We study the possibility to establish $L$-operator's formalism by Faddeev-Reshetikhin-Takhtajan-Semenov-Tian-Shansky (FRST) for quantized current algebras, that is, for quantum affine algebras in the ''new realization '' by V. Drinfeld with the corresponding Hopf algebra structure and for their Yangian counterpart. We establish this formalism using the twisting procedure by Tolstoy and the second author and explain the problems which FRST approach encounter for quantized current algebras. We show also that, for the case of $U_q(\hat {\frak sl}_n)$, entries of the L-operators of FRTS type give the Drinfeld current operators for the non-simple roots, which we discovered recently. As an application we deduce the commutation relations between these current operators for $U_q(\hat {\frak sl}_3)$.
The particles with a scattering matrix R(x) are defined as operators $Φ_i(z)$ satisfying the relation $ R_{i,j}^{j',i'}(x_1/x_2) Φ_{i'}(x_1)Φ_{j'}(x_2)= Φ_i(x_2)Φ_j(x_1)$. The algebra generated by those operators is called a Zamolochikov algebra. We construct a new Hopf algebra by adding half of the FRTS construction of a quantum affine algebra with this R(x). Then we double it to obtain a new Hopf algebra such that the full FRTS construction of a quantum affine algebra is a Hopf subalgebra inside. Drinfeld realization of quantum affine algebras is included as an example. This is a further gen…
We give a unified construction of quantum groups, q-Boson algebras and quantized Weyl algebras and an action of quantum groups on quantized Weyl algebras. This enables us to give a conceptual proof of the semi-simplicity of the category $\mathcal{O}(B_q)$ introduced by T.Nakashima and the classification of all simple objects in it.
We show that the finite-dimensional fundamental module over a quantized affine algebra is isomorphic to a Demazure module of a higest weight module of level one as a module over a quantized classical universal enveloping algebra.
Belov-Kanel and Kontsevich conjectured that the group of automorphisms of the n'th Weyl algebra and the group of polynomial symplectomorphisms of C^2 are canonically isomorphic. We discuss how this conjecture can be approached by means of (second) quantized Weyl algebras at roots of unity.
A Bernstein-Gelfand-Gelfand resolution for arbitrary Kac-Moody algebras and arbitrary subsets of the set of simple roots is proven. Moreover, quantum group analogs of the Bernstein-Gelfand-Gelfand resolution for symmetrizable Kac-Moody algebras are established. For quantized enveloping algebras with fixed complex nonzero deformation parameter q exactness is proven for all q which are not a root of unity.
We construct a monomial basis of the positive part of the quantized enveloping algebra associated to a finite-dimensional simple Lie algebra. As an application we give a simple proof of the existence and uniqueness of the canonical basis of the positive part.
Starting from bosonization, we study the operator that commute or commute up-to a total difference with of any quantized screen operator of a free field. We show that if there exists a operator in the form of a sum of two vertex operators which has the simplest correlation functions with the quantized screen operator, namely a function with one pole and one zero, then, the screen operator and this operator are uniquely determined, and this operator is the quantized virasoro algebra. For the case when the screen is a fermion, there are a family of this kind of operator, which give new algebraic…
For quantized universal enveloping algebras we construct weight modules by inducing representations of the centralizer of the Cartan subalgebra in the quantized universal enveloping algebra. The induced modules arising from finite-dimensional weight modules the centralizer algebra are studied. In particular, we study the induction of one-dimensional modules, and this is related to the study of commutative subalgebras of the centralizer algebra. For the special case of $U_q(\mathfrak{sl}(2,\mathbb{C}))$ we show that we get the admissible unitary representations corresponding to the non-compact …
For the Borel part of a quantized enveloping algebra we classify all right coideal subalgebras for which the intersection with the coradical is a Hopf algebra. The result is expressed in terms of characters of the subalgebras $U^+[w]$ of the quantized enveloping algebra, introduced by de Concini, Kac, and Procesi for any Weyl group element $w$. We explicitly determine all characters of $U^+[w]$ building on recent work by Yakimov on prime ideals of $U^+[w]$ which are invariant under a torus action. Key words: Quantum groups, coideal subalgebras
We introduce an integral form U of the quantized enveloping algebra of sl_2. The algebra U is just large enough so that the quasi-R-matrix is contained in a completion of U\otimes U. We study several completions of the algebra U, and determine their centers. This study is motivated by a study of integrality properties of the quantum sl_2 invariants of links and integral homology spheres.
We prove the decomposition conjecture of Leclerc and Thibon for the Schur algebra. We also give a new approach to the Lusztig conjecture for the dimension of the simple U(sl_k)-modules at roots of unity via canonical bases of the Hall algebra.
