Commuting Elements in Q-Deformed Heisenberg Algebras

by Hellstrom, Lars; Silvestrov, Sergei D.

• ISBN13: 9789810244033
• ISBN10: 9810244037
• Format: Hardcover
• Copyright: 2000-09-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

Noncommutative algebras, rings and other noncommutative objects, along with their more classical commutative counterparts, have become a key part of modern mathematics, physics and many other fields. The q-deformed Heisenberg algebras defined by deformed Heisenberg canonical commutation relations of quantum mechanics play a distinguished role as important objects in pure mathematics and in many applications in physics. The structure of commuting elements in an algebra is of fundamental importance for its structure and representation theory as well as for its applications. The main objects studied in this monograph are q-deformed Heisenberg algebras -- more specifically, commuting elements in q-deformed Heisenberg algebras.In this book the structure of commuting elements in q-deformed Heisenberg algebras is studied in a systematic way. Many new results are presented with complete proofs. Several appendices with some general theory used in other parts of the book include material on the Diamond lemma for ring theory, a theory of degree functions in arbitrary associative algebras, and some basic facts about q-combinatorial functions over an arbitrary field. The bibliography contains, in addition to references on q-deformed Heisenberg algebras, some selected references on related subjects and on existing and potential applications.The book is self-contained, as far as proofs and the background material are concerned. In addition to research and reference purposes, it can be used in a special course or a series of lectures on the subject or as complementary material to a general course on algebra. Specialists as well as doctoral and advanced undergraduate students in mathematics andphysics will find this book useful in their research and study.

Table of Contents

Preface	vii
Introduction	1	(18)
q-Deformed Heisenberg algebras	1	(4)
Some references and motivation	5	(9)
Contents by chapters	14	(1)
Conventions and notations	15	(4)
Immediate consequences of the commutation relations	19	(16)
The values of q	19	(1)
Reordering formulae	20	(10)
Simplifying commutation relations	30	(5)
Bases and normal form in H(q) and H(q, J)	35	(18)
The definition of H(q, J)	36	(2)
Three bases for H(q, J)	38	(10)
Comparison of the three bases	48	(1)
Computational aspects	49	(4)
Degree in and gradation of H(q, J)	53	(20)
Degree in H(q, J)	54	(8)
Grading H(q, J)	62	(6)
Some useful properties	68	(5)
Centralisers of elements in H(q, J)	73	(22)
General definitions and theorems	74	(3)
Classification of H(q, J)	77	(4)
The case qj of torsion type for some j ϵ J	81	(3)
The centre of H(q, J) when q is of strictly direct type on J	84	(2)
H(q, J) for q ϵ Q (J, K)	86	(9)
Centralisers of elements in H(q)	95	(22)
Classification of H(1)	95	(4)
H(q) when q is of free type	99	(5)
H(q) when q is of torsion type	104	(13)
Algebraic dependence of commuting elements in H(q) and H(q, n)	117	(12)
Representations of H(q, J) by q-difference operators	129	(12)
Appendix A The Diamond Lemma	141	(14)
A.1 Definitions and proofs	142	(7)
A.2 A short key to the notations	149	(2)
A.3 A few extra results	151	(4)
Appendix B Degree functions and gradations	155	(28)
B.1 General theory of degree functions	156	(6)
B.1.1 A generalisation of the Bernstein filtration	156	(4)
B.1.2 The basic properties	160	(2)
B.2 Degree and free algebras	162	(17)
B.2.1 Additivity of degree in free algebras	163	(5)
B.2.2 Some technical lemmas	168	(7)
B.2.3 Degree in a free algebra versus degree in a quotient	175	(4)
B.3 Gradations	179	(4)
Appendix C q-special combinatorics	183	(38)
C.1 Definitions and existence	184	(2)
C.2 Other properties	186	(10)
C.3 q-Stirling numbers	196	(19)
C.4 Extending the q-combinatorial functions	215	(6)
Appendix D Notes on notations	221	(2)
Bibliography	223	(32)
Index	255