Decomposition of (co)isotropic relations

We identify thirteen isomorphism classes of indecomposable coisotropic relations between Poisson vector spaces and show that every coisotropic relation between finite-dimensional Poisson vector spaces may be decomposed as a direct sum of multiples of these indecomposables. We also find a list of thirteen invariants, each of which is the dimension of a space constructed from the relation, such that the 13-vector of multiplicities and the 13-vector of invariants are related by an invertible matrix over $\mathbb Z$. It turns out to be simpler to do the analysis above for isotropic relations between presymplectic vector spaces. The coisotropic/Poisson case then follows by a simple duality argument.Comment: 9 pages. The final publication is available at Springer via http://dx.doi.org/10.1007/s11005-016-0863-5, in a special issue of Letters in Mathematical Physics dedicated to the memory of Louis Boutet de Monvel. A free, view-only version of the final publication is available under the following link http://rdcu.be/mFX

(Co)isotropic Pairs in Poisson and Presymplectic Vector Spaces

We give two equivalent sets of invariants which classify pairs of coisotropic subspaces of finite-dimensional Poisson vector spaces. For this it is convenient to dualize; we work with pairs of isotropic subspaces of presymplectic vector spaces. We identify ten elementary types which are the building blocks of such pairs, and we write down a matrix, invertible over $\mathbb{Z}$, which takes one 10-tuple of invariants to the other

Duality involutions, representations, and geometry

In this thesis we give an exposition of the theory of duality involutions, and within this context we present the results of two different research projects. Loosely speaking, a duality involution on a category C is a self-adjoint contravariant endofunctor of C. A prototypical example of such is the usual notion of duality for the finite dimensional vector spaces. We also consider duality involutions for bicategories, as defined by Shulman. The first project concerns classification problems in symplectic linear algebra. In this part, we discuss results regarding the symplectic group in its Lie algera, as well as work on systems of subspaces in symplectic vector spaces. In the language of duality involutions, symplectic structures are encoded as fixed point structures. The second project is about the Morita bicategory of finite-dimensional k-algebras and bimodules, and the representation pseudofunctor which sends an algebra to its category of representations. We show that this representation pseudofunctor is equivariant in a natural manner with respect to duality involutions which we define on its source and target