Higher categorified algebras versus bounded homotopy algebras

We define Lie 3-algebras and prove that these are in 1-to-1 correspondence with the 3-term Lie infinity algebras whose bilinear and trilinear maps vanish in degree (1,1) and in total degree 1, respectively. Further, we give an answer to a question of [Roy07] pertaining to the use of the nerve and normalization functors in the study of the relationship between categorified algebras and truncated sh algebras.Comment: 21 pages, 1 figur

Generalised Brownian Motion and Second Quantisation

A new approach to the generalised Brownian motion introduced by M. Bozejko and R. Speicher is described, based on symmetry rather than deformation. The symmetrisation principle is provided by Joyal's notions of tensorial and combinatorial species. Any such species V gives rise to an endofunctor F_V of the category of Hilbert spaces with contractions. A generalised Brownian motion is an algebra of creation and annihilation operators acting on F_V(H) for arbitrary Hilbert spaces H and having a prescription for the calculation of vacuum expectations in terms of a function t on pair partitions. The positivity is encoded by a *-semigroup of "broken pair partitions" whose representation space with respect to t is V. The existence of the second quantisation as functor Gamma_t from Hilbert spaces to noncommutative probability spaces is proved to be equivalent to the multiplicative property of the function t. For a certain one parameter interpolation between the fermionic and the free Brownian motion it is shown that the ``field algebras'' Gamma(K) are type II_1 factors when K is infinite dimensional.Comment: 33 pages, 5 figure

Representations of automorphism groups of finite O-modules of rank two

Let O be a complete discrete valuation domain with finite residue field. In this paper we describe the irreducible representations of the groups Aut(M) for any finite O-module M of rank two. The main emphasis is on the interaction between the different groups and their representations. An induction scheme is developed in order to study the whole family of these groups coherently. The results obtained depend on the ring O in a very weak manner, mainly through the degree of the residue field. In particular, a uniform description of the irreducible representations of GL(2,O/P^k) is obtained, where P is the maximal ideal of O.Comment: Final version, to appear in Advances in Mathematic

Cyclicity for categorified quantum groups

We equip the categorified quantum group attached to a KLR algebra and an arbitrary choice of scalars with duality functor which is cyclic, that is, such that f=f^** for all 2-morphisms f. This is accomplished via a modified diagrammatic formalism.Comment: 12 pages, xy-pic diagram