It is shown that for each 0 < lambda < 1, the free Araki-Woods factor of type III(lambda) cannot be written as a tensor product of two diffuse von Neumann algebras (i.e., is prime) and does not contain a Cartan subalgebra.

We prove a Poincaré lemma for a set of r smooth functions on a 2n-dimensional smooth manifold satisfying a commutation relation determined by r singular vector fields associated to a Cartan subalgebra of sp(2r, R). This result has a natural interpretation in terms of the cohomology associated to the infinitesimal deformation of a completely integrable system.

Let g be an affine Kac-Moody algebra with symmetric Cartan datum, n be the maximal nilpotent subalgebra of g. By the Hall algebra approach, we construct integral bases of the Z-form of the enveloping algebra U(n). In particular, the representation theory of tame quivers is essentially used in this paper.

We investigate a solvable SU(3)-XXZ-model modiied by elements of the Cartan-subalgebra. The nested Bethe-Ansatz is discussed in detail and the corresponding quantum spin model is stated. The coupled Bethe-equations are solved in the ther-modynamic limit. EEects of the modiication on the nite size scaling are derived.

The main point of the construction of spin Calogero type classical integrable systems based on dynamical r-matrices, developed by L.-C. Li and P. Xu, is reviewed. It is shown that non-Abelian dynamical r-matrices with variables in a reductive Lie algebra F and their Abelian counterparts with variables in a Cartan subalgebra of F lead essentially to the same models.

We obtain an estimate of Voiculescu’s (modified) free entropy dimension for generators of a II1-factor M with a subfactor N containing an abelian subalgebra A of finite multiplicity. It implies in particular that the interpolated free group subfactors of finite Jones index do not have abelian subalgebras of finite multiplicity or Cartan subalgebras.

Static spherically symmetric solutions in SU(N)-EYM and EYMD theories are classified by the node numbers of their non-trivial gauge field functions. With increasing node numbers, the solutions form sequences, tending to limiting solutions. The limiting solutions are based on subalgebras of su(N), consisting of a neutral non-abelian part and a charged abelian part, belonging to the Cartan subalg...

The trigonometric KZ equations associated with a Lie algebra g depend on a parameter λ ∈ h where h ⊂ g is the Cartan subalgebra. We suggest a system of dynamical difference equations with respect to λ compatible with the KZ equations. The dynamical equations are constructed in terms of intertwining operators of g -modules.

We consider the string, like point particles and branes, to be an irreducible representation of semi-direct product Cartan involution invariant subalgebra [Formula: see text] its vector representation. show that preserves string charges, little algebra, is essentially Borel text]. also known physical states carry a parts this algebra.