154 research outputs found
Two-parameter nonstandard deformation of 2x2 matrices
We introduce a two-parameter deformation of 2x2 matrices without imposing any
condition on the matrices and give the universal R-matrix of the nonstandard
quantum group which satisfies the quantum Yang-Baxter relation. Although in the
standard two-parameter deformation the quantum determinant is not central, in
the nonstandard case it is central. We note that the quantum group thus
obtained is related to the quantum supergroup by a
transformation.Comment: 10 page
Differential Geometry of the q-plane
Hopf algebra structure on the differential algebra of the extended -plane
is defined. An algebra of forms which is obtained from the generators of the
extended -plane is introduced and its Hopf algebra structure is given.Comment: 9 page
Modified Braid Equations for SO_q (3) and noncommutative spaces
General solutions of the equation with a maximal number of free
parameters in the specrtal decomposition of vector
matrices are implemented to construct modified braid equations (MBE). These
matrices conserve the given, standard, group relations of the nine elements of
T, but are not constrained to satisfy the standard braid equation (BE). Apart
from q and a normalisation factor our contains two free parameters,
instead of only one such parameter for deformed unitary algebras studied in a
previous paper [1] where the nonzero right hand side of the had a linear
term proprotional to . In the present case
the r.h.s. is, in general, nonliear. Several particular solutions are given
(Sec.2) and the general structure is analysed (App.A). Our formulation of the
problem in terms of projectors yield also two new solutions of standard
(nonmodified) braid equation (Sec.2) which are further discussed (App.B). The
noncommutative 3-spaces obtained by implementing such generalized
matrices are studied (Sec.3). The role of coboundary matrices (not
satisfying the standard BE) is explored. The MBE and Baxterization are
presented as complementary facets of the same basic construction, namely, the
general solution of equation (Sec.4). A new solution is presented
in this context. As a simple but remarkable particular case a nontrivial
solution of BE is obtained (App.B) for q=1. This solution has no free parameter
and is not obtainable by twisting the identity matrix. In the concluding
remarks (Sec.5), among other points, generalisation of our results to
is discussed.Comment: 18 pages, no figure
Quantum Jacobi-Trudi and Giambelli Formulae for from Analytic Bethe Ansatz
Analytic Bethe ansatz is executed for a wide class of finite dimensional
modules. They are labeled by skew-Young diagrams which, in
general, contain a fragment corresponding to the spin representation. For the
transfer matrix spectra of the relevant vertex models, we establish a number of
formulae, which are analogues of the classical ones due to
Jacobi-Trudi and Giambelli on Schur functions. They yield a full solution to
the previously proposed functional relation (-system), which is a Toda
equationComment: Plain Tex(macro included), 18 pages. 7 figures are compressed and
attache
Representations of the quantum matrix algebra
It is shown that the finite dimensional irreducible representaions of the
quantum matrix algebra ( the coordinate ring of ) exist only when both q and p are roots of unity. In this case th e space of
states has either the topology of a torus or a cylinder which may be thought of
as generalizations of cyclic representations.Comment: 20 page
Lie bialgebra contractions and quantum deformations of quasi-orthogonal algebras
Lie bialgebra contractions are introduced and classified. A non-degenerate
coboundary bialgebra structure is implemented into all pseudo-orthogonal
algebras starting from the one corresponding to . It allows
to introduce a set of Lie bialgebra contractions which leads to Lie bialgebras
of quasi-orthogonal algebras. This construction is explicitly given for the
cases . All Lie bialgebra contractions studied in this paper define
Hopf algebra contractions for the Drinfel'd-Jimbo deformations .
They are explicitly used to generate new non-semisimple quantum algebras as it
is the case for the Euclidean, Poincar\'e and Galilean algebras.Comment: 26 pages LATE
Algebraic Bethe Ansatz for deformed Gaudin model
The Gaudin model based on the sl_2-invariant r-matrix with an extra Jordanian
term depending on the spectral parameters is considered. The appropriate
creation operators defining the Bethe states of the system are constructed
through a recurrence relation. The commutation relations between the generating
function t(\lambda) of the integrals of motion and the creation operators are
calculated and therefore the algebraic Bethe Ansatz is fully implemented. The
energy spectrum as well as the corresponding Bethe equations of the system
coincide with the ones of the sl_2-invariant Gaudin model. As opposed to the
sl_2-invariant case, the operator t(\lambda) and the Gaudin Hamiltonians are
not hermitian. Finally, the inner products and norms of the Bethe states are
studied.Comment: 23 pages; presentation improve
Notes on two-parameter quantum groups, (II)
This paper is the sequel to [HP1] to study the deformed structures and
representations of two-parameter quantum groups
associated to the finite dimensional simple Lie algebras \mg. An equivalence
of the braided tensor categories \O^{r,s} and \O^{q} is explicitly
established.Comment: 21 page
Comments on Drinfeld Realization of Quantum Affine Superalgebra and its Hopf Algebra Structure
By generalizing the Reshetikhin and Semenov-Tian-Shansky construction to
supersymmetric cases, we obtain Drinfeld current realization for quantum affine
superalgebra . We find a simple coproduct for the quantum
current generators and establish the Hopf algebra structure of this super
current algebra.Comment: Some errors and misprints corrected and a remark in section 4
removed. 12 pages, Latex fil
Exact solution of A-D Temperley-Lieb Models
We solve for the spectrum of quantum spin chains based on representations of
the Temperley-Lieb algebra associated with the quantum groups {\cal U}_q(X_n }
for X_n = A_1,B_n,C_nD_n$. We employ a generalization of the coordinate
Bethe-Ansatz developed previously for the deformed biquadratic spin one chain.
As expected, all these models have equivalent spectra, i.e. they differ only in
the degeneracy of their eigenvalues. This is true for finite length and open
boundary conditions. For periodic boundary conditions the spectra of the lower
dimensional representations are containded entirely in the higher dimensional
ones. The Bethe states are highest weight states of the quantum group, except
for some states with energy zero
- …