Parafermion theory from the q-deformed algebra with q a root of unity

In this paper, we find the [Formula: see text]-deformed algebra with the finite- and infinite-dimensional Fock space and both the fermionic limit and the bosonic limit. Using the cardinality of set theory, we propose the Hamiltonian interpolating bosonic case and fermionic case, which enables us to construct the proper partition function and internal energy. As examples, we discuss the specific heat of free [Formula: see text] parafermion gas model and [Formula: see text] parafermion star.

NONLOCAL MATRIX GENERALIZATIONS OF N=2 SUPER VIRASORO ALGEBRA

We study the generalization of the second Gelfand-Dickey bracket to the superdifferential operators with matrix-valued coefficients. The associated matrix Miura transformation is derived. Using this bracket we work out a nonlocal and nonlinear N=2 superalgebra which contains the N=2 super Virasoro algebra as a subalgebra. The bosonic limit of this superalgebra is considered. We show that when the spin-1 fields in this bosonic algebra are set to zero the resulting Dirac bracket gives precisely the recently derived V2,2 algebra.