Within the last decade, semigroup theoretical methods have occurred
naturally in many aspects of ring theory, algebraic combinatorics,
representation theory and their applications. In particular,
motivated by noncommutative geometry and the theory of quantum
groups, there is a growing interest in the class of semigroup
algebras and their deformations.
This work presents a comprehensive treatment of the main results
and methods of the theory of Noetherian semigroup algebras. These
general results are then applied and illustrated in the context of
important classes of algebras that arise in a variety of areas and
have been recently intensively studied. Several concrete
constructions are described in full detail, in particular
intriguing classes of quadratic algebras and algebras related to
group rings of polycyclic-by-finite groups. These give new classes
of Noetherian algebras of small Gelfand-Kirillov dimension. The
focus is on the interplay between their combinatorics and the
algebraic structure. This yields a rich resource of examples that
are of interest not only for the noncommutative ring theorists, but
also for researchers in semigroup theory and certain aspects of
group and group ring theory. Mathematical physicists will find this
work of interest owing to the attention given to applications to
the Yang-Baxter equation.
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