Search

Splint is a decomposition of root system into union of root systems. Splint of root system for simple Lie algebra appears naturally in studies of (regular) embeddings of reductive subalgebras. Splint can be used to construct branching rules. We consider special embedding of Lie subalgebra to Lie algebra. We classify projections of algebra root systems using extended Dynkin diagrams and single out the conditions of splint appearance and coincidence of branching coefficients with weight multiplicities. While such a coincidence is not very common it is connected with Gelfand-Tsetlin basis., Comment: 13 pages, 3 figures, 2 tables

We study the decomposition problem for tensor powers of $B_2$-fundamental modules. To solve this problem singular weight technique and injection fan algorithms are applied. Properties of multiplicity coefficients are formulated in terms of multiplicity functions. These functions are constructed showing explicitly the dependence of multiplicity coefficients on the highest weight coordinates and the tensor power parameter. It is thus possible to study general properties of multiplicity coefficients for powers of the fundamental $B_2$- modules., Comment: 17 pages

Recurrent relations for branching coefficients are based on a special type of singular element decomposition. We show that this decomposition can be used to construct the parabolic Verma modules and finally to obtain the generalized Weyl-Verma formulas for characters. We demonstrate how branching coefficients can determine the generalized BGG resolution sequence., Comment: 15 pages, 2 figures, submitted to Theoretical and Mathematical Physics

Recurrent relations for branching coefficients in affine Lie algebras integrable highest weight modules are studied. The decomposition algorithm based on the injection fan technique is developed for the case of an arbitrary reductive subalgebra. In particular we consider the situation where the Weyl denominator becomes singular with respect to the subalgebra. We demonstrate that for any reductive subalgebra it is possible to define the injection fan and the analogue of the Weyl numerator - the tools that describe explicitly the recurrent properties of branching coefficients. Possible applications of the fan technique in CFT models are considered., Comment: LaTeX, 23 pages, 13 figures, section 2 is clarified, new lemma and theorem are introduced, figures are improved, submitted to Journal of Physics A: Mathematical and Theoretical

The recursion relations of branching coefficients $k_{\xi}^{(\mu)}$ for a module $L_{\frak{g}\downarrow \frak{h}}^{\mu}$ reduced to a Cartan subalgebra $\frak{h}$ are transformed in order to place the recursion shifts $\gamma \in \Gamma_{\frak{a}\subset \frak{h}}$ into the fundamental Weyl chamber. The new ensembles $F\Psi$ (the "folded fans") of shifts were constructed and the corresponding recursion properties for the weights belonging to the fundamental Weyl chamber were formulated. Being considered simultaneously for the set of string functions (corresponding to the same congruence class $\Xi_{v}$ of modules) the system of recursion relations constitute an equation $\mathbf{M}_{(u)}^{\Xi_{v}} \mathbf{m}_{(u)}^{\mu}= \delta_{(u)}^{\mu}$ where the operator $\mathbf{M}_{(u)}^{\Xi_{v}}$ is an invertible matrix whose elements are defined by the coordinates and multiplicities of the shift weights in the folded fans $F\Psi$ and the components of the vector $\mathbf{m}_{(u)}^{\mu}$ are the string function coefficients for $L^{\mu}$ enlisted up to an arbitrary fixed grade $u$. The examples are presented where the string functions for modules of $\frak{g}=A_{2}^{(1)}$ are explicitly constructed demonstrating that the set of folded fans provides a compact and effective tool to study the integrable highest weight modules., Comment: This is a contribution to the Special Issue on Kac-Moody Algebras and Applications, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

It is demonstrated that decompositions of integrable highest weight modules of a simple Lie algebra with respect to its reductive subalgebra obey the set of algebraic relations leading to the recursive properties for the corresponding branching coefficients. These properties are encoded in the special element (of the formal algebra E(g)) Gamma_(a ->g) that describes the injection and is called the fan. In the simplest case, when "a" is a Cartan subalgebra of g, the recursion procedure generates the weight diagram of the module L(g). When applied to a reduction of highest weight modules the recursion described by the fan provides a highly effective tool to obtain the explicit values of branching coefficients., Comment: 17 pages, 3 figures, to be published in Algebra i Analiz

Classical r-matrices with carriers g_c containing the Borel subalgebra b(+,g) of simple Lie algebra g are studied. Using the graphycal presentation of the dual Lie algebra g#(r) we show that such solutions of the CYBE always exist. To obtain the explicit form of these r-matrices we find the dual coordinates in which the adjoint action of g_c can be reduced. This gives us the detailed structure of the Jordanian r-matrices that are the candidates for enlarging the initial full chain r-matrix. We search the desired solution in the factorized form. This leads to the unique transformation: the initial chain is to be substituted by a special kind of peripheric r-matrice. To illustrate the method we consider the case of g=sl(11) in full details., Comment: 15 pages

