11,419 research outputs found
Rack and quandle homology
The theory of rack and quandle modules is developed - in particular a tensor
product is defined, and shown to satisfy an appropriate adjointness condition.
Notions of free rack and quandle modules are introduced, and used to define an
enveloping object (the `rack algebra' or `wring') for a given rack or quandle.
These constructions are then used to define homology and cohomology theories
for racks and quandles which contain all currently-known variants.Comment: 16 pages, LaTeX2e. Requires gtart.cls and diagram.sty (included
Super duality and Kazhdan-Lusztig polynomials
We establish a direct connection between the representation theories of Lie
algebras and Lie superalgebras (of type A) via Fock space reformulations of
their Kazhdan-Lusztig theories. As a consequence, the characters of
finite-dimensional irreducible modules of the general linear Lie superalgebra
are computed by the usual parabolic Kazhdan-Lusztig polynomials of type A. In
addition, we establish closed formulas for canonical and dual canonical bases
for the tensor product of any two fundamental representations of type A.Comment: v.2, substantially revised and streamlined, title modified, 45 page
Homological Algebra for Persistence Modules
We develop some aspects of the homological algebra of persistence modules, in
both the one-parameter and multi-parameter settings, considered as either
sheaves or graded modules. The two theories are different. We consider the
graded module and sheaf tensor product and Hom bifunctors as well as their
derived functors, Tor and Ext, and give explicit computations for interval
modules. We give a classification of injective, projective, and flat interval
modules. We state Kunneth theorems and universal coefficient theorems for the
homology and cohomology of chain complexes of persistence modules in both the
sheaf and graded modules settings and show how these theorems can be applied to
persistence modules arising from filtered cell complexes. We also give a
Gabriel-Popescu theorem for persistence modules. Finally, we examine categories
enriched over persistence modules. We show that the graded module point of view
produces a closed symmetric monoidal category that is enriched over itself.Comment: 41 pages, accepted by Foundations of Computational Mathematic
Geometry of q-Hypergeometric Functions as a Bridge between Yangians and Quantum Affine Algebras
The rational quantized Knizhnik-Zamolodchikov equation (qKZ equation)
associated with the Lie algebra is a system of linear difference
equations with values in a tensor product of Verma modules. We solve the
equation in terms of multidimensional -hypergeometric functions and define a
natural isomorphism between the space of solutions and the tensor product of
the corresponding quantum group Verma modules, where the parameter
is related to the step of the qKZ equation via .
We construct asymptotic solutions associated with suitable asymptotic zones
and compute the transition functions between the asymptotic solutions in terms
of the trigonometric -matrices. This description of the transition functions
gives a new connection between representation theories of Yangians and quantum
loop algebras and is analogous to the Kohno-Drinfeld theorem on the monodromy
group of the differential Knizhnik-Zamolodchikov equation.
In order to establish these results we construct a discrete Gauss-Manin
connection, in particular, a suitable discrete local system, discrete homology
and cohomology groups with coefficients in this local system, and identify an
associated difference equation with the qKZ equation.Comment: 66 pages, amstex.tex (ver. 2.1) and amssym.tex are required;
misprints are correcte
Logarithmic conformal field theory, log-modular tensor categories and modular forms
The two pillars of rational conformal field theory and rational vertex
operator algebras are modularity of characters on the one hand and its
interpretation of modules as objects in a modular tensor category on the other
one. Overarching these pillars is the Verlinde formula.
In this paper we consider the more general class of logarithmic conformal
field theories and -cofinite vertex operator algebras. We suggest that
their modular pillar are trace functions with insertions corresponding to
intertwiners of the projective cover of the vacuum, and that the categorical
pillar are finite tensor categories which are ribbon and whose
double is isomorphic to the Deligne product .
Overarching these pillars is then a logarithmic variant of Verlinde's
formula. Numerical data realizing this are the modular -matrix and modified
traces of open Hopf links.
The representation categories of -cofinite and logarithmic conformal
field theories that are fairly well understood are those of the -triplet algebras and the symplectic fermions. We illustrate the ideas in
these examples and especially make the relation between logarithmic Hopf links
and modular transformations explicit
Representations of vertex operator algebras and braided finite tensor categories
We discuss what has been achieved in the past twenty years on the
construction and study of a braided finite tensor category structure on a
suitable module category for a suitable vertex operator algebra. We identify
the main difficult parts in the construction, discuss the methods developed to
overcome these difficulties and present some further problems that still need
to be solved. We also choose to discuss three among the numerous applications
of the construction.Comment: 24 pages. One remark and one footnote are added and some mistakes of
terminology are correcte
Some open problems in mathematical two-dimensional conformal field theory
We discuss some open problems in a program of constructing and studying
two-dimensional conformal field theories using the representation theory of
vertex operator algebras.Comment: 16 pages. Typos are corrected and some sentences are adjusted. Final
version to appear in the proceedings of the Conference on Lie Algebras,
Vertex Operator Algebras, and Related Topics, held at University of Notre
Dame, Notre Dame, Indiana, August 14-18, 201
Associative-algebraic approach to logarithmic conformal field theories
We set up a strategy for studying large families of logarithmic conformal
field theories by using the enlarged symmetries and non--semi-simple
associative algebras appearing in their lattice regularizations (as discussed
in a companion paper). Here we work out in detail two examples of theories
derived as the continuum limit of XXZ spin-1/2 chains, which are related to
spin chains with supersymmetry algebras gl() and gl(),
respectively, with open (or free) boundary conditions in all cases. These
theories can also be viewed as vertex models, or as loop models. Their
continuum limits are boundary conformal field theories (CFTs) with central
charge and respectively, and in the loop interpretation they
describe dense polymers and the boundaries of critical percolation clusters,
respectively. We also discuss the case of dilute (critical) polymers as another
boundary CFT with . Within the supersymmetric formulations, these boundary
CFTs describe the fixed points of certain nonlinear sigma models that have a
supercoset space as the target manifold, and of Landau-Ginzburg field theories.
The submodule structures of indecomposable representations of the Virasoro
algebra appearing in the boundary CFT, representing local fields, are derived
from the lattice. A central result is the derivation of the fusion rules for
these fields
On the Commutative Algebra of Categories
We discuss what it means for a symmetric monoidal category to be a module
over a commutative semiring category. Each of the categories of (1) cartesian
monoidal categories, (2) semiadditive categories, and (3) connective spectra
can be recovered in this way as categories of modules over a commutative
semiring category (or -category in the last case). This language
provides a simultaneous generalization of the formalism of algebraic theories
(operads, PROPs, Lawvere theories) and stable homotopy theory, with essentially
a variant of algebraic K-theory bridging between the two.Comment: 51 pages; some proofs and notation clarified; this is the final
version to appear in Algebraic and Geometric Topolog
Affine Lie algebras and tensor categories
We review briefly the existing vertex-operator-algebraic constructions of
various tensor category structures on module categories for affine Lie
algebras. We discuss the results first conjectured in the work of Moore and
Seiberg that led us to the construction of the modular tensor category
structure in the positive integral level case. Then we review the existing
constructions and results in the following three cases: (i) the level plus the
dual Coxeter number is not a nonnegative rational number, (ii) the level is a
positive integer and (iii) the level is an admissible number. We also present
several open problems.Comment: 14 pages. For the proceedings of the "10th Seminar on Conformal Field
Theory: A conference on Vertex Algebras and Related Topics" held at Research
Institute for Mathematical Sciences, Kyoto University, Kyoto, April 23 - 27,
201
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