Two "quantum enveloping algebras", here denoted by $U(R)$ and $U^{\sim}(R)$, are associated in [FRTa] and [FRTb] to any Yang-Baxter operator R. The latter is only a bialgebra, in general; the former is a Hopf algebra.
In this paper, we study the pointed Hopf algebras $U(R_Q)$, where $R_Q$ is the Yang-Baxter operator associated with the multi-parameter deformation of $GL_n$ supplied in [AST]; cf also [S,Re].
Some earlier results concerning these Hopf algebras $U(R_Q)$ were obtained in [To,CLMT,CM]; a related (but different) Hopf algebra was studied in [DP].
The main new results obtained here concerning these quantum enveloping algebras are: 1)We list, in an extremely explicit form, those quantum enveloping algebras $U(R_Q)$ which are finite-dimensional--let ${\mathcal U}$ denote the collection of these. 2)We verify that the pointed Hopf algebras in ${\mathcal U}$ are quasitriangular and of Cartan type $A_n$ in the sense of Andruskiewich-Schneider.
3)We show that every $U(R_Q)$ is a Hopf quotient of a double cross-product (hence, as asserted in 2), is quasitriangular if finite-dimensional.) 4) CAUTION: These Hopf algebras are NOT always cocycle twists of the standard 1-parameter deformation. This somewhat surprising fact is an immediate consequence of the data furnished here-- clearly a cocycle twist will not convert an infinite-dimensional Hopf algebra to a finite-dimensional one! Furthermore, these Hopf algebras in ${\mathcal U}$ are (it is proved) not all cocycle twists of each other. 5)We discuss also the case when the quantum determinant is central in $A(R_Q)$, so it makes sense to speak of a $Q$-deformation of the special linear group.
In this article we establish explicit isomorphisms between three realizations of quantum twisted affine algebra $U_q(A_2^{(2)})$: the Drinfeld ("current") realization, the Chevalley realization and the so-called $RLL$ realization, investigated by Faddeev, Reshetikhin and Takhtajan.
In this paper, we present the hidden symmetry behind the Faddeev-Reshetikhin-Takhtajan-Semenov-Tian-Shansky realization of quantum affine superalgebras $U_q(\hat {osp}(1,2))$ and add the q-Serre relation to the Drinfeld realization of $U_q(\hat {osp}(1,2))$ (math.QA/9712011) derived from the FRTS realization.
The quantum GL(n) of Faddeev-Reshetikhin-Takhtajan and Dipper-Donkin are realized geometrically by using double partial flag varieties. As a consequence, the difference of these two Hopf algebras is caused by a twist of a cocycle in the multiplication.
We study the q-Clifford algebras Cl_q(N,c), called FRT-Clifford algebras, introduced by Faddeev, Reshetikhin and Takhtajan. It is shown that Cl_q(N,c) acts on the q-exterior algebra Λ(O_q^N). Moreover, explicit formulas for the embedding of U_q(so_N) into Cl_q(N,c) and its relation to the vector and spin representations of U_q(so_N) are given and proved. Key Words: q-Clifford algebra, Drinfeld-Jimbo algebra, spin representation
We prove that the reflection equation (RE) algebra $\La_R$ associated with a finite dimensional representation of a quasitriangular Hopf algebra $\Ha$ is twist-equivalent to the corresponding Faddeev-Reshetikhin-Takhtajan (FRT) algebra. We show that $\La_R$ is a module algebra over the twisted tensor square \twist{$\Ha$}{$\Ha$} and the double $\D(\Ha)$. We define FRT- and RE-type algebras and apply them to the problem of equivariant quantization on Lie groups and matrix spaces.
We develop the approach of Faddeev, Reshetikhin, Takhtajan [1] and of Majid [2] that enables one to associate a quasitriangular Hopf algebra to every regular invertible constant solution of the quantum Yang-Baxter equations. We show that such a Hopf algebra is actually a quantum double.
Fully braided analog of Faddeev-Reshetikhin-Takhtajan construction of quasitriangular bialgebra $A(X,R)$ is proposed. For given pairing $C$ factor-algebra $A(X,R;C)$ is a dual quantum braided group. Corresponding inhomogeneous quantum group is obtained as a result of generalized bosonization. Construction of first order bicovariant differential calculus is proposed.
