21 research outputs found
-Deformation of Poincar\'e Superalgebra with Classical Lorentz Subalgebra and its Graded Bicrossproduct Structure
The -deformed Poincar{\'e} superalgebra written in Hopf
superalgebra form is transformed to the basis with classical Lorentz subalgebra
generators. We show that in such a basis the -deformed Poincare
superalgebra can be written as graded bicrossproduct. We show that the
-deformed superalgebra acts covariantly on -deformed
chiral superspace.Comment: 13 pages, late
On epimorphisms and monomorphisms of Hopf algebras
We provide examples of non-surjective epimorphisms in the category
of Hopf algebras over a field, even with the additional requirement that
have bijective antipode, by showing that the universal map from a Hopf algebra
to its enveloping Hopf algebra with bijective antipode is an epimorphism in
\halg, although it is known that it need not be surjective. Dual results are
obtained for the problem of whether monomorphisms in the category of Hopf
algebras are necessarily injective. We also notice that these are automatically
examples of non-faithfully flat and respectively non-faithfully coflat maps of
Hopf algebras.Comment: 17 pages; changed the abstract, revised introduction, shortened some
proofs; to appear in J. Algebr
The monoidal centre as a limit
The centre of a monoidal category is a braided monoidal category. Monoidal
categories are monoidal objects (or pseudomonoids) in the monoidal bicategory
of categories. This paper provides a universal construction in a braided
monoidal bicategory that produces a braided monoidal object from any monoidal
object. Some properties and sufficient conditions for existence of the
construction are examined.Comment: 6 pages, pdf file of MacWrite documen
Torsors, herds and flocks
This paper presents non-commutative and structural notions of torsor. The two
are related by the machinery of Tannaka-Krein duality
Closed categories, star-autonomy, and monoidal comonads
This paper determines what structure is needed for internal homs in a
monoidal category C to be liftable to the category C^G of Eilenberg-Moore
coalgebras for a monoidal comonad G on C. We apply this to lift star-autonomy
with the view to recasting the definition of quantum groupoid.Comment: 25 page
Mackey functors on compact closed categories
We develop and extend the theory of Mackey functors as an application of
enriched category theory. We define Mackey functors on a lextensive category
\E and investigate the properties of the category of Mackey functors on \E.
We show that it is a monoidal category and the monoids are Green functors.
Mackey functors are seen as providing a setting in which mere numerical
equations occurring in the theory of groups can be given a structural
foundation. We obtain an explicit description of the objects of the Cauchy
completion of a monoidal functor and apply this to examine Morita equivalence
of Green functors
Skew monoidales, skew warpings and quantum categories
Kornel Szlach\'anyi recently used the term skew-monoidal category for a
particular laxified version of monoidal category. He showed that bialgebroids
with base ring could be characterized in terms of skew-monoidal
structures on the category of one-sided -modules for which the lax unit was
itself. We define skew monoidales (or skew pseudo-monoids) in any monoidal
bicategory . These are skew-monoidal categories when
is . Our main results are presented at the level of monoidal
bicategories. However, a consequence is that quantum categories in the sense of
Day-Street with base comonoid in a suitably complete braided monoidal
category are precisely skew monoidales in with unit coming from the counit of . Quantum groupoids are
those skew monoidales with invertible associativity constraint. In fact, we
provide some very general results connecting opmonoidal monads and skew
monoidales. We use a lax version of the concept of warping defined recently by
Booker-Street to modify monoidal structures.Comment: Minor changes and some renumbering in this versio
Categorical formulation of quantum algebras
We describe how dagger-Frobenius monoids give the correct categorical
description of certain kinds of finite-dimensional 'quantum algebras'. We
develop the concept of an involution monoid, and use it to construct a
correspondence between finite-dimensional C*-algebras and certain types of
dagger-Frobenius monoids in the category of Hilbert spaces. Using this
technology, we recast the spectral theorems for commutative C*-algebras and for
normal operators into an explicitly categorical language, and we examine the
case that the results of measurements do not form finite sets, but rather
objects in a finite Boolean topos. We describe the relevance of these results
for topological quantum field theory.Comment: 34 pages, to appear in Communications in Mathematical Physic