On a general analytical formula for U_q(su(3))-Clebsch-Gordan coefficients

We present the projection operator method in combination with the Wigner-Racah calculus of the subalgebra U_q(su(2)) for calculation of Clebsch-Gordan coefficients (CGCs) of the quantum algebra U_q(su(3)). The key formulas of the method are couplings of the tensor and projection operators and also a tensor form for the projection operator of U_q(su(3)). We obtain a very compact general analytical formula for the U_q(su(3)) CGCs in terms of the U_q(su(2)) Wigner 3nj-symbols.Comment: 9 pages, LaTeX; to be published in Yad. Fiz. (Phys. Atomic Nuclei), (2001

q-Analog of Gelfand-Graev Basis for the Noncompact Quantum Algebra U_q(u(n,1))

For the quantum algebra U_q(gl(n+1)) in its reduction on the subalgebra U_q(gl(n)) an explicit description of a Mickelsson-Zhelobenko reduction Z-algebra Z_q(gl(n+1),gl(n)) is given in terms of the generators and their defining relations. Using this Z-algebra we describe Hermitian irreducible representations of a discrete series for the noncompact quantum algebra U_q(u(n,1)) which is a real form of U_q(gl(n+1)), namely, an orthonormal Gelfand-Graev basis is constructed in an explicit form.Comment: Invited talk given by V.N.T. at XVIII International Colloquium "Integrable Systems and Quantum Symmetries", June 18--20, 2009, Prague, Czech Republi

Extremal projectors for contragredient Lie (super)symmetries (short review)

A brief review of the extremal projectors for contragredient Lie (super)symmetries (finite-dimensional simple Lie algebras, basic classical Lie superalgebras, infinite-dimensional affine Kac-Moody algebras and superalgebras, as well as their quantum $q$-analogs) is given. Some bibliographic comments on the applications of extremal projectors are presented.Comment: 21 pages, LaTeX; typos corrected, references adde

Mickelsson algebras and Zhelobenko operators

We construct a family of automorphisms of Mickelsson algebra, satisfying braid group relations. The construction uses 'Zhelobenko cocycle' and includes the dynamical Weyl group action as a particular case

Restoration of particle number as a good quantum number in BCS theory

As shown in previous work, number projection can be carried out analytically for states defined in a quasi-particle scheme when the states are expressed in a coherent state representation. The wave functions of number-projected states are well-known in the theory of orthogonal polynomials as Schur functions. Moreover, the functions needed in pairing theory are a particularly simple class of Schur functions that are easily constructed by means of recursion relations. It is shown that complete sets of states can be projected from corresponding quasi-particle states and that such states retain many of the properties of the quasi-particle states from which they derive. It is also shown that number projection can be used to construct a complete set of orthogonal states classified by generalized seniority for any nucleus.Comment: 21 pages, 2 figures, epsf.def style file for printing figure

РЕГИСТР КАК СРЕДСТВО УЛУЧШЕНИЯ КАЧЕСТВА МЕДИЦИНСКОЙ ПОМОЩИ БОЛЬНЫМ МУКОВИСЦИДОЗОМ

The creation of a register of mucoviscidosis (MV) patients is necessary for determining the epidemic situation in the region, evaluating the efficacy of therapeutic strategies and the quality of healthcare provided. The regional register of Yaroslavl contains information about 53 MV patients. The average age of these patients is 12.9 years, the amount of patients older than 18 is 22.7%, the average age of diagnosis is 3.4 years. The overall survival median is 26.8 years (by the beginning of 2012). The prevalence of the disease is 1:8005 newborns according to the results of neonatal screening. The amount of patients infected with Pseudomonas aeruginosa and Burkholderia cepacia is 30,2% and 2,5% respectively. The F508del mutation occurs in 43,4% of all cases. The next most prevalent mutations are N1303K, 394delTT, CFTRdele2,3 (21kb) (4,72% each), the number of unidentified mutations is only 8.49%. A comparative evaluation of therapeutic approaches (basic therapy) was conducted in the Yaroslavl region and in a number of European countries. Data obtained from the register allows to solve not only clinical and epidemiological problems, but also sort out organizational issues, plan medicine provisions and conduct medical and social rehabilitation.Создание регистра больных муковисцидозом (МВ) является необходимым для определения эпидемиологической ситуации в регионе, оценки эффективности терапевтических стратегий и качества оказания медицинской помощи. В Ярославском региональном регистре содержится информация о 53 больных МВ. Средний возраст больных 12,9 лет, количество пациентов старше 18 лет составляет 22,7%, средние сроки постановки диагноза 3,2 года. Общая медиана выживаемости на начало 2012 года — 26,8 лет. Распространенность заболевания согласно неонатальному скринингу составляет 1:8005 новорожденных. Число больных, инфицированных Pseudomonas aeruginosa и Burkholderia cepacia, — 30,2 и 2,5%, соответственно. Мутация F508del встречается в 43,4% случаев. Следующими по частоте являются мутации N1303K, 394delT, CFTRdele2,3 (21kb) (по 4,72%), число неидентифицированных мутаций составляет лишь 8,49%. Проведена сравнительная оценка терапевтических подходов (базисной терапии) в Ярославской области и ряде Европейских стран. Помимо клинико-эпидемиологических задач данные регистра позволяют решить организационные вопросы, планировать лекарственное обеспечение, осуществлять медико-социальную реабилитацию.

