Abstract
In low dimensional quantum field theories the global (gauge) symmetry can
in general not be described by an ordinary group but by some more
general algebraic object such as quantum groups or generalizations
thereof. In this thesis we construct 1+1 - dimensional lattice quantum
field theories - socalled quantum group spin chains and lattice
current algebras - whose global symmetry is given by some quantum
group at roots of unity.
The main problem in constructing these models stems from the fact that
the semisimple quotients of quantum groups at roots of unity are no
longer coassiciative and have to be described by weak quasi-quantum
groups. To solve this problem we introduce a new mathematical
construction, the so-called diagonal crossed product of an algebra M
with the dual of a quantum group G.
We give a natural generalization of this construction to the case
where G is a quasi-Hopf algebra in the sense of Drinfeld and, more
generally, also in the sense of Mack and Schomerus (i.e., where the
coproduct is non-unital).
In these cases our diagonal crossed product will still be an
associative algebra, even though
the analogue of an ordinary crossed product
in general is not well defined as an associative
algebra.
In the case M = G we obtain an explicit
definition of
the quantum double D(G) for (weak) quasi-Hopf algebras G.
We prove that
D(G) is itself a (weak) quasi-triangular quasi-Hopf algebra and
we give explicit formulas for the coproduct, the antipode and the
R-matrix. Moreover we show that any
diagonal crossed product naturally admits a
two-sided D(G)-coaction.
We then apply our formalism to construct quantum spin chains and
lattice current algebras based on a weak quasi-Hopf algebra as
iterated diagonal crossed products. This contains the important cases
of truncated quantum groups at roots of unity.
Both lattice models admit the quantum double D(G) as a localized
cosymmetry.
We investigate the representation theory of these
models. In particular we show that irreducible representations of
lattice current algebras (based on a semisimple weak quasi Hopf
algebra G) are in one-to-one
correspondence with the irreducible representations of the quantum
double D(G). | 