We translate effectively our earlier quantum constructions to the classical language and using Yang-Baxterisation of the Faddeev-Reshetikhin-Takhtajan algebra are able to construct Lax operators and associated $r$-matrices of classical integrable models. Thus new as well as known lattice systems of different classes are generated including new types of collective integrable models and canonical models with nonstandard $r$ matrices.
For every element w in the Weyl group of a simple Lie algebra g, De Concini, Kac, and Procesi defined a subalgebra U_q^w of the quantized universal enveloping algebra U_q(g). The algebra U_q^w is a deformation of the universal enveloping algebra U(n_+\cap w.n_-). We construct smash products of certain finite-type De Concini-Kac-Procesi algebras to obtain ones of affine type; we have analogous constructions in types A_n and D_n. We show that the multiplication in the affine type De Concini-Kac-Procesi algebras arising from this smash product construction can be twisted by a cocycle to produce c…
De Concini, Kac, and Procesi defined a family of subalgebras Uq[w] of the quantized enveloping algebra Uq(g) associated to elements w in the Weyl group of a simple Lie algebra g. These algebras are called quantum Schubert cell algebras. We show that, up to a mild cocycle twist, quotients of certain quantum Schubert cell algebras of types E_6 and E_6^{(1)} map isomorphically onto distinguished subalgebras of the Faddeev-Reshetikhin-Takhtajan universal bialgebra associated to the braiding on the quantum half-spin representation of Uq(so_{10}). We identify the quotients as those obtained by facto…
Let $H$ be a Hopf algebra that is $\mathbb Z$-graded as an algebra. We provide sufficient conditions for a 2-cocycle twist of $H$ to be a Zhang twist of $H$. In particular, we introduce the notion of a twisting pair for $H$ such that the Zhang twist of $H$ by such a pair is a 2-cocycle twist. We use twisting pairs to describe twists of Manin's universal quantum groups associated to quadratic algebras and provide twisting of solutions to the quantum Yang-Baxter equation via the Faddeev-Reshetikhin-Takhtajan construction.
This paper is a continuation of "Quantization of Lie bialgebras, III" (q-alg/9610030, revised version). In QLB-III, we introduced the Hopf algebra F(R)_\z associated to a quantum R-matrix R(z) with a spectral parameter, and a set of points \z=(z_1,...,z_n). This algebra is generated by entries of a matrix power series T_i(u), i=1,...,n,subject to Faddeev-Reshetikhin-Takhtajan type commutation relations, and is a quantization of the group GL_N[[t]]. In this paper we consider the quotient F_0(R)_\z of F(R)_\z by the relations \qdet_R(T_i)=1, where \qdet_R is the quantum determinant associated to…
A coloured braid group representation (CBGR) is constructed with the help of some modified universal ${\cal R}$-matrix, associated to $U_q(gl(2))$ quantised algebra. Explicit realisation of Faddeev-Reshetikhin-Takhtajan (FRT) algebra is built up for this CBGR and new solutions of quantum Yang-Baxter equation are subsequently found through Yang-Baxterisation of FRT algebra. These solutions are interestingly related to nonadditive type quantum $R$-matrix and have a nontrivial $q\rightarrow 1$ limit. Lax operators of several concrete integrable models, which may be considered as some `coloured' e…
We prove an integral version of the Schur--Weyl duality between the specialized Birman--Murakami--Wenzl algebra $B_n(-q^{2m+1},q)$ and the quantum algebra associated to the symplectic Lie algebra sp_{2m}. In particular, we deduce that this Schur--Weyl duality holds over arbitrary (commutative) ground rings, which answers a question of Lehrer and Zhang [Strongly multiplicity free modules for Lie algebras and quantum groups, J. Algebra (1) 306 (2006), 138--174] in the symplectic case. As a byproduct, we show that, as $Z[q,q^{-1}]$-algebra, the quantized coordinate algebra defined by Kashiwara is…
Coloured Hopf algebras, related to the coloured Yang-Baxter equation, are reviewed, as well as their duals. The special case of coloured quantum universal enveloping algebras provides a coloured extension of Drinfeld and Jimbo formalism. The universal $\cal T$-matrix is then generalized to the coloured context, and shown to lead to an algebraic formulation of the coloured RTT-relations, previously proposed by Basu-Mallick as part of a coloured extension of Faddeev, Reshetikhin, and Takhtajan approach to quantum groups and quantum algebras.
