39 research outputs found
Analytic Bethe Ansatz for Fundamental Representations of Yangians
We study the analytic Bethe ansatz in solvable vertex models associated with
the Yangian or its quantum affine analogue for and . Eigenvalue formulas are proposed for the transfer matrices
related to all the fundamental representations of . Under the Bethe
ansatz equation, we explicitly prove that they are pole-free, a crucial
property in the ansatz. Conjectures are also given on higher representation
cases by applying the -system, the transfer matrix functional relations
proposed recently. The eigenvalues are neatly described in terms of Yangian
analogues of the semi-standard Young tableaux.Comment: 45 pages, Plain Te
On a nonstandard two-parametric quantum algebra and its connections with and
A quantum algebra associated with a nonstandard
-matrix with two deformation parameters is studied and, in
particular, its universal -matrix is derived using Reshetikhin's
method. Explicit construction of the -dependent nonstandard -matrix
is obtained through a coloured generalized boson realization of the universal
-matrix of the standard corresponding to a
nongeneric case. General finite dimensional coloured representation of the
universal -matrix of is also derived. This
representation, in nongeneric cases, becomes a source for various
-dependent nonstandard -matrices. Superization of leads to the super-Hopf algebra . A contraction
procedure then yields a -deformed super-Heisenberg algebra
and its universal -matrix.Comment: 17pages, LaTeX, Preprint No. imsc-94/43 Revised version: A note added
at the end of the paper correcting and clarifying the bibliograph
Bicovariant Quantum Algebras and Quantum Lie Algebras
A bicovariant calculus of differential operators on a quantum group is
constructed in a natural way, using invariant maps from \fun\ to \uqg\ , given
by elements of the pure braid group. These operators --- the `reflection
matrix' being a special case --- generate algebras that
linearly close under adjoint actions, i.e. they form generalized Lie algebras.
We establish the connection between the Hopf algebra formulation of the
calculus and a formulation in compact matrix form which is quite powerful for
actual computations and as applications we find the quantum determinant and an
orthogonality relation for in .Comment: 38 page
Geometry of q-Hypergeometric Functions as a Bridge between Yangians and Quantum Affine Algebras
The rational quantized Knizhnik-Zamolodchikov equation (qKZ equation)
associated with the Lie algebra is a system of linear difference
equations with values in a tensor product of Verma modules. We solve the
equation in terms of multidimensional -hypergeometric functions and define a
natural isomorphism between the space of solutions and the tensor product of
the corresponding quantum group Verma modules, where the parameter
is related to the step of the qKZ equation via .
We construct asymptotic solutions associated with suitable asymptotic zones
and compute the transition functions between the asymptotic solutions in terms
of the trigonometric -matrices. This description of the transition functions
gives a new connection between representation theories of Yangians and quantum
loop algebras and is analogous to the Kohno-Drinfeld theorem on the monodromy
group of the differential Knizhnik-Zamolodchikov equation.
In order to establish these results we construct a discrete Gauss-Manin
connection, in particular, a suitable discrete local system, discrete homology
and cohomology groups with coefficients in this local system, and identify an
associated difference equation with the qKZ equation.Comment: 66 pages, amstex.tex (ver. 2.1) and amssym.tex are required;
misprints are correcte
More on quantum groups from the the quantization point of view
Star products on the classical double group of a simple Lie group and on
corresponding symplectic grupoids are given so that the quantum double and the
"quantized tangent bundle" are obtained in the deformation description.
"Complex" quantum groups and bicovariant quantum Lie algebras are discused from
this point of view. Further we discuss the quantization of the Poisson
structure on symmetric algebra leading to the quantized enveloping
algebra as an example of biquantization in the sense of Turaev.
Description of in terms of the generators of the bicovariant
differential calculus on is very convenient for this purpose. Finally
we interpret in the deformation framework some well known properties of compact
quantum groups as simple consequences of corresponding properties of classical
compact Lie groups. An analogue of the classical Kirillov's universal character
formula is given for the unitary irreducible representation in the compact
case.Comment: 18 page
On the Hopf algebras generated by the Yang-Baxter R-matrices
We reformulate the method recently proposed for constructing quasitriangular
Hopf algebras of the quantum-double type from the R-matrices obeying the
Yang-Baxter equations. Underlying algebraic structures of the method are
elucidated and an illustration of its facilities is given. The latter produces
an example of a new quasitriangular Hopf algebra. The corresponding universal
R-matrix is presented as a formal power series.Comment: 10 page
Link Invariants and Combinatorial Quantization of Hamiltonian Chern-Simons Theory
We define and study the properties of observables associated to any link in
(where is a compact surface) using the
combinatorial quantization of hamiltonian Chern-Simons theory. These
observables are traces of holonomies in a non commutative Yang-Mills theory
where the gauge symmetry is ensured by a quantum group. We show that these
observables are link invariants taking values in a non commutative algebra, the
so called Moduli Algebra. When these link invariants are pure
numbers and are equal to Reshetikhin-Turaev link invariants.Comment: 39, latex, 7 figure
Graph Invariants of Vassiliev Type and Application to 4D Quantum Gravity
We consider a special class of Kauffman's graph invariants of rigid vertex
isotopy (graph invariants of Vassiliev type). They are given by a functor from
a category of colored and oriented graphs embedded into a 3-space to a category
of representations of the quasi-triangular ribbon Hopf algebra . Coefficients in expansions of them with respect to () are
known as the Vassiliev invariants of finite type. In the present paper, we
construct two types of tangle operators of vertices. One of them corresponds to
a Casimir operator insertion at a transverse double point of Wilson loops. This
paper proposes a non-perturbative generalization of Kauffman's recent result
based on a perturbative analysis of the Chern-Simons quantum field theory. As a
result, a quantum group analog of Penrose's spin network is established taking
into account of the orientation. We also deal with the 4-dimensional canonical
quantum gravity of Ashtekar. It is verified that the graph invariants of
Vassiliev type are compatible with constraints of the quantum gravity in the
loop space representation of Rovelli and Smolin.Comment: 34 pages, AMS-LaTeX, no figures,The proof of thm.5.1 has been
improve
Braided Matrix Structure of the Sklyanin Algebra and of the Quantum Lorentz Group
Braided groups and braided matrices are novel algebraic structures living in
braided or quasitensor categories. As such they are a generalization of
super-groups and super-matrices to the case of braid statistics. Here we
construct braided group versions of the standard quantum groups . They
have the same FRT generators but a matrix braided-coproduct \und\Delta
L=L\und\tens L where , and are self-dual. As an application, the
degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices
; it is a braided-commutative bialgebra in a braided category. As a
second application, we show that the quantum double D(\usl) (also known as
the `quantum Lorentz group') is the semidirect product as an algebra of two
copies of \usl, and also a semidirect product as a coalgebra if we use braid
statistics. We find various results of this type for the doubles of general
quantum groups and their semi-classical limits as doubles of the Lie algebras
of Poisson Lie groups.Comment: 45 pages. Revised (= much expanded introduction
Classical integrable lattice models through quantum group related formalism
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 -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 matrices.Comment: 7 pages; Talk presented at NEEDS'93 (Gallipoli,Italy