9789810244033

Commuting Elements in Q-Deformed Heisenberg Algebras

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  • ISBN13:

    9789810244033

  • ISBN10:

    9810244037

  • Format: Hardcover
  • Copyright: 2000-09-01
  • Publisher: WORLD SCIENTIFIC PUB CO INC
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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

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