Mathematics – Quantum Algebra
Scientific paper
2001-04-03
Mathematics
Quantum Algebra
28 pages, LaTeX; lectures given at the NATO Advanced Study Institute "Special Functions 2000"
Scientific paper
In these lectures we develop the projection operator method for quantum groups. Here the term "quantum groups" means q-deformed universal enveloping algebras of contragredient Lie (super)algebras of finite growth. Contains of the lectures can be divided on two parts. Basis fragments of the first part are: combinatorial structure of root systems, the q-analog of the Cartan-Weyl basis, the extremal projector and the universal R-matrix for any contragredient Lie (super)algebra of finite growth. The explicit expressions for the extremal projectors and the universal R-matrices are ordered products of special q-series depending on noncommutative Cartan-Weyl generators. In second part we consider some applications of the extremal projectors. Here we use the projector operator method to develop the theory of the Clebsch-Gordan coefficients for the quantum algebras U_q(su(2)) and U_q(su(3)). In particular, we give a very compact general formula for the canonical U_q(su(3))\supset U_q(su(2)) Clebsch-Gordan coefficients in terms of the U_q(su(2)) Wigner 3nj-symbols which are connected with the basic hyperheometric series. Then we apply the projection operator method for the construction of the q-analog of the Gelfand-Tsetlin basis for U_q(su(n)). Finally using analogy between the extremal projector p(U_q(sl(2))) of the quantum algebra U_q(sl(2)) and the \delta(x)-function we introduce 'adjoint extremal projectors' p^{(n)}(U_q(sl(2)) (n=1,2,...) which are some generalizations of the extremal projector p(U_q(sl(2))), and which are analogies of the derivatives of the \delta(x)-function, \delta^{(n)}(x).
No associations
LandOfFree
Projection operator method for quantum groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Projection operator method for quantum groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Projection operator method for quantum groups will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-173639