Finite-Dimensional Representations of Lie Algebras and Completely Integrable Systems

Abstract

We introduce a class of supersymmetric cycles in spacetimes of the form AdS times a sphere or T 1,1 which can be considered as generalizations of the giant gravitons. Branes wrapped on these cycles preserve 1/2, 1/4 or 1/8 of the supersymmetry. On the CFT side these configurations correspond to superpositions of the large number of BPS states.

The Algebra of Hamiltonian Systems

“…Theorem 2.9 (Trofimov [1980]). Suppose that (h 1 , ... , h.; W) is an S-representation of G. Then a) all the functions h 1 , ... , h. are in involution on all coadjoint orbits of G; b) the shifts of the h; are in involution on all coadjoint orbits of G, i.e.…”

“…The operation of shifting the argument can be combined with the operation of Iifting functions used in Section 2.1 (see Trofimov [1980]). Wehave the following proposition.…”

A note on the classification of nine dimensional Lie algebras with nontrivial Levi decomposition

“…• Levi decomposition: L p,q 9,24 = sl (2, R) − → ⊕ R g 6,35 • Describing representation: R = D 1 2 ⊕ 4D 0 • Structure tensor: • codim g [g, g] = 1.…”

“…Although we use the classical analytical approach, direct integration of the invariants is a cumbersome task. In order to obtain the invariants, we consider the theory of semi-invariants for the coadjoint representation [34,35] and the reduction to subalgebras the invariants of which can be computed easily or are already known. The procedure is based in the labelling of representations using subgroups chains (so called missing label operator problem) developed for arbitrary Lie algebras in [25].…”

Algebraic structure of certain integrable hamiltonian systems

“…their common level surface determines the orbit of general position), and the remaining ~(n-r) functions are no longer constant on the orbits. We recall that the symplectic form just described is, generally, degenerate on the whole algebra (unlike the orbits [8,9] .…”