains the vertex representation of the two-parameter quantum affine algebras of ADEADE types as special subalgebras.