191 research outputs found
De Rham Cohomology and Hodge decomposition for Quantum Groups
Let G=G(t,z) be one of the N^2-dimensional bicovariant first order
differential calculi for the quantum groups GL_q(N), SL_q(N), O_q(N), or
Sp_q(N), where q is a transcendental complex number and z is a regular
parameter. It is shown that the de Rham cohomology of Woronowicz' external
algebra G^ coincides with the de Rham cohomologies of its left-coinvariant, its
right-coinvariant and its (twosided) coinvariant subcomplexes. In the cases
GL_q(N) and SL_q(N) the cohomology ring is isomorphic to the coinvariant
external algebra G^_{inv} and to the vector space of harmonic forms. We prove a
Hodge decomposition theorem in these cases. The main technical tool is the
spectral decomposition of the quantum Laplace-Beltrami operator.
Keywords: quantum groups, bicovariant differential calculi, de Rham
cohomology, Laplace-Beltrami operator, Hodge theoryComment: LaTeX2e, 40 page
Drinfeld second realization of the quantum affine superalgebras of via the Weyl groupoid
We obtain Drinfeld second realization of the quantum affine superalgebras
associated with the affine Lie superalgebra . Our results are
analogous to those obtained by Beck for the quantum affine algebras. Beck's
analysis uses heavily the (extended) affine Weyl groups of the affine Lie
algebras. In our approach the structures are based on a Weyl groupoid.Comment: 40 pages, 1 figure. close to the final version to appear in RIMS
Kokyuroku Bessatsu (Besstsu) B8 (2008) 171-21
Braided racks, Hurwitz actions and Nichols algebras with many cubic relations
We classify Nichols algebras of irreducible Yetter-Drinfeld modules over
groups such that the underlying rack is braided and the homogeneous component
of degree three of the Nichols algebra satisfies a given inequality. This
assumption turns out to be equivalent to a factorization assumption on the
Hilbert series. Besides the known Nichols algebras we obtain a new example. Our
method is based on a combinatorial invariant of the Hurwitz orbits with respect
to the action of the braid group on three strands.Comment: v2: 35 pages, 6 tables, 14 figure
On Nichols algebras over SL(2,Fq) and GL(2,Fq)
We compute necessary conditions on Yetter-Drinfeld modules over the groups
SL(2,Fq) and GL(2,Fq) to generate finite dimensional Nichols algebras. This is
a first step towards a classification of pointed Hopf algebras with a group of
group-likes isomorphic to one of these groups.Comment: Major exposition revision, including referees remarks. To appear in
J. Math. Phys. 13 page
The rain forest is a human creation.
The Amazon rain forest was, before the Europeans came, as cultivated as forests anywhere else in the world. It was not 'virgin'. The native people still living within the forest are remnants descended from highly structured societies with suitably developed agriculture and food systems. A vast number of foods including guaraná, açaà and manioc (above) have been cultivated for thousands of years. This revolutionises understanding of the nature and value of Amazonia, the imperative need to protect and strengthen its own identity, and to learn yet again that in agro-ecology is the salvation of the planet
Dirac Operators on Quantum Projective Spaces
We construct a family of self-adjoint operators D_N which have compact
resolvent and bounded commutators with the coordinate algebra of the quantum
projective space CP_q(l), for any l>1 and 0<q<1. They provide 0^+ dimensional
equivariant even spectral triples. If l is odd and N=(l+1)/2, the spectral
triple is real with KO-dimension 2l mod 8.Comment: 54 pages, no figures, dcpic, pdflate
Differential and holomorphic differential operators on noncommutative algebras
Abstract This paper deals with sheaves of differential operators on noncommutative algebras, in a manner related to the classical theory of D-modules. The sheaves are defined by quotienting the tensor algebra of vector fields (suitably deformed by a covariant derivative). As an example we can obtain enveloping algebra like relations for Hopf algebras with differential structures which are not bicovariant. Symbols of differential operators are defined, but not studied. These sheaves are shown to be in the center of a category of bimodules with flat bimodule covariant derivatives. Also holomorphic differential operators are considered
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