For a class of pointed Hopf algebras including the quantized enveloping algebras, we discuss cleft extensions, cocycle deformations and the second cohomology. We present such a non-standard method of computing the abelian second cohomology that derives information from the non-abelian second cohomology classifying cleft extensions. As a sample computation, a quantum analogue of Whitehead's second lemma for Lie-algebra cohomology is proved.
We use the fusion construction in the twisted quantum affine algebras to obtain a unified method to deform the wedge product for classical Lie algebras. As a byproduct we uniformly realize all non-spin fundamental modules for quantized enveloping algebras of classical types, and show that they admit natural crystal bases as modules for the (derived) twisted quantum affine algebra. These crystal bases are parametrized in terms of the $q$-wedge products.
We construct braid group actions on coideal subalgebras of quantized enveloping algebras which appear in the theory of quantum symmetric pairs. In particular, we construct an action of the semidirect product of Z^n and the classical braid group in n strands on the coideal subalgebra corresponding to the symmetric pair (sl_{2n}(C), sp_{2n}(C)). This proves a conjecture by Molev and Ragoucy. We expect similar actions to exist for all symmetric Lie algebras. The given actions are inspired by Lusztig's braid group action on quantized enveloping algebras and are defined explicitly on generators. Br…
For a quantized enveloping algebra of a complex semisimple Lie algebra with deformation parameter not a root of unity, we classify all homogeneous right coideal subalgebras. Any such right coideal subalgebra is determined uniquely by a triple consisting of two elements of the Weyl group and a subset of the set of simple roots satisfying some natural conditions. The essential ingredients of the proof are the Lusztig automorphisms and the classification of homogeneous right coideal subalgebras of the Borel Hopf subalgebras of quantized enveloping algebras obtained previously by H.-J. Schneider a…
A generalized Hopf algebra structure for the positive (negative) part of the Drinfeld-Jimbo quantum group of type A_n is established without make any use of the usual deformation of the abelian part of sl_{n+1}.
We study the representations of the W-algebra W(g) associated to an arbitrary finite-dimensional simple Lie algebra g via the quantized Drinfeld-Sokolov reductions. The characters of irreducible representations with non-degenerate highest weights are expressed by Kazhdan-Lusztig polynomials. The irreduciblity conjecture of Frenkel, Kac and Wakimoto is proved completely for the "-" reduction and partially for the "+" reduction. In particular, the existence of the minimal series representations (= the modular invariant representations) of W(g) is proved.
In this paper we describe the right coideal subalgebras containing all group-like elements of the two-parameter quantum groups Uq(g) and uq(g), where g is a simple Lie algebra of type G2. As a consequence, we determine that there are precisely 60 different right coideal subalgebras containing all group-like elements.
In this paper, we extend the Drinfeld current realization of quantum affine algebras $U_q(\hat {\gg})$ and of the Yangians in several directions: we construct current operators for non-simple roots of ${\gg}$, define a new braid group action in terms of the current operators and describe the universal R-matrix for the corresponding ``Drinfeld'' comultiplication in the form of infinite product and in the form of certain integrals over current operators.
We consider the BGG category $\mathcal{O}$ of a quantized universal enveloping algebra $U_q(\mathfrak{g})$. We call a module $M\in \mathcal{O}$ tensor-closed if $M\otimes N\in\mathcal{O}$ for any $N\in \mathcal{O}$. In this paper we prove that $M\in \mathcal{O}$ is tensor-closed if and only if $M$ is finite dimensional. The method used in this paper applies to the unquantized case as well.
We develop an operator commutant version of the First Fundamental Theorem of invariant theory for the general linear quantum group $U_q(\mathfrak{gl}_n)$ by using a double centralizer property inside a quantized Clifford algebra. In particular, we show that $U_q(\mathfrak{gl}_m)$ generates the centralizer of the $U_q(\mathfrak{gl}_n)$-action on the tensor product of braided exterior algebras $\bigwedge_q(\mathbb{C}^n)^{\otimes m}$. We obtain a multiplicity-free decomposition of the $U_q(\mathfrak{gl}_n) \otimes U_q(\mathfrak{gl}_m)$-module $\bigwedge_q(\mathbb{C}^n)^{\otimes m} \cong \bigwedge…
Let $\frak{g}$ be a finite dimensional simple complex Lie algebra and $U=U_q(\frak{g})$ the quantized enveloping algebra (in the sense of Jantzen) with $q$ being generic. In this paper, we show that the center $Z(U_q(\frak{g}))$ of the quantum group $U_q(\frak{g})$ is isomorphic to a monoid algebra, and that $Z(U_q(\frak{g}))$ is a polynomial algebra if and only if $\frak{g}$ is of type $A_1, B_n, C_n, D_{2k+2}, E_7, E_8, F_4$ or $G_2.$ Moreover, in case $\frak{g}$ is of type $D_{n}$ with $n$ odd, then $Z(U_q(\frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra in $n+1$ vari…
We study certain Poisson structures related to quantized enveloping algebras. In particular, we give a description of the Poisson structure of a certain manifold associated to the ring of differential operators.
