570 research outputs found
Double affine Hecke algebras and affine flag manifolds, I
This lecture reviews the classification of simple modules of double affine
Hecke algebras via the K-theory of Steinberg varieties of affine typeComment: 52 page
Frobenius-Schur Indicators and Exponents of Spherical Categories
We obtain two formulae for the higher Frobenius-Schur indicators: one for a
spherical fusion category in terms of the twist of its center and the other one
for a modular tensor category in terms of its twist. The first one is a
categorical generalization of an analogous result by Kashina, Sommerhauser, and
Zhu for Hopf algebras, and the second one extends Bantay's 2nd indicator
formula for a conformal field theory to higher degree. These formulae imply the
sequence of higher indicators of an object in these categories is periodic. We
define the notion of Frobenius-Schur (FS-)exponent of a pivotal category to be
the global period of all these sequences of higher indicators, and we prove
that the FS-exponent of a spherical fusion category is equal to the order of
the twist of its center. Consequently, the FS-exponent of a spherical fusion
category is a multiple of its exponent, in the sense of Etingof, by a factor
not greater than 2. As applications of these results, we prove that the
exponent and the dimension of a semisimple quasi-Hopf algebra H have the same
prime divisors, which answers two questions of Etingof and Gelaki affirmatively
for quasi-Hopf algebras. Moreover, we prove that the FS-exponent of H divides
dim(H)^4. In addition, if H is a group-theoretic quasi-Hopf algebra, the
FS-exponent of H divides dim(H)^2, and this upper bound is shown to be tight.Comment: 32p. LaTex file with macros and figures. Some typos and Thm 8.4 in v2
have been corrected. The current Thm 8.4 is a combined result of Thms 8.4 and
8.5 in version
Quasideterminants
The determinant is a main organizing tool in commutative linear algebra. In
this review we present a theory of the quasideterminants defined for matrices
over a division algebra. We believe that the notion of quasideterminants should
be one of main organizing tools in noncommutative algebra giving them the same
role determinants play in commutative algebra.Comment: amstex; final version; to appear in Advances in Mat
Rational K-matrices for finite-dimensional representations of quantum affine algebras
Let be a complex simple Lie algebra. We prove that every
finite-dimensional representation of the (untwisted) quantum affine algebra
gives rise to a family of spectral K-matrices, namely
solutions of Cherednik's generalized reflection equation, which depends upon
the choice of a quantum affine symmetric pair . Moreover, we prove that every irreducible representation
over remains generically irreducible under restriction to
. From the latter result, we deduce that every obtained
K-matrix can be normalized to a matrix-valued rational function in a
multiplicative parameter, known in the study of quantum integrability as a
trigonometric K-matrix. Finally, we show that our construction recovers many of
the known solutions of the standard reflection equation and gives rise to a
large class of new solutions.Comment: 37 page
Modules of covariants in modular invariant theory
Let the finite group act linearly on the vector space over the field
of arbitrary characteristic. If is a subgroup the extension of
invariant rings is studied using modules of covariants.
An example of our results is the following. Let be the subgroup of
generated by the reflections in . A classical theorem due to Serre says that
if is a free -module then . We generalize this result as
follows. If is a free -module then is generated by and
, and the invariant ring is free over and
generated as an algebra by -invariants and -invariants.Comment: 36 pages, proofs of main theorems have been improve
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