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Noncommutative Polynomial Algebras of Solvable Type and Their
Modules is the first book to systematically introduce the basic
constructive-computational theory and methods developed for
investigating solvable polynomial algebras and their modules. In
doing so, this book covers: A constructive introduction to solvable
polynomial algebras and Groebner basis theory for left ideals of
solvable polynomial algebras and submodules of free modules The new
filtered-graded techniques combined with the determination of the
existence of graded monomial orderings The elimination theory and
methods (for left ideals and submodules of free modules) combining
the Groebner basis techniques with the use of Gelfand-Kirillov
dimension, and the construction of different kinds of elimination
orderings The computational construction of finite free resolutions
(including computation of syzygies, construction of different kinds
of finite minimal free resolutions based on computation of
different kinds of minimal generating sets), etc. This book is
perfectly suited to researchers and postgraduates researching
noncommutative computational algebra and would also be an ideal
resource for teaching an advanced lecture course.
"Presents the structure of algebras appearing in representation
theory of groups and algebras with general ring theoretic methods
related to representation theory. Covers affine algebraic sets and
the nullstellensatz, polynomial and rational functions, projective
algebraic sets. Groebner basis, dimension of algebraic sets, local
theory, curves and elliptic curves, and more."
This monograph strives to introduce a solid foundation on the usage
of Groebner bases in ring theory by focusing on noncommutative
associative algebras defined by relations over a field K. It also
reveals the intrinsic structural properties of Groebner bases,
presents a constructive PBW theory in a quite extensive context
and, along the routes built via the PBW theory, the book
demonstrates novel methods of using Groebner bases in determining
and recognizing many more structural properties of algebras, such
as the Gelfand-Kirillov dimension, Noetherianity, (semi-)primeness,
PI-property, finiteness of global homological dimension, Hilbert
series, (non-)homogeneous p-Koszulity, PBW-deformation, and regular
central extension.With a self-contained and constructive Groebner
basis theory for algebras with a skew multiplicative K-basis,
numerous illuminating examples are constructed in the book for
illustrating and extending the topics studied. Moreover,
perspectives of further study on the topics are prompted at
appropriate points. This book can be of considerable interest to
researchers and graduate students in computational (computer)
algebra, computational (noncommutative) algebraic geometry;
especially for those working on the structure theory of rings,
algebras and their modules (representations).
"Presents the structure of algebras appearing in representation
theory of groups and algebras with general ring theoretic methods
related to representation theory. Covers affine algebraic sets and
the nullstellensatz, polynomial and rational functions, projective
algebraic sets. Groebner basis, dimension of algebraic sets, local
theory, curves and elliptic curves, and more."
This self-contained monograph is the first to feature the intersection of the structure theory of noncommutative associative algebras and the algorithmic aspect of Groebner basis theory. A double filtered-graded transfer of data in using noncommutative Groebner bases leads to effective exploitation of the solutions to several structural-computational problems, e.g., an algorithmic recognition of quadric solvable polynomial algebras, computation of GK-dimension and multiplicity for modules, and elimination of variables in noncommutative setting. All topics included deal with algebras of (q-)differential operators as well as some other operator algebras, enveloping algebras of Lie algebras, typical quantum algebras, and many of their deformations.
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