We define admissible quasi-Hopf quantized universal enveloping (QHQUE) algebras by h-adic valuation conditions. We show that any QHQUE algebra is twist-equivalent to an admissible one. We prove a related statement: any associator is twist-equivalent to a Lie associator. We attach a quantized formal series algebra to each admissible QHQUE algebra and study the resulting Poisson algebras.
We study the algebra $U_ζ$ obtained via Lusztig's `integral' form [Lu 1, 2] of the generic quantum algebra for the Lie algebra $\frak {g=sl}_2$ modulo the two-sided ideal generated by $K^l-1$. We show that $U_ζ$ is a smash product of the quantum deformation of the restricted universal enveloping algebra $\bold u_ζ$ of $\frak g$ and the ordinary universal enveloping algebra $U$ of $\frak g$, and we compute the primitive (= prime) ideals of $\Uz$. Next we describe a decomposition of $\bold u_ζ$ into the simple $U$- submodules, which leads to an explicit formula for the center and the indecomposa…
We develop a method to give presentations of quantized function algebras of complex reductive groups. In particular, we give presentations of quantized function algebras of automorphism groups of finite dimensional simple complex Lie algebras.
We define two subalgebras which can be seen as the quantization of the coordinate rings of the unipotent radical of the standard positive (respectively negative) Borel subgroup of $SL_{n+1}$. We give a presentation for these algebras and show they are isomorphic. Moreover, we show that these algebras are the conivariants of a natural $O_q(T)$-coaction on $O_q(B^\pm)$. Finally, using a dual pairing, we show that these algebras are isomorphic to the quantum nilpotent Lie subalgebra of $U_q(\mathfrak{sl}_{n+1})$.
In this article the quantized matrix algebras as in the title have been studied at a root of unity. A full classification of simple modules over such quantized matrix algebras of rank $2$ along with a class of finite dimensional indecomposable modules are given.
In this paper, we give a criterion on the semisimplicity of quantized walled Brauer algebras $\mathscr B_{r,s}$ and classify its simple modules over an arbitrary field $κ$.
We obtain a presentation by generators and relations for generalized Schur algebras and their quantizations. This extends earlier results obtained in the type A case. The presentation is compatible with Lusztig's modified form of a quantized enveloping algebra. We show that generalized Schur algebras inherit a canonical basis from the modified form of the quantized enveloping algebra, that this gives them a cellular structure, and thus they are quasihereditary over a field.
Holonomy R-matrices parametrized by finite-dimensional representations are constructed for quantized universal enveloping algebras of simple Lie algebras at roots of 1.
We obtain a presentation of quantum Schur algebras (over the field Q(v)) by generators and relations. This presentation is compatible with the usual presentation of the quantized universal enveloping algebra of the Lie algebra gl(2). We also locate the ``integral'' form of the quantum Schur algebra within the presented algebra and show it has a basis which is closely related to Lusztig's basis of the integral form of the quantized enveloping algebra.
We give a systematic description of many monomial bases for a given quantized enveloping algebra and of many integral monomial bases for the associated Lusztig $\mathbb Z[v,v^{-1}]$-form. The relations between monomial bases, PBW bases and canonical bases are also discussed.
The Dolbeault complex of a quantized compact Hermitian symmetric space is expressed in terms of the Koszul complex of a braided symmetric algebra of Berenstein and Zwicknagl. This defines a spectral triple quantizing the Dolbeault-Dirac operator associated to the canonical spin^c structure.
In this paper, we establish the connection between the quantized W-algebra of ${\frak sl}(2,1)$ and quantum parafermions of $U_q(\hat {\frak sl}(2))$ that a shifted product of the two quantum parafermions of $U_q(\hat {\frak sl}(2))$ generates the quantized W-algebra of ${\frak sl}(2,1)$.
Let $r,s$ be two parameters chosen from $\C^{\ast}$ such that $r^{m}s^{n}=1$ implies $m=n=0$. We compute the derivations of the two-parameter quantized enveloping algebra $\U$ and calculate its first degree Hochschild cohomology group. We further determine the group of algebra automorphisms for the two-parameter Hopf algebra $\V$. As a result, we determine the group of Hopf algebra automorphisms for $\V$.
In this note we solve the isomorphism problem for the multiparameter quantized Weyl algebras, in the case when none of the deformation parameters q_i is a root of unity, over an arbitrary field.
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…
In this paper, we study and classify Hilbert space representations of cross product *-algebras of the quantized enveloping algebra $U_q(e_2)$ with the coordinate algebras $O(E_q(2))$ of the quantum motion group and $O(\C_q)$ of the complex plane, and of the quantized enveloping algebra $U_q(su_{1,1})$ with the coordinate algebras $O(SU_q(1,1))$ of the quantum group $SU_q(1,1)$ and $O(U_q)$ of the quantum disc. Invariant positive functionals and the corresponding Heisenberg representations are explicitely described.
