Let g = k + p be a complexified Cartan decomposition of a complex semisimple Lie algebra g and let K be the subgroup of the adjoint group of g corresponding to k. If H is an irreducible Harish-Chandra module of U (g), then H is completely determined by the finite-dimensional action of the centralizer U (g) K on any one fixed primary k component in H. This original approach of Harish-Chandra to ...

Cartan subalgebras, automorphisms and the UCT problem Selçuk Barlak This talk deals with the connection between the UCT problem for separable, nuclear C*-algebras and Cartan subalgebras, that is, MASAs that admit faithful conditional expectations and whose normalizers generate the ambient C*-algebras. We present a characterization, in terms of Cartan subalgebras, of the UCT for fixed point alge...

A new N = 4 superconformal algebra (SCA) is presented. Its internal affine Lie algebra is based on the seven-dimensional Lie algebra su(2)⊕ g, where g should be identified with a fourdimensional non-reductive Lie algebra. Thus, it is the first known example of what we choose to call a non-reductive SCA. It contains a total of 16 generators and is obtained by a nontrivial Inönü-Wigner contractio...

Duality transformations play an important role in modern theoretical physics. In Seiberg-Witten theory [1] electric-magnetic duality (for a recent review of electric-magnetic duality see [2] and references therein) is a basic ingredient in obtaining the exact form of the low-energy effective action. Hence, duality is a crucial tool in studying non-perturbative physics. Any truly non-perturbativ...

Let L = H(2r; n) be a graded Lie algebra of Hamiltonian type in the Cartan type series over an algebraically closed field of characteristic p > 2. In the generalized restricted Lie algebra setup, any irreducible representation of L corresponds uniquely to a (generalized) p-character χ. When the height of χ is no more than min{pni − pni−1 | i = 1, 2, . . . , 2r} − 2, the corresponding irreducibl...

We investigate the Spin(7) holonomy metric of cohomogeneity one with the principal orbit SU(3)/U(1). A choice of U(1) in the two dimensional Cartan subalgebra is left as free and this allows manifest Σ3 = W (SU(3)) (= the Weyl group) symmetric formulation. We find asymptotically locally conical (ALC) metrics as octonionic gravitational instantons. Complex projective space CP(2) that is a supers...

The dimensional reduction of heterotic supergravity with gauge fields truncated to the Cartan subalgebra exhibits a continuous O(d, d+ 16;R) global symmetry, related to the O(d, d+16;Z) T-duality of heterotic strings on a d-torus. The O(d, d+16;R) symmetry is not present, however, if the supergravity reduction is done including the full set of E8×E8 or SO(32) gauge fields. We analyze which dual...

Let £ be a split semisimple Lie algebra over a field of characteristic zero and <£ = 3C+ ]C«eA«£a be the rootspace decomposition of <£ relative to a splitting Cartan subalgebra 5C, where the subset A of 3C* is the corresponding root-system. Fix a simple system of roots {c*i, «2, • • • , ai}, for which the positive (resp. negative) roots are denoted by A+ (resp. A_). For a EA let Ra be the W...

Let $L$ be a finite-dimensional Lie algebra. We say a subalgebra $H$ of $L$ is permutably complemented in $L$ if there is a subalgebra $K$ of $L$ such that $L=H+K$ and $Hcap K=0$. Also, if every subalgebra of $L$ is permutably complemented in $L$, then $L$ is called completely factorisable. In this article, we consider the influence of these concepts on the structure of a Lie algebra, in partic...

Using the gauge transformations of the Classical Dynamical YangBaxter Equation introduced by P. Etingof and A. Varchenko in [EV], we reduce the classification of dynamical r-matrices on a commutative subalgebra l of a Lie algebra g to a purely algebraic problem under the assumption that l admits a g-invariant complement, where g is the centralizer of l. We then describe, for a simple complex Li...