Twist deformation U_F(g) is equivalent to the quantum group Fun_d(G#) and has two preferred bases: the one originating from U(g) and that of the coordinate functions on the dual Lie group G#. The costructure of the Hopf algebra U_F(g) is analized in terms of group G#. The weight diagram of the adjoint representation of the algebra g# is constructed in terms of the root system L(g). The explicit form of the g --> g# transformation can be obtained for any simple Lie algebra g and the factorizable chain F of extended Jordanian twists. The dual group approach is used to find new solutions of the twist equations. The parametrized family of extended Jordanian deformations for U(sl(3)) is constructed and studied in terms of SL(3)#. New realizations of the parabolic twist are found., Comment: 17 pages

The quantization properties of composite peripheric twists are studied. Peripheric chains of extended twists are constructed for U(sl(N)) in order to obtain composite twists with sufficiently large carrier subalgebras. It is proved that the peripheric chains can be enlarged with additional Reshetikhin and Jordanian factors. This provides the possibility to construct new solutions to Drinfeld equations and, thus, to quantize new sets of Lie-Poisson structures. When the Jordanian additional factors are used the carrier algebras of the enlarged peripheric chains are transformed into algebras of motion of the form G_{JB}^{P}={G}_{H}\vdash {G}_{P}. The factor algebra G_{H} is a direct sum of Borel and contracted Borel subalgebras of lower dimensions. The corresponding omega--form is a coboundary. The enlarged peripheric chains F_{JB}^{P} represent the twists that contain operators external with respect to the Lie-Poisson structure. The properties of new twists are illustrated by quantizing r-matrices for the algebras U(sl(3)), U(sl(4)) and U(sl(7))., Comment: 24 pages, LaTeX

The problem of constructing the explicit form for full twist deformations of simple Lie algebras g with twist carriers containing the maximal nilpotent subalgebra N^+(g) is studied. Our main tool is the sequence of regular subalgebras g_i in U(g) that become primitive under the action of extended jordanian twists F_E: U(g) -> U_E(g). It is demonstrated that the structure of the sequence {g_i} is defined by the extended Dynkin diagram of algebra g. To construct the injection of g_i in U_E(g) the special deformations of algebras U_E(g) are performed. They are reduced to the (cohomologically trivial) twists F_s. Thus it is proved that full chains of twists can be written in the canonical form F_B = F'_N ... F'_2F'_1. The links F'_i in such chains must contain not only the extended twists F_E but also the factors F_s whose form depend on the type of the series of classical algebra g. The explicit forms of universal R-matrices (and the R-matrices in the fundamental representations) corresponding to full chains of twists for classical simple Lie algebras are found. The properties of the construction are illustrated by the example of full chain of extended twists for algebra sp(3)., Comment: 27 pages, LaTex 2e

The twist deformations for simple Lie algebras U(g) whose twisting elements F are known explicitly are usually defined on the carrier subspace injected in the Borel subalgebra B^+(g). We solve the problem of creating the parabolic twist F_P whose carrier algebra P not only covers B^+(g) but also intersects nontrivially with B^-(g). This algebra P is the parabplic subalgebra in sl(3) and has the structure of the algebra of two-dimensional motions. The parabolic twist is explicitly constructed as a composition of the well known extended jordanian twist F_EJ and the new factor F_D. The latter can be considered as a special version of the jordanian twist. The twisted costructure is found for U(P) and the corresponding universal R-matrix is presented., Comment: 9 pages, Latex 2e

Serious difficulties arise in the construction of chains of twists for symplectic Lie algebras. Applying the canonical chains of extended twists to deform the Hopf algebras U(sp(N)) one is forced to deal only with improper chains (induced by the U(sl(N)) subalgebras). In the present paper this problem is solved. For chains of regular injections the sets of maximal extended jordanian twists F_{E,k} are considered. We prove that there exists for U(sp(N)) the twist F_{B,k} composed of the factors F_{E,k}. It is demonstrated that the twisting procedure deforms the space of the primitive subalgebra sp(N-1). The recursive algorithm for such deformation is found. This construction generalizes the results obtained for orthogonal classical Lie algebras and demonstrates the universality of primitivization effect for regular chains of subalgebras. For the chain of maximal length the twists F_{B,k,max} become full, their carriers contain the Borel subalgebra B(sp(N)). Using such twisting procedures one can obtain the explicit quantizations for a wide class of classical r-matrices. As an example the full chain of extended twists for U(sp(3)) is considered., Comment: 10 pages, LaTeX-2e, to be published in the Proceedings of XXIII International Colloquium on Group-Theoretical Methods in Physics, Dubna, 31.07-05.08, 2000

