ar X iv : m at h / 05 05 06 3 v 2 [ m at h . SG ] 2 5 Se p 20 06 1 A symplectic approach to van den Ban ’ s convexity theorem

ar X iv : m at h / 05 05 06 3 v 1 [ m at h . SG ] 3 M ay 2 00 5 A SYMPLECTIC REALIZATION OF VAN DEN BAN ’ S CONVEXITY THEOREM

Let G be a complex semisimple Lie group and τ a complex an-tilinear involution that commutes with the Cartan involution. If H denotes the connected subgroup of τ-fixed points in G, and K is maximally compact, each H-orbit in G/K can be equipped with a Poisson structure as described by Evens and Lu. We consider symplectic leaves of certain such H-orbits and show that an application of the symple...

ar X iv : m at h / 05 06 06 8 v 1 [ m at h . G R ] 3 J un 2 00 5 ALGORITHM FOR LANG ’ S THEOREM

We give an efficient algorithm for Lang's Theorem in split connected reductive groups defined over finite fields of characteristic greater than 3. This algorithm can be used to construct many important structures in finite groups of Lie type. We use an algorithm for computing a Chevalley basis for a split reductive Lie algebra, which is of independent interest.

ar X iv : m at h / 05 06 22 3 v 2 [ m at h . A C ] 1 3 Se p 20 05 COMBINATORIAL SECANT VARIETIES

The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Given any projective variety and any term order, we explore whether the initial ideal o...

ar X iv : m at h / 04 05 17 6 v 4 [ m at h . R T ] 1 6 Ju n 20 06 QUANTIZED SYMPLECTIC OSCILLATOR ALGEBRAS OF RANK ONE

A quantized symplectic oscillator algebra of rank 1 is a PBW deformation of the smash product of the quantum plane with Uq(sl2). We study its representation theory, and in particular, its category O.

ar X iv : m at h / 06 05 35 5 v 1 [ m at h . G R ] 1 3 M ay 2 00 6 Axes in Outer Space