scholarly journals COIDEAL SUBALGEBRAS IN QUANTUM AFFINE ALGEBRAS

2003 ◽  
Vol 15 (08) ◽  
pp. 789-822 ◽  
Author(s):  
A. I. MOLEV ◽  
E. RAGOUCY ◽  
P. SORBA
Keyword(s):  
Algebra Structure ◽  
Quantum Affine Algebra ◽  
Affine Algebra ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Defining Relations ◽  
Quantized Enveloping Algebras ◽  
Central Elements ◽  
Quantized Enveloping Algebra ◽  
Quantum Determinant

We introduce two subalgebras in the type A quantum affine algebra which are coideals with respect to the Hopf algebra structure. In the classical limit q→1 each subalgebra specializes to the enveloping algebra [Formula: see text], where [Formula: see text] is a fixed point subalgebra of the loop algebra [Formula: see text] with respect to a natural involution corresponding to the embedding of the orthogonal or symplectic Lie algebra into [Formula: see text]. We also give an equivalent presentation of these coideal subalgebras in terms of generators and defining relations which have the form of reflection-type equations. We provide evaluation homomorphisms from these algebras to the twisted quantized enveloping algebras introduced earlier by Gavrilik and Klimyk and by Noumi. We also construct an analog of the quantum determinant for each of the algebras and show that its coefficients belong to the center of the algebra. Their images under the evaluation homomorphism provide a family of central elements of the corresponding twisted quantized enveloping algebra.

scholarly journals TWO DESCRIPTIONS OF THE QUANTUM AFFINE ALGEBRA Uv() VIA HALL ALGEBRA APPROACH

2011 ◽  
Vol 54 (2) ◽  
pp. 283-307 ◽  
Author(s):  
IGOR BURBAN ◽  
OLIVIER SCHIFFMANN
Keyword(s):  
Integral Form ◽  
Derived Categories ◽  
Quantum Affine Algebra ◽  
Hall Algebra ◽  
Affine Algebra ◽  
Coherent Sheaves ◽  
Enveloping Algebra ◽  
Algebra Approach ◽  
Quantized Enveloping Algebra

AbstractWe compare the reduced Drinfeld doubles of the composition subalgebras of the category of representations of the Kronecker quiver and the category of coherent sheaves on ℙ1. Using this approach, we show that the Drinfeld–Beck isomorphism for the quantized enveloping algebra Uv() is a corollary of an equivalence between the derived categories Db(Rep()) and Db(Coh(ℙ1)). This technique allows to reprove several results on the integral form of Uv().

scholarly journals TWO PERMANENTS IN THE UNIVERSAL ENVELOPING ALGEBRAS OF THE SYMPLECTIC LIE ALGEBRAS

2009 ◽  
Vol 20 (03) ◽  
pp. 339-368 ◽  
Author(s):  
MINORU ITOH
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Universal Enveloping Algebra ◽  
Irreducible Representations ◽  
General Linear ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Central Elements ◽  
Orthogonal Lie Algebra

This paper presents new generators for the center of the universal enveloping algebra of the symplectic Lie algebra. These generators are expressed in terms of the column-permanent and it is easy to calculate their eigenvalues on irreducible representations. We can regard these generators as the counterpart of central elements of the universal enveloping algebra of the orthogonal Lie algebra given in terms of the column-determinant by Wachi. The earliest prototype of all these central elements is the Capelli determinants in the universal enveloping algebra of the general linear Lie algebra.

scholarly journals SYMMETRIES AND INVARIANTS OF TWISTED QUANTUM ALGEBRAS AND ASSOCIATED POISSON ALGEBRAS

2008 ◽  
Vol 20 (02) ◽  
pp. 173-198 ◽  
Author(s):  
A. I. MOLEV ◽  
E. RAGOUCY
Keyword(s):  
Poisson Bracket ◽  
Quantum Algebras ◽  
Poisson Algebras ◽  
Triangular Matrices ◽  
Polynomial Invariants ◽  
Enveloping Algebra ◽  
Quantized Enveloping Algebras ◽  
Quantized Enveloping Algebra ◽  
Group B ◽  
Mathematical Literature

We construct an action of the braid group BN on the twisted quantized enveloping algebra [Formula: see text] where the elements of BN act as automorphisms. In the classical limit q → 1, we recover the action of BN on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action preserves the Poisson bracket on the space of polynomials which was introduced by Nelson and Regge in their study of quantum gravity and rediscovered in the mathematical literature. Furthermore, we construct a Poisson bracket on the space of polynomials associated with another twisted quantized enveloping algebra [Formula: see text]. We use the Casimir elements of both twisted quantized enveloping algebras to reproduce and construct some well-known and new polynomial invariants of the corresponding Poisson algebras.

scholarly journals A quantum cluster algebra approach to representations of simply laced quantum affine algebras

2020 ◽  
Author(s):  
Léa Bittmann
Keyword(s):  
Cluster Algebra ◽  
Irreducible Representations ◽  
Algebra Structure ◽  
Quantum Affine Algebra ◽  
Affine Algebra ◽  
Grothendieck Ring ◽  
Finite Dimensional ◽  
Quantum Cluster Algebra ◽  
Algebra Approach ◽  
Quantum Cluster

AbstractWe establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the (q, t)-characters of certain irreducible representations, among which fundamental representations, are obtained as quantum cluster variables. This approach gives a new algorithm to compute these (q, t)-characters. As an application, we prove that the quantum Grothendieck ring of a larger category of representations of the Borel subalgebra of the quantum affine algebra, defined in a previous work as a quantum cluster algebra, contains indeed the well-known quantum Grothendieck ring of the category of finite-dimensional representations. Finally, we display our algorithm on a concrete example.

scholarly journals EXTENDED QUANTUM ENVELOPING ALGEBRAS OF (2)

2009 ◽  
Vol 51 (3) ◽  
pp. 441-465 ◽  
Author(s):  
WU ZHIXIANG
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Cyclic Group ◽  
Hopf Algebras ◽  
Infinite Cyclic Group ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Quantized Enveloping Algebra ◽  
Usual Quantum ◽  
Cocommutative Hopf Algebra

AbstractIn present paper we define a new kind of quantized enveloping algebra of (2). We denote this algebra by Ur,t, where r, t are two non-negative integers. It is a non-commutative and non-cocommutative Hopf algebra. If r = 0, then the algebra Ur,t is isomorphic to a tensor product of the algebra of infinite cyclic group and the usual quantum enveloping algebra of (2) as Hopf algebras. The representation of this algebra is studied.

scholarly journals Quantum groups via cyclic quiver varieties I

2015 ◽  
Vol 152 (2) ◽  
pp. 299-326 ◽  
Author(s):  
Fan Qin
Keyword(s):  
Quantum Groups ◽  
Bilinear Form ◽  
Canonical Basis ◽  
Natural Basis ◽  
Grothendieck Ring ◽  
Quiver Varieties ◽  
Enveloping Algebra ◽  
Dual Canonical Basis ◽  
Quantized Enveloping Algebra ◽  
Cyclic Quiver

We construct the quantized enveloping algebra of any simple Lie algebra of type $\mathbb{A}\mathbb{D}\mathbb{E}$ as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig’s bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as a geometric counterpart of Bridgeland’s recent work for type $\mathbb{A}\mathbb{D}\mathbb{E}$.

Sign in / Sign up

Export Citation Format

Share Document