Morphisms and order ideals of toric posets

Toric posets are cyclic analogues of finite posets. They can be viewed combinatorially as equivalence classes of acyclic orientations generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper we study toric intervals, morphisms, and order ideals, and we provide a connection to cyclic reducibility and conjugacy in Coxeter groups.Comment: 28 pages, 8 figures. A 12-page "extended abstract" version appears as [v2

Toric Hyperkahler Varieties

Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces by symplectic torus actions. A toric hyperkahler variety is a complete intersection in a Lawrence toric variety. Both varieties are non-compact, and they share the same cohomology ring, namely, the Stanley-Reisner ring of a matroid modulo a linear system of parameters. Familiar applications of toric geometry to combinatorics, including the Hard Lefschetz Theorem and the volume polynomials of Khovanskii-Pukhlikov, are extended to the hyperkahler setting. When the matroid is graphic, our construction gives the toric quiver varieties, in the sense of Nakajima.Comment: 32 pages, Latex; minor corrections and a reference adde

Sup-lattice 2-forms and quantales

A 2-form between two sup-lattices L and R is defined to be a sup-lattice bimorphism L x R -> 2. Such 2-forms are equivalent to Galois connections, and we study them and their relation to quantales, involutive quantales and quantale modules. As examples we describe applications to C*-algebras.Comment: 30 pages. Contains more detailed background section and corrections of several typos and mistake