78 research outputs found
Quantum double of {\rm U}_q((\ksl_2)^{\leq 0})
Let be the quantized enveloping algebra associated to the
simple Lie algebra . In this paper, we study the quantum double of
the Borel subalgebra of . We construct an
analogue of Kostant--Lusztig -form for and show that it is
a Hopf subalgebra. We prove that, over an algebraically closed field, every
simple -module is the pullback of a simple -module through
certain surjection from onto , and the category of finite
dimensional weight -modules is equivalent to a direct sum of
copies of the category of finite dimensional weight
-modules. As an application, we recover (in a conceptual way)
Chen's results as well as Radford's results on the quantum double of Taft
algebra. Our main results allow a direct generalization to the quantum double
of the Borel subalgebra of the quantized enveloping algebra associated to
arbitrary Cartan matrix
Finite-dimensional quasi-Hopf algebras of Cartan type
In this paper, we present a general method for constructing
finite-dimensional quasi-Hopf algebras from finite abelian groups and braided
vector spaces of Cartan type. The study of such quasi-Hopf algebras leads to
the classification of finite-dimensional radically graded basic quasi-Hopf
algebras over abelian groups with dimensions not divisible by and
associators given by abelian -cocycles. As special cases , the small
quasi-quantum groups are introduced and studied. Many new explicit examples of
finite-dimensional genuine quasi-Hopf algebras are obtained
The Green rings of the generalized Taft Hopf algebras
In this paper, we investigate the Green ring of the generalized
Taft algebra , extending the results of Chen, Van Oystaeyen and Zhang
in \cite{Coz}. We shall determine all nilpotent elements of the Green ring
. It turns out that each nilpotent element in can be
written as a sum of indecomposable projective representations. The Jacobson
radical of is generated by one element, and its
rank is . Moreover, we will present all the finite dimensional
indecomposable representations over the complexified Green ring of
Our analysis is based on the decomposition of the tensor product of
indecomposable representations and the observation of the solutions for the
system of equations associated to the generating relations of the Green ring
.Comment: To appear in Contemporary Math. AMS, Proceeding of Hopf Algebra
conference in Almeria, 201
Local cohomology associated to the radical of a group action on a noetherian algebra
An arbitrary group action on an algebra results in an ideal
of . This ideal fits into the classical
radical theory, and will be called the radical of the group action. If is a
noetherian algebra with finite GK-dimension and is a finite group, then the
difference between the GK-dimensionsof and that of is
called the pertinency of the group action. We provide some methods to find
elements of the radical, which helps to calculate the pertinency of some
special group actions. The -adic local cohomology of is
related to the singularities of the invariant subalgebra . We establish an
equivalence between the quotient category of the invariant and that of
the skew group ring through the torsion theory associated to the radical
. With the help of the equivalence, we show that the invariant
subalgebra will inherit certain Cohen-Macaulay property from .Comment: 26 page
Braided autoequivalences and quantum commutative bi-Galois objects
Let be a quasitriangular weak Hopf algebra over a field . We show
that there is a braided monoidal equivalence between the Yetter-Drinfeld module
category over and the category of comodules over some
braided Hopf algebra in the category . Based on this
equivalence, we prove that every braided bi-Galois object over the braided
Hopf algebra defines a braided autoequivalence of the category
if and only if is quantum commutative. In case is
semisimple over an algebraically closed field, i.e. the fusion case, then every
braided autoequivalence of trivializable on
is determined by such a quantum commutative Galois object. The quantum
commutative Galois objects in form a group measuring the Brauer
group of as studied in [20] in the Hopf algebra case
Reconstruction of tensor categories from their structure invariants
In this paper, we study tensor (or monoidal) categories of finite rank over
an algebraically closed field . Given a tensor category
, we have two structure invariants of : the Green
ring (or the representation ring) and the Auslander algebra
of . We show that a Krull-Schmit abelian tensor
category of finite rank is uniquely determined (up to tensor
equivalences) by its two structure invariants and the associated associator
system of . In fact, we can reconstruct the tensor category
from its two invarinats and the associator system. More general,
given a quadruple satisfying certain conditions, where is
a -ring of rank , is a finite dimensional -algebra with a complete set of primitive orthogonal idempotents,
is an algebra map from to an algebra
constructed from and , and is a family of
"invertible" matrices over , we can construct a Krull-Schmidt and abelian
tensor category over such that is the Green ring
of and is the Auslander algebra of . In this case,
has finitely many indecomposable objects (up to isomorphisms) and
finite dimensional Hom-spaces. Moreover, we will give a necessary and
sufficient condition for such two tensor categories to be tensor equivalent
Cocycle Deformations and Brauer Group Isomorphisms
Let be a Hopf algebra over a commutative ring with unity and
be a cocycle on . In this paper, we
show that the Yetter-Drinfeld module category of the cocycle deformation Hopf
algebra is equivalent to the Yetter-Drinfeld module category of
. As a result of the equivalence, the "quantum Brauer" group BQ is
isomorphic to BQ. Moreover, the group \Gal(\HR) constructed
in \cite{Z} is studied under a cocycle deformation.Comment: 33 pages, no figure
The Calabi-Yau property of Hopf algebras and braided Hopf algebras
Let be a finite dimensional semisimple Hopf algebra and a braided
Hopf algebra in the category of Yetter-Drinfeld modules over . When is a
Calabi-Yau algebra, a necessary and sufficient condition for R#H to be a
Calabi-Yau Hopf algebra is given. Conversely, when is the group algebra of
a finite group and the smash product R#H is a Calabi-Yau algebra, we give a
necessary and sufficient condition for the algebra to be a Calabi-Yau
algebra.Comment: 30 page
Calabi-Yau pointed Hopf algebras of finite Cartan type
We study the Calabi-Yau property of pointed Hopf algebra U(\mc{D},\lmd) of
finite Cartan type. It turns out that this class of pointed Hopf algebras
constructed by N. Andruskiewitsch and H.-J. Schneider contains many Calabi-Yau
Hopf algebras. To give concrete examples of new Calabi-Yau Hopf algebras, we
classify the Calabi-Yau pointed Hopf algebras U(\mc{D},\lmd) of dimension
less than 5.Comment: 42 pages, introduction slightly revise
Calabi-Yau Nichols algebras of Hecke type
Let be a Nichols algebra of Hecke type. In this paper, we show that if
is Noetherian and of finite global dimension, then has a rigid
dualizing complex. We then give a necessary and sufficient condition for to
be a Calabi-Yau algebra.Comment: no figure
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