We show how to construct a graded locally compact Hausdorff \'etale groupoid from C*-algebra carrying coaction of discrete group, together with suitable abelian subalgebra. call this the extended Weyl groupoid. When is trivial and subalgebra Cartan, our agrees Renault's prove that if G second-countable grading interior trivially isotropy torsion free, then its reduced isomorphic as G. In partic...

In this paper, we construct a family of non-weight modules over the untwisted N=2 superconformal algebras. Those when regarded as Cartan subalgebra (modulo center) are free rank 2. We give classification isomorphism classes such modules. Moreover, all submodules precisely determined. particular, those not simple. The corresponding simple quotient classified. Furthermore, these restricted N=1 al...

Let g be a complex simple Lie algebra, and h ⊂ g be a Cartan subalgebra. In the end of 1990s, B. Kostant defined two filtrations on h, one using the Clifford algebras and the odd analogue of the Harish-Chandra projection hcodd : Cl(g) → Cl(h), and the other one using the canonical isomorphism ȟ = h∗ (here ȟ is the Cartan subalgebra in the simple Lie algebra ǧ corresponding to the dual root syst...

where h is any additive mapping of G into F (see [1, p. 138]). Each of these algebras is a simple Lie algebra, for which u0 spans a one-dimensional Cartan subalgebra, with one-dimensional root spaces spanned by the ux. We shall prove that under certain hypotheses, these are the only Lie algebras over F of rank one. Kaplansky in [5] has proved that any restricted simple Lie algebra over F of ran...

The natural extension to the SU(5) Georgi-Glashow grand unification model is enlarge gauge symmetry group. In this work, SU(7) group examined. Cartan subalgebra determined along with their commutation relations. associated roots and weights of algebra are derived discussed. raising lowering operators explicitly constructed presented. Higher dimensional representations developed by graphical as ...

We prove that an inductive limit of aperiodic noncommutative Cartan inclusions is a inclusion whenever the connecting maps are injective, preserve normalisers and entwine conditional expectations. show under additional assumption subalgebra either essentially separable, simple or Type I we get in limit. Consequently, subsume case where building block subalgebras commutative provide proof theore...

Chevalley’s theorem states that for any simple finite dimensional Lie algebra g: (1) the restriction homomorphism of the algebra of polynomials S(g∗) −→ S(h∗) onto the Cartan subalgebra h induces an isomorphism S(g∗)g ∼= S(h∗)W , where W is the Weyl group of g; (2) each g-invariant polynomial is a linear combination of the polynomials trρ(x)k , where ρ is a finite dimensional representation of ...

We show that O(2, 2) transformation of SU(2) WZNW model gives rise to marginal deformation of this model by the operator d 2 zJ(z) ¯ J (¯ z) where J, ¯ J are U(1) currents in the Cartan subalgebra. Generalization of this result to other WZNW theories is discussed. We also consider O(3, 3) transformation of the product of an SU(2) WZNW model and a gauged SU(2) WZNW model. The three parameter set...

In 1984, Victor Kac [K4] suggested an approach to a description of central elements of a completion of U(g) for any Kac-Moody Lie algebra g. The method is based on a recursive procedure. Each step is reduced to a system of linear equations over a certain subalgebra of meromorphic functions on the Cartan subalgebra. The determinant of the system coincides with the Shapovalov determinant for g. W...

The Cartan subalgebra of the quantum loop algebra Uq(Lsl2) is generated by a family of mutually commuting operators, responsible for the l-weight decomposition of finite dimensional Uq(Lsl2)-modules. The natural Jordan filtration induced by these operators is generically non-trivial on l-weight spaces of dimension greater than one. We derive, for every standard module of Uq(Lsl2), the dimension...