q-Analog of Gelfand-Graev Basis for the Noncompact Quantum Algebra Uq(u(n,1))

For the quantum algebra Uq(gl(n+1)) in its reduction on the subalgebra Uq(gl(n)) an explicit description of a Mickelsson-Zhelobenko reduction Z-algebra Zq(gl(n+1),gl(n)) is given in terms of the generators and their defining relations. Using this Z-algebra we describe Hermitian irreducible representations of a discrete series for the noncompact quantum algebra Uq(u(n,1)) which is a real form of Uq(gl(n+1)), namely, an orthonormal Gelfand-Graev basis is constructed in an explicit form

Deformations of the Boson sp(4,R)sp(4,R) Representation and its Subalgebras

The boson representation of the sp(4,R) algebra and two distinct deformations of it, are considered, as well as the compact and noncompact subalgebras of each. The initial as well as the deformed representations act in the same Fock space. One of the deformed representation is based on the standard q-deformation of the boson creation and annihilation operators. The subalgebras of sp(4,R) (compact u(2) and three representations of the noncompact u(1,1) are also deformed and are contained in this deformed algebra. They are reducible in the action spaces of sp(4,R) and decompose into irreducible representations. The other deformed representation, is realized by means of a transformation of the q-deformed bosons into q-tensors (spinor-like) with respect to the standard deformed su(2). All of its generators are deformed and have expressions in terms of tensor products of spinor-like operators. In this case, an other deformation of su(2) appears in a natural way as a subalgebra and can be interpreted as a deformation of the angular momentum algebra so(3). Its representation is reducible and decomposes into irreducible ones that yields a complete description of the same

Feigin-Frenkel center in types B, C and D

For each simple Lie algebra g consider the corresponding affine vertex algebra V_{crit}(g) at the critical level. The center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of Feigin and Frenkel about two decades ago. However, only recently simple formulas for the generators of the center were found for the Lie algebras of type A following Talalaev's discovery of explicit higher Gaudin Hamiltonians. We give explicit formulas for generators of the centers of the affine vertex algebras V_{crit}(g) associated with the simple Lie algebras g of types B, C and D. The construction relies on the Schur-Weyl duality involving the Brauer algebra, and the generators are expressed as weighted traces over tensor spaces and, equivalently, as traces over the spaces of singular vectors for the action of the Lie algebra sl_2 in the context of Howe duality. This leads to explicit constructions of commutative subalgebras of the universal enveloping algebras U(g[t]) and U(g), and to higher order Hamiltonians in the Gaudin model associated with each Lie algebra g. We also introduce analogues of the Bethe subalgebras of the Yangians Y(g) and show that their graded images coincide with the respective commutative subalgebras of U(g[t]).Comment: 29 pages, constructions of Pfaffian-type Sugawara operators and commutative subalgebras in universal enveloping algebras are adde

Weight bases of Gelfand-Tsetlin type for representations of classical Lie algebras

This paper completes a series devoted to explicit constructions of finite-dimensional irreducible representations of the classical Lie algebras. Here the case of odd orthogonal Lie algebras (of type B) is considered (two previous papers dealt with C and D types). A weight basis for each representation of the Lie algebra o(2n+1) is constructed. The basis vectors are parametrized by Gelfand--Tsetlin-type patterns. Explicit formulas for the matrix elements of generators of o(2n+1) in this basis are given. The construction is based on the representation theory of the Yangians. A similar approach is applied to the A type case where the well-known formulas due to Gelfand and Tsetlin are reproduced.Comment: 29 pages, Late