Irreducibility of some quantum representations of mapping class groups

Introduction to quantum representations of mapping class groups

We provide an (almost) self-contained construction of the Witten-Reshetikhin-Turaev representations of the mapping class group. We describe its properties including its Hermitian structure, irreducibility and integrality (at prime level). The construction of these notes relies only on skein theory (Kauffman Bracket) and does not use surgery techniques. We hope that they will be accessible to non-specialists.

量子仿射代数的Whittaker构造

Whittaker constructions for quantum affine algebras

本文通过抛物诱导从Heisenberg李代数的不可约Whittaker模构造了无扭仿射Kac-Moody李代数的虚Whittaker模，建立了不可约性判据，并推广到量子仿射代数情形，同时证明了$\mathcal{U}_q(A_1^{(1)})$上一类非量子形变模的不可约性。

One-particle irreducibility with initial correlations

In quantum field theory (QFT), the vacuum expectation value of a normal product of creation and annihilation operators is always zero. This simple property paves the way to the classical treatment of perturbative QFT. This is no longer the case in the presence of initial correlations, that is if the vacuum is replaced by a general state. As a consequence, the combinatorics of correlated systems such as the ones occurring in many-body physics is more complex than that of quantum field theory and the general theory has made very slow progress. Similar observations hold in statistical physics or …

Twisted Hamiltonian Lie Algebras and Their Multiplicity-Free Representations

We construct a class of new Lie algebras by generalizing the one-variable Lie algebras generated by the quadratic conformal algebras (or corresponding Hamiltonian operators) associated to Poisson algebras and a quasi-derivation found by Xu. These algebras can be viewed as certain twists of Xu's generalized Hamiltonian Lie algebras. The simplicity of these algebras is completely determined. Moreover, we construct a family of multiplicity-free representations of these Lie algebras and prove their irreducibility.

Casimir Operators and Monodromy Representations of Generalised Braid Groups

Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one-parameter family of flat connections D on h with values in any finite-dimensional h-module V and simple poles on the root hyperplanes. The corresponding monodromy representation of the braid group B of type g is a deformation of the action of (a finite extension of) W on V. The residues of D are the Casimirs of the sl(2)-subalgebras of g corresponding to its roots. The irreducibility of a subspace U of V under these implies that, for generic values of the parameter, the braid group B acts irred…

Local representations of the (1+1) quantum extended Galilei algebra

We present the q-deformed counterpart of the local representations of the (1+1) extended Galilei group. These representations act on the space of wavefunctions defined in the space-time. As in the classical case the q-local representations are reducible and the condition of irreducibility is given by the q-Casimir equation that in the limit of the deformation parameter going to zero becomes the Schroedinger equation of a free particle.