3 research outputs found
Automorphisms of the fine grading of sl(n,C) associated with the generalized Pauli matrices
We consider the grading of by the group of
generalized Pauli matrices. The grading decomposes the Lie algebra into
one--dimensional subspaces. In the article we demonstrate that the normalizer
of grading decomposition of in is the group , where is the cyclic group of order . As an
example we consider graded by and all contractions
preserving that grading. We show that the set of 48 quadratic equations for
grading parameters splits into just two orbits of the normalizer of the grading
in
Representations of the q-deformed algebra U'_q(so_4)
We study the nonstandard -deformation of the universal
enveloping algebra obtained by deforming the defining relations
for skew-symmetric generators of . This algebra is used in
quantum gravity and algebraic topology. We construct a homomorphism of
to the certain nontrivial extension of the Drinfeld--Jimbo
quantum algebra and show that this homomorphism
is an isomorphism. By using this homomorphism we construct irreducible finite
dimensional representations of the classical type and of the nonclassical type
for the algebra . It is proved that for not a root of
unity each irreducible finite dimensional representation of
is equivalent to one of these representations. We prove that every finite
dimensional representation of for not a root of unity is
completely reducible. It is shown how to construct (by using the homomorphism
) tensor products of irreducible representations of .
(Note that no Hopf algebra structure is known for .) These
tensor products are decomposed into irreducible constituents.Comment: 28 pages, LaTe
Representations of the cyclically symmetric q-deformed algebra
An algebra homomorphism from the nonstandard q-deformed (cyclically
symmetric) algebra to the extension of the Hopf
algebra is constructed. Not all irreducible representations of
can be extended to representations of . Composing
the homomorphism with irreducible representations of
we obtain representations of . Not all of these representations of
are irreducible. Reducible representations of are
decomposed into irreducible components. In this way we obtain all irreducible
representations of when is not a root of unity. A part of these
representations turns into irreducible representations of the Lie algebra
so when . Representations of the other part have no classical
analogue. Using the homomorphism it is shown how to construct tensor
products of finite dimensional representations of . Irreducible
representations of when is a root of unity are constructed.
Part of them are obtained from irreducible representations of by means of the homomorphism .Comment: 28 pages, LaTe