1,718 research outputs found
On multigraded generalizations of Kirillov-Reshetikhin modules
We study the category of Z^l-graded modules with finite-dimensional graded
pieces for certain Z+^l-graded Lie algebras. We also consider certain Serre
subcategories with finitely many isomorphism classes of simple objects. We
construct projective resolutions for the simple modules in these categories and
compute the Ext groups between simple modules. We show that the projective
covers of the simple modules in these Serre subcategories can be regarded as
multigraded generalizations of Kirillov-Reshetikhin modules and give a
recursive formula for computing their graded characters
Extensions and block decompositions for finite-dimensional representations of equivariant map algebras
Suppose a finite group acts on a scheme and a finite-dimensional Lie
algebra . The associated equivariant map algebra is the Lie
algebra of equivariant regular maps from to . The irreducible
finite-dimensional representations of these algebras were classified in
previous work with P. Senesi, where it was shown that they are all tensor
products of evaluation representations and one-dimensional representations. In
the current paper, we describe the extensions between irreducible
finite-dimensional representations of an equivariant map algebra in the case
that is an affine scheme of finite type and is reductive.
This allows us to also describe explicitly the blocks of the category of
finite-dimensional representations in terms of spectral characters, whose
definition we extend to this general setting. Applying our results to the case
of generalized current algebras (the case where the group acting is trivial),
we recover known results but with very different proofs. For (twisted) loop
algebras, we recover known results on block decompositions (again with very
different proofs) and new explicit formulas for extensions. Finally,
specializing our results to the case of (twisted) multiloop algebras and
generalized Onsager algebras yields previously unknown results on both
extensions and block decompositions.Comment: 41 pages; v2: minor corrections, formatting changed to match
published versio
Macdonald Polynomials and level two Demazure modules for affine
We define a family of symmetric polynomials indexed by a pair of dominant integral weights. The polynomial is the specialized Macdonald polynomial and we prove that is the graded character of a level two Demazure module associated to the affine Lie algebra . Under suitable conditions on (which includes the case when or ) we prove that is Schur positive and give explicit formulae for them in terms of Macdonald polynomials
A Generalized Q-operator for U_q(\hat(sl_2)) Vertex Models
In this paper, we construct a Q-operator as a trace of a representation of
the universal R-matrix of over an infinite-dimensional
auxiliary space. This auxiliary space is a four-parameter generalization of the
q-oscillator representations used previously. We derive generalized T-Q
relations in which 3 of these parameters shift. After a suitable restriction of
parameters, we give an explicit expression for the Q-operator of the 6-vertex
model and show the connection with Baxter's expression for the central block of
his corresponding operator.Comment: 22 pages, Latex2e. This replacement is a revised version that
includes a simple explicit expression for the Q matrix for the 6-vertex mode
Dorey's Rule and the q-Characters of Simply-Laced Quantum Affine Algebras
Let Uq(ghat) be the quantum affine algebra associated to a simply-laced
simple Lie algebra g. We examine the relationship between Dorey's rule, which
is a geometrical statement about Coxeter orbits of g-weights, and the structure
of q-characters of fundamental representations V_{i,a} of Uq(ghat). In
particular, we prove, without recourse to the ADE classification, that the rule
provides a necessary and sufficient condition for the monomial 1 to appear in
the q-character of a three-fold tensor product V_{i,a} x V_{j,b} x V_{k,c}.Comment: 30 pages, latex; v2, to appear in Communications in Mathematical
Physic
Universal Baxterization for -graded Hopf algebras
We present a method for Baxterizing solutions of the constant Yang-Baxter
equation associated with -graded Hopf algebras. To demonstrate the
approach, we provide examples for the Taft algebras and the quantum group
.Comment: 8 page
A family of tridiagonal pairs and related symmetric functions
A family of tridiagonal pairs which appear in the context of quantum
integrable systems is studied in details. The corresponding eigenvalue
sequences, eigenspaces and the block tridiagonal structure of their matrix
realizations with respect the dual eigenbasis are described. The overlap
functions between the two dual basis are shown to satisfy a coupled system of
recurrence relations and a set of discrete second-order difference
equations which generalize the ones associated with the Askey-Wilson orthogonal
polynomials with a discrete argument. Normalizing the fundamental solution to
unity, the hierarchy of solutions are rational functions of one discrete
argument, explicitly derived in some simplest examples. The weight function
which ensures the orthogonality of the system of rational functions defined on
a discrete real support is given.Comment: 17 pages; LaTeX file with amssymb. v2: few minor changes, to appear
in J.Phys.A; v3: Minor misprints, eq. (48) and orthogonality condition
corrected compared to published versio
R-matrices of three-state Hamiltonians solvable by Coordinate Bethe Ansatz
We review some of the strategies that can be implemented to infer an
-matrix from the knowledge of its Hamiltonian. We apply them to the
classification achieved in arXiv:1306.6303, on three state -invariant
Hamiltonians solvable by CBA, focusing on models for which the -matrix is
not trivial.
For the 19-vertex solutions, we recover the -matrices of the well-known
Zamolodchikov--Fateev and Izergin--Korepin models. We point out that the
generalized Bariev Hamiltonian is related to both main and special branches
studied by Martins in arXiv:1303.4010, that we prove to generate the same
Hamiltonian. The 19-vertex SpR model still resists to the analysis, although we
are able to state some no-go theorems on its -matrix.
For 17-vertex Hamiltonians, we produce a new -matrix.Comment: 22 page
Difference L operators related to q-characters
We introduce a factorized difference operator L(u) annihilated by the
Frenkel-Reshetikhin screening operator for the quantum affine algebra
U_q(C^{(1)}_n). We identify the coefficients of L(u) with the fundamental
q-characters, and establish a number of formulas for their higher analogues.
They include Jacobi-Trudi and Weyl type formulas, canceling tableau sums,
Casorati determinant solution to the T-system, and so forth. Analogous
operators for the orthogonal series U_q(B^{(1)}_n) and U_q(D^{(1)}_n) are also
presented.Comment: 25 pages, LaTeX2e, no figur
Lorentz invariant and supersymmetric interpretation of noncommutative quantum field theory
In this paper, using a Hopf-algebraic method, we construct deformed
Poincar\'e SUSY algebra in terms of twisted (Hopf) algebra. By adapting this
twist deformed super-Poincar\'e algrebra as our fundamental symmetry, we can
see the consistency between the algebra and non(anti)commutative relation among
(super)coordinates and interpret that symmetry of non(anti)commutative QFT is
in fact twisted one. The key point is validity of our new twist element that
guarantees non(anti)commutativity of space. It is checked in this paper for N=1
case. We also comment on the possibility of noncommutative central charge
coordinate. Finally, because our twist operation does not break the original
algebra, we can claim that (twisted) SUSY is not broken in contrast to the
string inspired SUSY in N=1 non(anti)commutative superspace.Comment: 15 pages, LaTeX. v3:One section added, typos corrected, to appear in
Int. J. Mod. Phys.
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