50 research outputs found
Weak Bialgebras of Fractions
We construct the algebra of fractions of a Weak Bialgebra relative to a
suitable denominator set of group-like elements that is `almost central', a
condition we introduce in the present article which is sufficient in order to
guarantee existence of the algebra of fractions and to render it a Weak
Bialgebra. The monoid of all group-like elements of a coquasi-triangular Weak
Bialgebra, for example, forms a suitable set of denominators as does any monoid
of central group-like elements of an arbitrary Weak Bialgebra. We use this
technique in order to construct new Weak Bialgebras whose categories of
finite-dimensional comodules relate to SL2-fusion categories in the same way as
GL(2) relates to SL(2).Comment: 10 pages; LaTeX 2e with xypic diagram
Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras
We study a special sort of 2-dimensional extended Topological Quantum Field
Theories (TQFTs) which we call open-closed TQFTs. These are defined on
open-closed cobordisms by which we mean smooth compact oriented 2-manifolds
with corners that have a particular global structure in order to model the
smooth topology of open and closed string worldsheets. We show that the
category of open-closed TQFTs is equivalent to the category of knowledgeable
Frobenius algebras. A knowledgeable Frobenius algebra (A,C,i,i^*) consists of a
symmetric Frobenius algebra A, a commutative Frobenius algebra C, and an
algebra homomorphism i:C->A with dual i^*:A->C, subject to some conditions.
This result is achieved by providing a generators and relations description of
the category of open-closed cobordisms. In order to prove the sufficiency of
our relations, we provide a normal form for such cobordisms which is
characterized by topological invariants. Starting from an arbitrary such
cobordism, we construct a sequence of moves (generalized handle slides and
handle cancellations) which transforms the given cobordism into the normal
form. Using the generators and relations description of the category of
open-closed cobordisms, we show that it is equivalent to the symmetric monoidal
category freely generated by a knowledgeable Frobenius algebra. Our formalism
is then generalized to the context of open-closed cobordisms with labeled free
boundary components, i.e. to open-closed string worldsheets with D-brane labels
at their free boundaries.Comment: 47 pages; LaTeX2e with xypic and pstricks macros; corrected typo
2-Groups, trialgebras and their Hopf categories of representations
AbstractA strict 2-group is a 2-category with one object in which all morphisms and all 2-morphisms have inverses. 2-Groups have been studied in the context of homotopy theory, higher gauge theory and Topological Quantum Field Theory (TQFT). In the present article, we develop the notions of trialgebra and cotrialgebra, generalizations of Hopf algebras with two multiplications and one comultiplication or vice versa, and the notion of Hopf categories, generalizations of monoidal categories with an additional functorial comultiplication. We show that each strict 2-group has a ‘group algebra’ which is a cocommutative trialgebra, and that each strict finite 2-group has a ‘function algebra’ which is a commutative cotrialgebra. Each such commutative cotrialgebra gives rise to a symmetric Hopf category of corepresentations. In the semisimple case, this Hopf category is a 2-vector space according to Kapranov and Voevodsky. We also show that strict compact topological 2-groups are characterized by their C*-cotrialgebras of ‘complex-valued functions’, generalizing the Gel'fand representation, and that commutative cotrialgebras are characterized by their symmetric Hopf categories of corepresentations, generalizing Tannaka–Kreıˇn reconstruction. Technically, all these results are obtained using ideas from functorial semantics, by studying models of the essentially algebraic theory of categories in various base categories of familiar algebraic structures and the functors that describe the relationships between them
Open-closed TQFTs extend Khovanov homology from links to tangles
We use a special kind of 2-dimensional extended Topological Quantum Field
Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov
homology from links to arbitrary tangles, not necessarily even. For every plane
diagram of an oriented tangle, we construct a chain complex whose homology is
invariant under Reidemeister moves. The terms of this chain complex are modules
of a suitable algebra A such that there is one action of A or A^op for every
boundary point of the tangle. We give examples of such algebras A for which our
tangle homology theory reduces to the link homology theories of Khovanov, Lee,
and Bar-Natan if it is evaluated for links. As a consequence of the Cardy
condition, Khovanov's graded theory can only be extended to tangles if the
underlying field has finite characteristic. In all cases in which the algebra A
is strongly separable, i.e. for Bar-Natan's theory in any characteristic and
for Lee's theory in characteristic other than 2, we also provide the required
algebraic operation for the composition of oriented tangles. Just as Khovanov's
theory for links can be recovered from Lee's or Bar-Natan's by a suitable
spectral sequence, we provide a spectral sequence in order to compute our
tangle extension of Khovanov's theory from that of Bar-Natan's or Lee's theory.
Thus, we provide a tangle homology theory that is locally computable and still
strong enough to recover characteristic p Khovanov homology for links.Comment: 56 pages, LaTeX2e with xypic and pstricks macro
On characteristic equations, trace identities and Casimir operators of simple Lie algebras
Two approaches are developed to exploit, for simple complex or compact real
Lie algebras g, the information that stems from the characteristic equations of
representation matrices and Casimir operators. These approaches are selected so
as to be viable not only for `small' Lie algebras and suitable for treatment by
computer algebra. A very large body of new results emerges in the forms, a) of
identities of a tensorial nature, involving structure constants etc. of g, b)
of trace identities for powers of matrices of the adjoint and defining
representations of g, c) of expressions of non-primitive Casimir operators of g
in terms of primitive ones. The methods are sufficiently tractable to allow not
only explicit proof by hand of the non-primitive nature of the quartic Casimir
of g2, f4, e6, but also e.g. of that of the tenth order Casimir of f4.Comment: 39 pages, 8 tables, late