1,990 research outputs found
Symplectic Leaves of Complex Reductive Poisson-Lie Groups
All factorizable Lie bialgebra structures on complex reductive Lie algebras
were described by Belavin and Drinfeld. We classify the symplectic leaves of
the full class of corresponding connected Poisson-Lie groups. A formula for
their dimensions is also proved.Comment: 46 pages, LaTeX2e, Theorem 1.10 proved in the general case, minor
misprints correcte
Affine Jacquet functors and Harish-Chandra categories
We define an affine Jacquet functor and use it to describe the structure of
induced affine Harish-Chandra modules at noncritical levels, extending the
theorem of Kac and Kazhdan [KK] on the structure of Verma modules in the
Bernstein-Gelfand-Gelfand categories O for Kac-Moody algebras. This is combined
with a vanishing result for certain extension groups to construct a block
decomposition of the categories of affine Harish-Chandra modules of Lian and
Zuckerman [LZ]. The latter provides an extension of the works of Rocha-Caridi,
Wallach [RW] and Deodhar, Gabber, Kac [DGK] on block decompositions of BGG
categories for Kac-Moody algebras. We also prove a compatibility relation
between the affine Jacquet functor and the Kazhdan-Lusztig tensor product. A
modification of this is used to prove that the affine Harish-Chandra category
is stable under fusion tensoring with the Kazhdan-Lusztig category (a case of
our finiteness result [Y]) and will be further applied in studying translation
functors for Kac-Moody algebras, based on the fusion tensor product.Comment: 29 pages, AMS-Latex, v2 contains several minor change
A classification of H-primes of quantum partial flag varieties
We classify the invariant prime ideals of a quantum partial flag variety
under the action of the related maximal torus. As a result we construct a
bijection between them and the torus orbits of symplectic leaves of the
standard Poisson structure on the corresponding flag variety. It was previously
shown by K. Goodearl and the author that the latter are precisely the Lusztig
strata of the partial flag variety.Comment: 14 pages, AMSLatex. to appear in Proc. AM
The Launois-Lenagan conjecture
In this note we prove the Launois-Lenagan conjecture on the classification of
the automorphism groups of the algebras of quantum matrices R_q[M_n] of square
shape for all positive integers n, base fields K, and deformation parameters q
\in K^* which are not roots of unity.Comment: 7 pages, AMS Latex, v. 2 contains a shorter proof of the result,
minor changes in the final v.
Strata of prime ideals of De Concini-Kac-Procesi algebras and Poisson geometry
To each simple Lie algebra g and an element w of the corresponding Weyl group
De Concini, Kac and Procesi associated a subalgebra U^w_- of the quantized
universal enveloping algebra U_q(g), which is a deformation of the universal
enveloping algebra U(n_- \cap w(n_+)) and a quantization of the coordinate ring
of the Schubert cell corresponding to w. The torus invariant prime ideals of
these algebras were classified by M\'eriaux and Cauchon [25], and the author
[30]. These ideals were also explicitly described in [30]. They index the the
Goodearl-Letzter strata of the stratification of the spectra of U^w_- into
tori. In this paper we derive a formula for the dimensions of these strata and
the transcendence degree of the field of rational Casimirs on any open
Richardson variety with respect to the standard Poisson structure [15].Comment: 15 pages, AMS-Latex, v. 2 contains extended Sect.
Rigidity of quantum tori and the Andruskiewitsch-Dumas conjecture
We prove the Andruskiewitsch-Dumas conjecture that the automorphism group of
the positive part of the quantized universal enveloping algebra
of an arbitrary finite dimensional simple Lie algebra g
is isomorphic to the semidirect product of the automorphism group of the Dynkin
diagram of g and a torus of rank equal to the rank of g. The key step in our
proof is a rigidity theorem for quantum tori. It has a broad range of
applications. It allows one to control the (full) automorphism groups of large
classes of associative algebras, for instance quantum cluster algebras.Comment: 31 pages, AMS Latex, v.3 contains an application to the isomorphism
problem for the algebras U_q^+(g) suggested by L. Scott, minor changes in v.
