36 research outputs found
Hidden Symmetry of the Differential Calculus on the Quantum Matrix Space
A standard bicovariant differential calculus on a quantum matrix space is considered. The principal result of this work is in observing
that the is in fact a
-module differential algebra.Comment: 5 page
Coadditive differential complexes on quantum groups and quantum spaces
A regular way to define an additive coproduct (or ``coaddition'') on the
q-deformed differential complexes is proposed for quantum groups and quantum
spaces related to the Hecke-type R-matrices. Several examples of braided
coadditive differential bialgebras (Hopf algebras) are presented.Comment: 9 page
Differential Hopf algebra structures on the universal enveloping algebra of a Lie algebra
A Class of Bicovariant Differential Calculi on Hopf Algebras
We introduce a large class of bicovariant differential calculi on any quantum
group , associated to -invariant elements. For example, the deformed
trace element on recovers Woronowicz' calculus. More
generally, we obtain a sequence of differential calculi on each quantum group
, based on the theory of the corresponding braided groups . Here
is any regular solution of the QYBE.Comment: 16 page
Covariant differential complexes on quantum linear groups
We consider the possible covariant external algebra structures for Cartan's
1-forms on GL_q(N) and SL_q(N). We base upon the following natural postulates:
1. the invariant 1-forms realize an adjoint representation of quantum group;
2. all monomials of these forms possess the unique ordering.
For the obtained external algebras we define the exterior derivative
possessing the usual nilpotence condition, and the generally deformed version
of Leibniz rules. The status of the known examples of GL_q(N)-differential
calculi in the proposed classification scheme, and the problems of
SL_q(N)-reduction are discussed.Comment: 23 page
All bicovariant differential calculi on Glq(3,C) and SLq(3,C)
All bicovariant first order differential calculi on the quantum group
GLq(3,C) are determined. There are two distinct one-parameter families of
calculi. In terms of a suitable basis of 1-forms the commutation relations can
be expressed with the help of the R-matrix of GLq(3,C). Some calculi induce
bicovariant differential calculi on SLq(3,C) and on real forms of GLq(3,C). For
generic deformation parameter q there are six calculi on SLq(3,C), on SUq(3)
there are only two. The classical limit q-->1 of bicovariant calculi on
SLq(3,C) is not the ordinary calculus on SL(3,C). One obtains a deformation of
it which involves the Cartan-Killing metric.Comment: 24 pages, LaTe
Tannaka-Krein duality for Hopf algebroids
We develop the Tannaka-Krein duality for monoidal functors with target in the
categories of bimodules over a ring. The \coend of such a functor turns out
to be a Hopf algebroid over this ring. Using the result of a previous paper we
characterize a small abelian, locally finite rigid monoidal category as the
category of rigid comodules over a transitive Hopf algebroid.Comment: 25 pages, final version, to appear in Israel Journal of Mathematic
Rigidity and defect actions in Landau-Ginzburg models
Studying two-dimensional field theories in the presence of defect lines
naturally gives rise to monoidal categories: their objects are the different
(topological) defect conditions, their morphisms are junction fields, and their
tensor product describes the fusion of defects. These categories should be
equipped with a duality operation corresponding to reversing the orientation of
the defect line, providing a rigid and pivotal structure. We make this
structure explicit in topological Landau-Ginzburg models with potential x^d,
where defects are described by matrix factorisations of x^d-y^d. The duality
allows to compute an action of defects on bulk fields, which we compare to the
corresponding N=2 conformal field theories. We find that the two actions differ
by phases.Comment: 53 pages; v2: clarified exposition of pivotal structures, corrected
proof of theorem 2.13, added remark 3.9; version to appear in CM
-Deformed Grassmann Field and the Two-dimensional Ising Model
In this paper we construct the exact representation of the Ising partition
function in the form of the -invariant functional integral for the
lattice free -fermion field theory (). It is shown that the
-fermionization allows one to re-express the partition function of the
eight-vertex model in external field through functional integral with
four-fermion interaction. To construct these representations, we define a
lattice -deformed Grassmann bispinor field and extend the Berezin
integration rules to this field. At we obtain the lattice
-fermion field which allows us to fermionize the two-dimensional Ising
model. We show that the Gaussian integral over -Grassmann variables is
expressed through the -deformed Pfaffian which is equal to square root
of the determinant of some matrix at .Comment: 24 pages, LaTeX; minor change