Automorphisms of the fine grading of sl(n,C) associated with the generalized Pauli matrices

We consider the grading of $sl(n,\mathbb{C})$ by the group $\Pi_n$ of generalized Pauli matrices. The grading decomposes the Lie algebra into $n^2-1$ one--dimensional subspaces. In the article we demonstrate that the normalizer of grading decomposition of $sl(n,\mathbb{C})$ in $\Pi_n$ is the group $SL(2, \mathbb{Z}_n)$, where $\mathbb{Z}_n$ is the cyclic group of order $n$. As an example we consider $sl(3,\mathbb{C})$ graded by $\Pi_3$ 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 $\Pi_3$

Representations of the q-deformed algebra U'_q(so_4)

We study the nonstandard $q$-deformation $U'_q({\rm so}_4)$ of the universal enveloping algebra $U({\rm so}_4)$ obtained by deforming the defining relations for skew-symmetric generators of $U({\rm so}_4)$. This algebra is used in quantum gravity and algebraic topology. We construct a homomorphism $\phi$ of $U'_q({\rm so}_4)$ to the certain nontrivial extension of the Drinfeld--Jimbo quantum algebra $U_q({\rm sl}_2)^{\otimes 2}$ 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 $U'_q({\rm so}_4)$. It is proved that for $q$ not a root of unity each irreducible finite dimensional representation of $U'_q({\rm so}_4)$ is equivalent to one of these representations. We prove that every finite dimensional representation of $U'_q({\rm so}_4)$ for $q$ not a root of unity is completely reducible. It is shown how to construct (by using the homomorphism $\phi$) tensor products of irreducible representations of $U'_q({\rm so}_4)$. (Note that no Hopf algebra structure is known for $U'_q({\rm so}_4)$.) These tensor products are decomposed into irreducible constituents.Comment: 28 pages, LaTe

Representations of the cyclically symmetric q-deformed algebra soq(3)so_q(3)

An algebra homomorphism $\psi$ from the nonstandard q-deformed (cyclically symmetric) algebra $U_q(so_3)$ to the extension ${\hat U}_q(sl_2)$ of the Hopf algebra $U_q(sl_2)$ is constructed. Not all irreducible representations of $U_q(sl_2)$ can be extended to representations of ${\hat U}_q(sl_2)$. Composing the homomorphism $\psi$ with irreducible representations of ${\hat U}_q(sl_2)$ we obtain representations of $U_q(so_3)$. Not all of these representations of $U_q(so_3)$ are irreducible. Reducible representations of $U_q(so_3)$ are decomposed into irreducible components. In this way we obtain all irreducible representations of $U_q(so_3)$ when $q$ is not a root of unity. A part of these representations turns into irreducible representations of the Lie algebra so$_3$ when $q\to 1$. Representations of the other part have no classical analogue. Using the homomorphism $\psi$ it is shown how to construct tensor products of finite dimensional representations of $U_q(so_3)$. Irreducible representations of $U_q(so_3)$ when $q$ is a root of unity are constructed. Part of them are obtained from irreducible representations of ${\hat U}_q(sl_2)$ by means of the homomorphism $\psi$.Comment: 28 pages, LaTe