Quantized Weyl algebras at roots of unity
We classify the centers of the quantized Weyl algebras that are PI and derive
explicit formulas for the discriminants of these algebras over a general class
of polynomial central subalgebras. Two different approaches to these formulas
are given: one based on Poisson geometry and deformation theory, and the other
using techniques from quantum cluster algebras. Furthermore, we classify the PI
quantized Weyl algebras that are free over their centers and prove that their
discriminants are locally dominating and effective. This is applied to solve
the automorphism and isomorphism problems for this family of algebras and their
tensor products.Comment: 27 pages, AMS Late
-vectors of 2-Calabi--Yau categories and Borel subalgebras of
We develop a general framework for -vectors of 2-Calabi--Yau categories,
which deals with cluster tilting subcategories that are not reachable from each
other and contain infinitely many indecomposable objects. It does not rely on
iterative sequences of mutations.
We prove a categorical (infinite-rank) generalization of the
Nakanishi--Zelevinsky duality for -vectors and establish two formulae for
the effective computation of -vectors -- one in terms of indices and the
other in terms of dimension vectors for cluster tilted algebras.
In this framework, we construct a correspondence between the -vectors of
the cluster categories of type due to
Igusa--Todorov and the roots of the Borel subalgebras of
. Contrary to the finite dimensional case, the Borel
subalgebras of are not conjugate to each other. On
the categorical side, the cluster tilting subcategories of
exhibit different homological properties. The
correspondence builds a bridge between the two classes of objects.Comment: This is the final version, which has been accepted for publication in
Selecta Mathematica. 37 page
Quantum Schubert cells via representation theory and ring theory
We resolve two questions of Cauchon and Meriaux on the spectra of the quantum
Schubert cell algebras U^-[w]. The treatment of the first one unifies two very
different approaches to Spec U^-[w], a ring theoretic one via deleting
derivations and a representation theoretic one via Demazure modules. The
outcome is that now one can combine the strengths of both methods. As an
application we solve the containment problem for the Cauchon-Meriaux
classification of torus invariant prime ideals of U^-[w]. Furthermore, we
construct explicit models in terms of quantum minors for the Cauchon quantum
affine space algebras constructed via the procedure of deleting derivations
from all quantum Schubert cell algebras U^-[w]. Finally, our methods also give
a new, independent proof of the Cauchon-Meriaux classification.Comment: 29 pages, AMS Latex, minor changes in v.
Symmetric pairs for Nichols algebras of diagonal type via star products
We construct symmetric pairs for Drinfeld doubles of pre-Nichols algebras of
diagonal type and determine when they possess an Iwasawa decomposition. This
extends G. Letzter's theory of quantum symmetric pairs. Our results can be
uniformly applied to Kac-Moody quantum groups for a generic quantum parameter,
for roots of unity in respect to both big and small quantum groups, to quantum
supergroups and to exotic quantum groups of ufo type. We give a second
construction of symmetric pairs for Heisenberg doubles in the above generality
and prove that they always admit an Iwasawa decomposition.
For symmetric pair coideal subalgebras with Iwasawa decomposition in the
above generality we then address two problems which are fundamental already in
the setting of quantum groups. Firstly, we show that the symmetric pair coideal
subalgebras are isomorphic to intrinsically defined deformations of partial
bosonizations of the corresponding pre-Nichols algebras. To this end we develop
a general notion of star products on N-graded connected algebras which provides
an efficient tool to prove that two deformations of the partial bosonization
are isomorphic. The new perspective also provides an effective algorithm for
determining the defining relations of the coideal subalgebras.
Secondly, for Nichols algebras of diagonal type, we use the linear
isomorphism between the coideal subalgebra and the partial bosonization to give
an explicit construction of quasi K-matrices as sums over dual bases. We show
that the resulting quasi K-matrices give rise to weakly universal K-matrices in
the above generality.Comment: 54 page
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