We show that the 1+1-dimensional quantum superplane introduced by Manin is a quantum supergroup according to the Faddeev-Reshetikhin-Takhtajan approach. We give its supermatrix element, its corresponding R-matrix and its Hopf structure. This new point of view allows us, first, to realize its dual Hopf superalgebra starting from postulated initial pairings. Second, we construct a right-invariant differential calculus on it and then deduce the corresponding quantum Lie superalgebra which as a commutation super-algebra appears classical, and as Hopf structure is a non-cocommutative q-deformed one…
We provide a Faddeev-Reshetikhin-Takhtajan's RTT approach to the quantum group Fun(GL_{r,s}(n)) and the quantum enveloping algebra U_{r,s}(gl_n) corresponding to the two-parameter R-matrix. We prove that the quantum determinant det_{r,s}T is a quasi-central element in Fun(GL_{r,s}(n)) generalizing earlier results of Dipper-Donkin and Du-Parshall-Wang. The explicit formulation provides an interpretation of the deforming parameters, and the quantized algebra U_{r,s}(R) is identified to U_{r,s}(gl_n) as the dual algebra. We then construct n-1 quasi-central elements in U_{r,s}(R) which are analogu…
We provide an alternative approach to the Faddeev-Reshetikhin-Takhtajan presentation of the quantum group U_q(g), with L-operators as generators and relations ruled by an R-matrix. We look at U_q(g) as being generated by the quantum Borel subalgebras U_q(b_+) and U_q(b_-), and we use the standard presentation of the latters as quantum function algebras. When g = gl(n) these Borel quantum function algebras are generated by the entries of a triangular q-matrix, thus eventually U_q(gl(n)) is generated by the entries of an upper triangular and a lower triangular q-matrix, which share the same diag…
We investigate the algebra $F_q(N)$ introduced by Faddeev, Reshetikhin and Takhadjian. In case $q$ is a primitive root of unity the degree, the center, and the set of irreducible representations are found. The Poisson structure is determined and the De Concini-Kac-Procesi Conjecture is proved for this case. In the case of $q$ generic, the primitive ideals are described. A related algebra studied by Oh is also treated.
We briefly report our application of a version of noncommutative geometry to the quantum Euclidean space $R^N_q$, for any $N \ge 3$; this space is covariant under the action of the quantum group $SO_q(N)$, and two covariant differential calculi are known on it. More precisely, we describe how to construct in a Cartan `moving-frame formalism' the metric, two covariant derivatives, the Dirac operator, the frame, the inner derivations dual to the frame elements, for both of these calculi. The components of the frame elements in the basis of differentials provide a `local realization' of the Fadde…
We extend the notion of dually conjugate Hopf (super)algebras to the coloured Hopf (super)algebras ${\cal H}^c$ that we recently introduced. We show that if the standard Hopf (super)algebras ${\cal H}_q$ that are the building blocks of ${\cal H}^c$ have Hopf duals ${\cal H}_q^*$, then the latter may be used to construct coloured Hopf duals ${\cal H}^{c*}$, endowed with coloured algebra and antipode maps, but with a standard coalgebraic structure. Next, we review the case where the ${\cal H}_q$'s are quantum universal enveloping algebras of Lie (super)algebras $U_q(g)$, so that the correspondin…
We review the Reshetikhin-Turaev approach to construction of non-compact knot invariants involving R-matrices associated with infinite-dimensional representations, primarily those made from Faddeev's quantum dilogarithm. The corresponding formulas can be obtained from modular transformations of conformal blocks as their Kontsevich-Soibelman monodromies and are presented in the form of transcendental integrals, where the main issue is manipulation with integration contours. We discuss possibilities to extract more explicit and handy expressions which can be compared with the ordinary (compact) …
Every fusion category C that is k-linear over a suitable field k, is the category of finite-dimensional comodules of a Weak Hopf Algebra H. This Weak Hopf Algebra is finite-dimensional, cosemisimple and has commutative bases. It arises as the universal coend with respect to the long canonical functor ω:C->Vect_k. We show that H is a quotient H=H[G]/I of a Weak Bialgebra H[G] which has a combinatorial description in terms of a finite directed graph G that depends on the choice of a generator M of C and on the fusion coefficients of C. The algebra underlying H[G] is the path algebra of the quive…