The quantum deformation of the Jordanian twist F_qJ for the standard quantum Borel algebra U_q(B) is constructed. It gives the family U_qJ(B) of quantum algebras depending on parameters x and h. In a generic point these algebras represent the hybrid (standard-nonstandard) quantization. The quantum Jordanian twist can be applied to the standard quantization of any Kac-Moody algebra. The corresponding classical r-matrix is a linear combination of the Drinfeld- Jimbo and the Jordanian ones. The obtained two-parametric families of Hopf algebras are smooth and for the limit values of the parameters the standard and nonstandard quantizations are recovered. The twisting element F_qJ also has the correlated limits, in particular when q tends to unity it acquires the canonical form of the Jordanian twist. To illustrate the properties of the quantum Jordanian twist we construct the hybrid quantizations for U(sl(2)) and for the corresponding affine algebra U(hat(sl(2))). The universal quantum R-matrix and its defining representation are presented., Comment: 12 pages, Latex

We show that for some Hopf subalgebras in U_F(so(M)) nontrivially deformed by a twist F it is possible to find the nonlinear primitive copies. This enlarges the possibilities to construct chains of twists. For orthogonal algebra U(so(M)) we present a method to compose the full chains with carrier space as large as the Borel subalgebra B(so(M)). These chains can be used to construct the new deformed Yangians., Comment: Latex, 7p. Talk given at 9th International Colloquium "Quantum Groups and Integrable Systems", Prague, 22-24 June 2000

The nontrivial subspaces with primitive coproducts are found in the deformed universal enveloping algebras. They can form carrier spaces for additional Jordanian twists. The latter can be used to construct sequences of twists for algebras whose root systems contain long series of roots. The corresponding twist for the so(5) algebra is given explicitly., Comment: Latex, 12p

It is demonstrated how chains of twists for classical Lie algebras induce the new twist deformations (the deformed Yangians) that quantize the generalized rational solutions of the classical Yang-Baxter equation. For the case of Y(g) with g=so(2N+1) the explicit expression for the corresponding R-matrix is given in the defining representation., Comment: 8p.,Latex, to be published in the Proceedings of the SQS'99 Conference (Dubna, July 99)

Multidimensional Heisenberg algebras, whose creation and annihilation operators are the N-dimensional vectors, can be injected into simple Lie algebras g. It is demonstrated that the spectrum of their deformations can be investigated using chains of extended Jordanian twists applied to U(g). In the case of U(sl(N)) (for N>5) the two-dimensional Heisenberg subalgebras have nine deformed costructures connected by four "internal" and "external" twists composing the commutative diagram., Comment: 12 pages, Latex, to be published in the Proceedings of the XIV-th Symposium, Karpacz, Sept. 99

The properties of the set {L} of extended jordanian twists for algebra sl(3) are studied. Starting from the simplest algebraic construction --- the peripheric Hopf algebra U_ P'(0,1)(sl(3)) --- we construct explicitly the complete family of extended twisted algebras {U_ E(\theta)(sl(3))} corresponding to the set of 4-dimensional Frobenius subalgebras {L(\theta)} in sl(3). It is proved that the extended twisted algebras with different values of the parameter \theta are connected by a special kind of Reshetikhin twist. We study the relations between the family {U_E(\theta)(sl(3))} and the one-dimensional set {U_DJR(\lambda)(sl(3))} produced by the standard Reshetikhin twist from the Drinfeld--Jimbo quantization U_DJ(sl(3)). These sets of deformations are in one-to-one correspondence: each element of {U_E(\theta)(sl(3))} can be obtained by a limiting procedure from the unique point in the set {U_DJR(\lambda)(sl(3))}., Comment: 14 pages, LaTeX 209

The properties of the set L of extended jordanian twists are studied. It is shown that the boundaries of L contain twists whose characteristics differ considerably from those of internal points. The extension multipliers of these "peripheric" twists are factorizable. This leads to simplifications in the twisted algebra relations and helps to find the explicit form for coproducts. The peripheric twisted algebra U(sl(4)) is obtained to illustrate the construction. It is shown that the corresponding deformation U_{P}(sl(4)) cannot be connected with the Drinfeld--Jimbo one by a smooth limit procedure. All the carrier algebras for the extended and the peripheric extended twists are proved to be Frobenius., Comment: 16 pages, LaTeX 209. Some misprints have been corrected and new Comments added