45 research outputs found
Squared Hopf algebras and reconstruction theorems
Given an abelian k-linear rigid monoidal category V, where k is a perfect
field, we define squared coalgebras as objects of cocompleted V tensor V
(Deligne's tensor product of categories) equipped with the appropriate notion
of comultiplication. Based on this, (squared) bialgebras and Hopf algebras are
defined without use of braiding.
If V is the category of k-vector spaces, squared (co)algebras coincide with
conventional ones. If V is braided, a braided Hopf algebra can be obtained from
a squared one.
Reconstruction theorems give equivalence of squared co- (bi-, Hopf) algebras
in V and corresponding fibre functors to V (which is not the case with other
definitions). Finally, squared quasitriangular Hopf coalgebra is a solution to
the problem of defining quantum groups in braided categories.Comment: Latex2e, 31 pages, to appear in the Proceedings of Banach Center
Minisemester on Quantum Groups, November 199
Unital A∞-categories
Ми доводимо, що три означення унiтальностi для A∞-категорiй запропонованi Любашенком, Концевичем i Сойбельманом, та Фукая є еквiвалентними.We prove that three definitions of unitality for A∞-categories suggested by Lyubashenko, by Kontsevich and Soibelman, and by Fukaya are equivalent
Homotopy equivalence of normalized and unnormalized complexes, revisited
We consider the unnormalized and normalized complexes of a simplicial or a cosimplicial object coming from the DoldśKan correspondence for an idempotent complete additive category (kernels and cokernels are not required). The normalized complex is defined as the image of certain idempotent in the unnormalized complex. We prove that this idempotent is homotopic to identity via homotopy which is expressed via faces and degeneracies. Hence, the normalized and unnormalized complex are homotopy isomorphic to each other. We provide explicit formulae for the homotopy
The homotopy theory of dg-categories and derived Morita theory
The main purpose of this work is the study of the homotopy theory of
dg-categories up to quasi-equivalences. Our main result provides a natural
description of the mapping spaces between two dg-categories and in
terms of the nerve of a certain category of -bimodules. We also prove
that the homotopy category is cartesian closed (i.e. possesses
internal Hom's relative to the tensor product). We use these two results in
order to prove a derived version of Morita theory, describing the morphisms
between dg-categories of modules over two dg-categories and as the
dg-category of -bi-modules. Finally, we give three applications of our
results. The first one expresses Hochschild cohomology as endomorphisms of the
identity functor, as well as higher homotopy groups of the \emph{classifying
space of dg-categories} (i.e. the nerve of the category of dg-categories and
quasi-equivalences between them). The second application is the existence of a
good theory of localization for dg-categories, defined in terms of a natural
universal property. Our last application states that the dg-category of
(continuous) morphisms between the dg-categories of quasi-coherent (resp.
perfect) complexes on two schemes (resp. smooth and proper schemes) is
quasi-equivalent to the dg-category of quasi-coherent complexes (resp. perfect)
on their product.Comment: 50 pages. Few mistakes corrected, and some references added. Thm.
8.15 is new. Minor corrections. Final version, to appear in Inventione
Curved Koszul duality theory
38 pagesInternational audienceWe extend the bar-cobar adjunction to operads and properads, not necessarily augmented. Due to the default of augmentation, the objects of the dual category are endowed with a curvature. We handle the lack of augmentation by extending the category of coproperads to include objects endowed with a curvature. As usual, the bar-cobar construction gives a (large) cofibrant resolution for any properad, such as the properad encoding unital and counital Frobenius algebras, a notion which appears in 2d-TQFT. We also define a curved Koszul duality theory for operads or properads presented with quadratic, linear and constant relations, which provides the possibility for smaller relations. We apply this new theory to study the homotopy theory and the cohomology theory of unital associative algebras
Classification of the quantum deformation of the superalgebra
We present a classification of the possible quantum deformations of the
supergroup and its Lie superalgebra . In each case, the
(super)commutation relations and the Hopf structures are explicitly computed.
For each matrix, one finds two inequivalent coproducts whether one chooses
an unbraided or a braided framework while the corresponding structures are
isomorphic as algebras. In the braided case, one recovers the classical algebra
for suitable limits of the deformation parameters but this is no
longer true in the unbraided case.Comment: 23p LaTeX2e Document - packages amsfonts,subeqn - misprints and
errors corrected, one section adde
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
The structure of quantum Lie algebras for the classical series B_l, C_l and D_l
The structure constants of quantum Lie algebras depend on a quantum
deformation parameter q and they reduce to the classical structure constants of
a Lie algebra at . We explain the relationship between the structure
constants of quantum Lie algebras and quantum Clebsch-Gordan coefficients for
adjoint x adjoint ---> adjoint. We present a practical method for the
determination of these quantum Clebsch-Gordan coefficients and are thus able to
give explicit expressions for the structure constants of the quantum Lie
algebras associated to the classical Lie algebras B_l, C_l and D_l.
In the quantum case also the structure constants of the Cartan subalgebra are
non-zero and we observe that they are determined in terms of the simple quantum
roots. We introduce an invariant Killing form on the quantum Lie algebras and
find that it takes values which are simple q-deformations of the classical
ones.Comment: 25 pages, amslatex, eepic. Final version for publication in J. Phys.
A. Minor misprints in eqs. 5.11 and 5.12 correcte
Factorizable ribbon quantum groups in logarithmic conformal field theories
We review the properties of quantum groups occurring as Kazhdan--Lusztig dual
to logarithmic conformal field theory models. These quantum groups at even
roots of unity are not quasitriangular but are factorizable and have a ribbon
structure; the modular group representation on their center coincides with the
representation on generalized characters of the chiral algebra in logarithmic
conformal field models.Comment: 27pp., amsart++, xy. v2: references added, some other minor addition
Hopf Categories
We introduce Hopf categories enriched over braided monoidal categories. The
notion is linked to several recently developed notions in Hopf algebra theory,
such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. We
generalize the fundamental theorem for Hopf modules and some of its
applications to Hopf categories.Comment: 47 pages; final version to appear in Algebras and Representation
Theor