1,470 research outputs found
The center of
We determine the center of a localization of by the covariant elements
(non-mutable elements) by means of constructions and results from quantum
cluster algebras. In our set-up, is any finite-dimensional
complex Lie algebra and is any element in the Weyl group . The
non-zero complex parameter is mostly assumed not to be a root of unity, but
our method also gives many details in case is a primitive root of unity. We
point to a new and very useful direction of approach to a general set of
problems which we exemplify here by obtaining the result that the center is
determined by the null space of . Further, we use this to give a
generalization to double Schubert Cell algebras where the center is proved to
be given by . Another family of
quadratic algebras is also considered and the centers determined.Comment: 28 pages LaTeX. Relevant references as well as a new section relating
to the root-of-unity case have been added. Now in print with minor change
Quantized Dirac Operators
We determine what should correspond to the Dirac operator on certain
quantized hermitian symmetric spaces and what its properties are. A new insight
into the quantized wave operator is obtained.Comment: To appear in the Proceedings of the Quantum Groups And Integrable
Systems meeting in Prag, June 22-24 2000. To be published with the
Czechoslovak Journal of Physi
Special classes of homomorphisms between generalized Verma modules for
We study homomorphisms between quantized generalized Verma modules
for . There is a natural notion of degree for such
maps, and if the map is of degree , we write .
We examine when one can have a series of such homomorphisms
, where
denotes the map . If, classically, , then and . The answer is then that must be
one-sided in the sense that either or
(non-exclusively). There are further demands on if we insist on
homomorphisms. However, it is also
interesting to loosen this to considering only homomorphisms, in which case the conditions on
disappear. By duality, there result have implications on covariant quantized
differential operators. We finish by giving an explicit, though sketched,
determination of the full set of
homomorphisms .Comment: 10 pages proceedings of Group 32, Prague 201
Quantized Heisenberg Space
We investigate the algebra introduced by Faddeev, Reshetikhin and
Takhadjian. In case is a primitive root of unity the degree, the center,
and the set of irreducible representations are found. The Poisson structure is
determined and the De Concini-Kac-Procesi Conjecture is proved for this case.
In the case of generic, the primitive ideals are described. A related
algebra studied by Oh is also treated.Comment: 20 pages LaTeX documen
Quantized rank R matrices
First some old as well as new results about P.I. algebras, Ore extensions,
and degrees are presented. Then quantized matrices as well as
quantized factor algebras of are analyzed. The latter are the
quantized function algebra of rank matrices obtained by working modulo the
ideal generated by all quantum subdeterminants and a
certain localization of this algebra is proved to be isomorphic to a more
manageable one. In all cases, the quantum parameter is a primitive th roots
of unity. The degrees and centers of the algebras are determined when is a
prime and the general structure is obtained for arbitrary .Comment: 18 pages with 3 eps figures. Some proofs in Section 5 have been
changed and a remark has been remove
The exponential nature and positivity
In the present article, a basis of the coordinate algebra of the
multi-parameter quantized matrix is constructed by using an elementary method
due to Lusztig. The construction depends heavily on an anti-automorphism, the
bar action. The exponential nature of the bar action is derived which provides
an inductive way to compute the basis elements. By embedding the basis into the
dual basis of Lusztig's canonical basis of , the positivity
properties of the basis as well as the positivity properties of the canonical
basis of the modified quantum enveloping algebra of type , which has been
conjectured by Lusztig, are proved
Double-partition Quantum Cluster Algebras
A family of quantum cluster algebras is introduced and studied. In general,
these algebras are new, but subclasses have been studied previously by other
authors. The algebras are indexed by double partitions or double flag
varieties. Equivalently, they are indexed by broken lines . By grouping
together neighboring mutations into quantum line mutations we can mutate from
the cluster algebra of one broken line to another. Compatible pairs can be
written down. The algebras are equal to their upper cluster algebras. The
variables of the quantum seeds are given by elements of the dual canonical
basis.
This is the final version, where some arguments have been expanded and/or
improved and several typos corrected. Full bibliographic details: Journal of
Algebra (2012), pp. 172-203 DOI information: 10.1016/j.jalgebra.2012.09.015Comment: LaTeX 39 page
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