We quantize the coordinate ring of the moduli space of B-bundles on the elliptic curve. Here B is a Borel subgroup of some semisimple Lie group. We construct some representations of these algebras and study intertwining operators for these representations. We apply our constructions to produce some objects: the elliptic Belavin R-matrix, the quantization of the algebra of functions on the Grassmannian, some generalized elliptic R-matrix. We consider also the affine case and write down the explicit formula for commuting elements.
We propose a quantum analogue of a Tits-Kantor-Koecher algebra with a Jordan torus as an coordinated algebra by looking at the vertex operator construction over a Fock space.
We study a monoid associated to complex semisimple Lie algebras, called the quantic monoid. Its monoid ring is shown to be isomorphic to a degenerate quantized enveloping algebra. Moreover, we provide normal forms and a straightening algorithm for this monoid. All these results are proved by a realization in terms of representations of quivers, namely as the monoid of generic extensions of a quiver with automorphism.
A quantized symplectic oscillator algebra of rank 1 is a PBW deformation of the smash product of the quantum plane with U_q(sl_2). We study its representation theory, and in particular, its category O.
Explicit formulas of the universal $R$-matrix are given for all quantized nontwisted rank 3 affine Lie algebras $U_q(A_2^{(1)})\,,~U_q(C_2^{(1)})$ and $U_q(G_2^{(1)})$.
Recently, Kuniba, Okado and Yamada proved that the transition matrix of PBW-type bases of the positive-half of a quantized universal enveloping algebra $U_q(\mathfrak{g})$ coincides with a matrix coefficients of the intertwiner between certain irreducible modules over the corresponding quantized coordinate ring $A_q(\mathfrak{g})$, introduced by Soibelman. In the present article, we give a new proof of their result, by using representation theory of the $q$-boson algebra, and the Drinfeld paring of $U_q(\mathfrak{g})$.
It is known that a generalized $q$-Schur algebra may be constructed as a quotient of a quantized enveloping algebra $\UU$ or its modified form $\dot{\UU}$. On the other hand, we show here that both $\UU$ and $\dot{\UU}$ may be constructed within an inverse limit of a certain inverse system of generalized $q$-Schur algebras. Working within the inverse limit $\hat{\UU}$ clarifies the relation between $\dot{\UU}$ and $\UU$. This inverse limit is a $q$-analogue of the linear dual $R[G]^*$ of the coordinate algebra of a corresponding linear algebraic group $G$.
In this paper, we establish explicit relationship between decomposition numbers of quantized walled Brauer algebras and those for either Hecke algebras associated to certain symmetric groups or (rational) $q$-Schur algebras over a field $κ$. This enables us to use Ariki's result \cite{Ar} and Varagnolo-Vasserot's result \cite{VV} to compute such decomposition numbers via inverse Kazhdan-Lusztig polynomials associated with affine Weyl groups of type $A$ if the ground field is $\mathbb C$.
Let B(\infty) be the crystal corresponding to the nilpotent part of a quantized Kac-Moody algebra. We suggest a general way to represent B(\infty) as the set of integer solutions of a system of linear inequalities. As an application, we treat in a unified manner all Kac-Moody algebras of rank 2 (sharpening the result by Kashiwara), as well as the algebras of types A_n and A_{n-1}^{(1)}.
The modified quantized enveloping algebra $\dot{\mathbf{U}}$ has a remarkable canonical basis, which was introduced by Lusztig. In this paper, we give an explicit description of all elements of the canonical basis of $\dot{\mathbf{U}}$ for type $A_{2}$.
We will give an explicit description of the center of the De Concini-Kac type specialization of a quantized enveloping algebra at an even root of unity. The case of an odd root of unity was already dealt with by De Concini-Kac-Procesi. Our description in the even case is similar to but a little more complicated than the odd case.
A natural family of quantized matrix algebras is introduced. It includes the two best studied such. Located inside ${\s U}_q(A_{2n-1})$, it consists of quadratic algebras with the same Hilbert series as polynomials in $n^2$ variables. We discuss their general properties and investigate some members of the family in great detail with respect to associated varieties, degrees, centers, and symplectic leaves. Finally, the space of rank r matrices becomes a Poisson submanifold, and there is an associated tensor category of $\rank\leq r$ matrices.
By using cocycle deformation, we construct a certain class of Hopf algebras, containing the quantized enveloping algebras and their analogues, from what we call pre-Nichols algebras. Our construction generalizes in some sense the known construction by (generalized) quantum doubles, but unlike in the known situation, it saves us from difficulties in checking complicated defining relations.
