Reductions of N -wave type equations related to simple Lie algebras and the hierarchy of their Hamiltonian structures are studied. The reduction group GR is realized as a subgroup of the Weyl group of the corresponding algebra. Some of the reduced equations are of physical interest. 1. Preliminary. The analysis and the classification of all reductions for the nonlinear evolution equations solva...

We prove that II1 factors M have a unique (up to unitary conjugacy) crossproduct type decomposition around “core subfactors” N ⊂ M satisfying the property HT of ([P1]) and a certain “torsion freeness” condition. In particular, this shows that isomorphism of factors of the form Lα(Z)⋊Γ, for torsion free, non-amenable subgroups Γ ⊂ SL(2,Z) and α = e, t 6∈ Q, implies isomorphism of the correspondi...

Let G be a simple and simply connected complex linear algebraic group, with Lie algebra g. Let ρ : G → AutV be an irreducible finite-dimensional representation of G, and let ρ∗ : g → EndV be the induced representation of g. The goal of this paper is to study ρ∗, and in particular to give normal forms for the action of ρ∗(X) for regular elements X of g. Of course, when X is semisimple, the actio...

For each simply-laced Dynkin graph ∆ we realize the simple complex Lie algebra of type ∆ as a quotient algebra of the complex degenerate composition Lie algebra L(A) 1 of a domestic canonical algebra A of type ∆ by some ideal I of L(A) 1 that is defined via the Hall algebra of A, and give an explicit form of I. Moreover, we show that each root space of L(A) 1 /I has a basis given by the coset o...

A finite dimensional Lie algebra f is Frobenius if there is a linear Frobenius functional F : f→ C such that the skew bilinear form BF defined by BF (x, y) = F ([x, y]) is non-degenerate. The principal element of f is then the unique element F̂ such that F (x) = F ([F̂ , x]); it depends on the choice of functional. However, if f is a subalgebra of a simple Lie algebra g and not an ideal of any la...

in this note, we characterize chebyshev subalgebras of unital jb-algebras. we exhibit that if b is chebyshev subalgebra of a unital jb-algebra a, then either b is a trivial subalgebra of a or a= h r .l, where h is a hilbert space

Basis tensor gauge theory (BTGT) is a vierbein analog reformulation of ordinary theories in which the field describes Wilson line. After brief review BTGT, we clarify Lorentz group representation properties associated with variables used for its quantization. In particular, show that starting from an SO(1,3) satisfying Lorentz-invariant U(1,3) matrix constraints, BTGT introduces frame choice to...

Using the belongs to relation (∈) and quasi-coincidence with relation (q) between fuzzy points and fuzzy sets, the concept of (α, β)fuzzy subalgebras where α, β are any two of {∈, q, ∈ ∨ q, ∈∧ q} with α 6=∈∧ q was introduced, and related properties were investigated in [3]. As a continuation of the paper [3], in this paper, the notion of a fuzzy subalgebra with thresholds is introduced, and its...

We consider correlation functions for the Wess-Zumino-Witten model on the torus with the insertion of a Cartan element; mathematically this means that we consider the function of the form F = Tr((1 (z 1) : : : n (z n)q ?@ e h) where i are intertwiners between Verma modules and evaluation modules over an aane Lie algebra ^ g, @ is the grading operator in a Verma module and h is in the Cartan sub...

We introduce a quasitriangular Hopf algebra or ‘quantum group’ U(B), the double-bosonisation, associated to every braided group B in the category of Hmodules over a quasitriangular Hopf algebra H, such that B appears as the ‘positive root space’, H as the ‘Cartan subalgebra’ and the dual braided group B∗ as the ‘negative root space’ of U(B). The choice B = f recovers Lusztig’s construction of U...