16 research outputs found
On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization
We study classical twists of Lie bialgebra structures on the polynomial
current algebra , where is a simple complex
finite-dimensional Lie algebra. We focus on the structures induced by the
so-called quasi-trigonometric solutions of the classical Yang-Baxter equation.
It turns out that quasi-trigonometric -matrices fall into classes labelled
by the vertices of the extended Dynkin diagram of . We give
complete classification of quasi-trigonometric -matrices belonging to
multiplicity free simple roots (which have coefficient 1 in the decomposition
of the maximal root). We quantize solutions corresponding to the first root of
.Comment: 41 pages, LATE
Convex Bases of PBW type for Quantum Affine Algebras
This note has two purposes. First we establish that the map defined in [L,
(a)] is an isomorphism for certain admissible sequences. Second we
show the map gives rise to a convex basis of Poincar\'e--Birkhoff--Witt (PBW)
type for \bup, an affine untwisted quantized enveloping algebra of
Drinfeld and Jimbo. The computations in this paper are made possible by
extending the usual braid group action by certain outer automorphisms of the
algebra.Comment: 7 pages, to appear in Comm. Math. Phy
Cremmer-Gervais r-matrices and the Cherednik Algebras of type GL2
We give an intepretation of the Cremmer-Gervais r-matrices for sl(n) in terms
of actions of elements in the rational and trigonometric Cherednik algebras of
type GL2 on certain subspaces of their polynomial representations. This is used
to compute the nilpotency index of the Jordanian r-matrices, thus answering a
question of Gerstenhaber and Giaquinto. We also give an interpretation of the
Cremmer-Gervais quantization in terms of the corresponding double affine Hecke
algebra.Comment: 6 page
Weakly coupled N=4 Super Yang-Mills and N=6 Chern-Simons theories from u(2|2) Yangian symmetry
In this paper we derive the universal R-matrix for the Yangian Y(u(2|2)),
which is an abstract algebraic object leading to rational solutions of the
Yang-Baxter equation on representations. We find that on the fundamental
representation the universal R-matrix reduces to the standard rational R-matrix
R = R_0(1 + P/u), where the scalar prefactor is surprisingly simple compared to
prefactors one finds e.g. for sl(n) R-matrices. This leads precisely to the
S-matrix giving the Bethe Ansatz of one-loop N = 4 Super Yang-Mills theory and
two-loop N = 6 Chern-Simons theory.Comment: 16 page
Finite-dimensional representations of the quantum superalgebra and related q-identities
Explicit expressions for the generators of the quantum superalgebra
acting on a class of irreducible representations are given. The
class under consideration consists of all essentially typical representations:
for these a Gel'fand-Zetlin basis is known. The verification of the quantum
superalgebra relations to be satisfied is shown to reduce to a set of
-number identities.Comment: 12 page
Two-parameter Quantum Affine Algebra , Drinfel'd Realization and Quantum Affine Lyndon Basis
We further define two-parameter quantum affine algebra
after the work on the finite cases
(see [BW1], [BGH1], [HS] & [BH]), which turns out to be a Drinfel'd double. Of
importance for the quantum {\it affine} cases is that we can work out the
compatible two-parameter version of the Drinfel'd realization as a quantum
affinization of and establish the Drinfel'd isomorphism
Theorem in the two-parameter setting, via developing a new combinatorial
approach (quantum calculation) to the quantum {\it affine} Lyndon basis we
present (with an explicit valid algorithm based on the use of Drinfel'd
generators).Comment: 31 page
Once again about quantum deformations of D=4 Lorentz algebra: twistings of q-deformation
This paper together with the previous one (arXiv:hep-th/0604146) presents the
detailed description of all quantum deformations of D=4 Lorentz algebra as Hopf
algebra in terms of complex and real generators. We describe here in detail two
quantum deformations of the D=4 Lorentz algebra o(3,1) obtained by twisting of
the standard q-deformation U_{q}(o(3,1)). For the first twisted q-deformation
an Abelian twist depending on Cartan generators of o(3,1) is used. The second
example of twisting provides a quantum deformation of Cremmer-Gervais type for
the Lorentz algebra. For completeness we describe also twisting of the Lorentz
algebra by standard Jordanian twist. By twist quantization techniques we obtain
for these deformations new explicit formulae for the deformed coproducts and
antipodes of the o(3,1)-generators.Comment: 17 page
Coherent States for Quantum Compact Groups
Coherent states are introduced and their properties are discussed for all
simple quantum compact groups. The multiplicative form of the canonical element
for the quantum double is used to introduce the holomorphic coordinates on a
general quantum dressing orbit and interpret the coherent state as a
holomorphic function on this orbit with values in the carrier Hilbert space of
an irreducible representation of the corresponding quantized enveloping
algebra. Using Gauss decomposition, the commutation relations for the
holomorphic coordinates on the dressing orbit are derived explicitly and given
in a compact R--matrix formulation (generalizing this way the --deformed
Grassmann and flag manifolds). The antiholomorphic realization of the
irreducible representations of a compact quantum group (the analogue of the
Borel--Weil construction) are described using the concept of coherent state.
The relation between representation theory and non--commutative differential
geometry is suggested.}Comment: 25 page
Deformation of Yangian Y(sl(2))
A quantization of a non-standard rational solution of CYBE for sit is given explicitly. We obtain the quantization with the help of a twisting of the usual Yangian Y (sl(2)). This quantum object (deformed Yangian Y-eta,Y-xi(sl(2))) is a two-parametric deformation of the universal enveloping algebra U(sl(2)[u]) of the polynomial current algebra sl(2)[u]. We consider the pseudotriangular structure on Y-eta,Y-xi(sl(2)), the quantum double DYeta,xi(sl(2)), its the universal R-matrix and also the RTT-realization of Y-eta,Y-xi(sl(2))