33 research outputs found
QUANTIZATION OF A CLASS OF PIECEWISE AFFINE TRANSFORMATIONS ON THE TORUS
We present a unified framework for the quantization of a family of discrete
dynamical systems of varying degrees of "chaoticity". The systems to be
quantized are piecewise affine maps on the two-torus, viewed as phase space,
and include the automorphisms, translations and skew translations. We then
treat some discontinuous transformations such as the Baker map and the
sawtooth-like maps. Our approach extends some ideas from geometric quantization
and it is both conceptually and calculationally simple.Comment: no. 28 pages in AMSTE
Classification of simple linearly compact n-Lie superalgebras
We classify simple linearly compact n-Lie superalgebras with n>2 over a field
F of characteristic 0. The classification is based on a bijective
correspondence between non-abelian n-Lie superalgebras and transitive Z-graded
Lie superalgebras of the form L=\oplus_{j=-1}^{n-1} L_j, such that L_{-1}=g,
where dim L_{n-1}=1, L_{-1} and L_{n-1} generate L, and [L_j, L_{n-j-1}] =0 for
all j, thereby reducing it to the known classification of simple linearly
compact Lie superalgebras and their Z-gradings. The list consists of four
examples, one of them being the n+1-dimensional vector product n-Lie algebra,
and the remaining three infinite-dimensional n-Lie algebras.Comment: Final version to appear in Communications in Mathematical Physic
Cyclotomic Gaudin models: construction and Bethe ansatz
This is a pre-copyedited author produced PDF of an article accepted for publication in Communications in Mathematical Physics, Benoit, V and Young, C, 'Cyclotomic Gaudin models: construction and Bethe ansatz', Commun. Math. Phys. (2016) 343:971, first published on line March 24, 2016. The final publication is available at Springer via http://dx.doi.org/10.1007/s00220-016-2601-3 © Springer-Verlag Berlin Heidelberg 2016To any simple Lie algebra and automorphism we associate a cyclotomic Gaudin algebra. This is a large commutative subalgebra of generated by a hierarchy of cyclotomic Gaudin Hamiltonians. It reduces to the Gaudin algebra in the special case . We go on to construct joint eigenvectors and their eigenvalues for this hierarchy of cyclotomic Gaudin Hamiltonians, in the case of a spin chain consisting of a tensor product of Verma modules. To do so we generalize an approach to the Bethe ansatz due to Feigin, Frenkel and Reshetikhin involving vertex algebras and the Wakimoto construction. As part of this construction, we make use of a theorem concerning cyclotomic coinvariants, which we prove in a companion paper. As a byproduct, we obtain a cyclotomic generalization of the Schechtman-Varchenko formula for the weight function.Peer reviewe