On models of irreducible representations of certain lie algebras and special functions

ON MODELS OF IRREDUCIBLE

REPRESENTATIONS OF CERTAIN LIE ALGEBRAS AND SPECIAL FUNCTIONS

by VIVEK SAHAI

Thesis submitted in fulfilment of the requirements of the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

INDIAN INSTITUTE OF TECHNOLOGY, DELHI

1989

CERTIFICATE

This is to certify that the thesis entitled "ON MODELS OF IRREDUCIBLE REPRESENTATIONS OF CERTAIN LIE ALGEBRAS AND SPECIAL FUNCTIONS'', which is being submitted by Mr. Vivek Sahai for the award of the degree, DOCTOR OF PHILOSOPHY, to the Indian Institute of Technology, Delhi, is a bonafide record of research work done under my guidance and supervision.

The thesis has reached the stage of fuifitment of the requirements and regulations related to the degree. The results obtained in this thesis have not been submitted to any other Institute or University for the award of any degree or diploma.

H. L/ Manocha Su rvisor.

IN MEMORY OF

MY GRANDFATHER

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SY140PSIS

Lie theory - the theory of Lie groups, Lie algebras and their applications - is a fundamental part of mathematics. It has been the focus of burgeoning research effort, and is now seen to encompass a tremendous spectrum of mathematical areas. Research works of Lie, Car tan, Killing and Weyl have led to its recent most elegant form which is extremely useful in many applications of mathematics. Indeed, the applications of Lie theory are astonishing in their pervasiveness and sometimes in their expectedness.

The past four decades have witnessed mathematicians showing keen interest in relating Lie theory to hypergeometric functions of one and more variables. Though, to some, expressions 'Lie theory and special functions' sound like 'East and West', the works of Weisner, Vilenkin, Miller, Kalnins, Manocha and a few others have eloquently demonstrated that there is a natural and deep connection between Lie theory and special functions. The scenario at present is that Lie theory has tremendously enriched the theory of special functions, while special functions have repaid it by bringing 'concreteness' in the 'abstractness' of Lie theory. Indeed, the two are moving hand in hand with prospects of further strengthening of ties between the two. One aspect of Lie theory which has made a tremendous contribution to this development is the representation theory of Lie groups and Lie algebras. To be precise, the connection between Lie theory and special functions has been by and large based on the following sequence :

1. Construction of a model of representation, preferably irreducible, of a Lie algebra on the representation space involving special functions.

2. Exponention of the above model so as to have a model of representation of the corresponding Lie group.

3. Exploitation of the model in (2) for obtaining identities involving special function appearing in CD.

As can be seen, the most important step in the above sequence is constructing a model of representation of a Lie algebra with special functions appearing in the representation space. To accomplish this, there is a need for developing theory on this.

Indeed, theory on this has been developed by Weisner, Vilenkin, Miller, Kalnins and Manocha, to quote a few. However, there is a need for enriching and growing this theory further. It is precisely this role of developing theory and constructing models which the proposed thesis takes upon itself.

The thesis will consist of six chapters. A brief insight of each chapter is given below :

Chapter f. Introduction

This chapter is introductory in nature and provides all the basic information needed for the subsequent chapters. It mainly contains two sections. The first section deals with Lie groups and Lie algebras, while the second one is devoted to special functions. An effort is made to make the thesis self contained as far as possible.

Chapter 2. Models based on fractional differentiation

A theorem is established which suggests various irreducible representations of the special linear algebra sLC2,C). Guided by

this theorem, models of representations are constructed, with special functions 2F1 and /F/ appearing in the representation space. Sensing that these models have the potential to induce new models, theory of fractional calculus is introduced and then invoked. This helps in constructing new models, with hypergeometric functionsk+1

k+1 F

and appearing in the k+1 representation spaces. Later, the new models are fully exploited for obtaining identities, believed to be new.

Chapter 3. Models based on Mellin transformation

Two theorems are reproduced which provide guidelines for constructing models of irreducible representations of the Lie algebras si.C2,C) and oscillator algebra §C0,1). Based on these, models of representations are constructed, with the Laguerre polynomials L

m

CoD Cx) appearing in the representation space. To

translate these models into new ones, Mellin transformation is introduced. This helps in constructing new models in terms of difference differential operators, with the polynomials SmC a3

Cp;X), defined in terms of F D

Cn)

' serving as the representation space. Through exponentiation, these models lead to new interesting results and identities.

Chapter 4. Models based on Euler transformation

Based on a theorem due to Miller, models of irreducible

representations of siC2,C) and (0,13 in one variable are introduced. Through another theorem, which is established, these models are upgraded into those involving two variables. A transformation based on Euler integral is introduced, which transforms the above models into deeper ones with hypergeometric functions appearing in the representation space. This exercise

leads to recurrence relations in terms of difference operators.

Chapter 5. Models of q-representations

Taking a cue from the work of Manocha, theory involving g-representations is developed. Based on this theory, models of irreducible g-representations of C0,0), § q(0,1) and g (1,0) are constructed. As g -4 1 all these models reduce to those of new ones.

Chapter 6. Models of irreducible unitary representation of compact Lie group SUC83

The compact Lie group SU(2) and its real Lie algebra suC2) are introduced. An attempt is made to construct a model of an irreducible unitary representation of SUC2) on a finite dimensional Hilbert space involving hypergeometric functions gl.

To accomplish this, firstly a model of a finite-dimensional irreducible representation of sLC2,C) in one variable is constructed. A theorem is established which upgrades this model to a two variable model. Since suC2) is a real part of s/C2,C), meaning that s1C2,C) is the complexification of suC2), restriction of this model to suC2) gives a finite-dimensional irreducible representation of suC2) on a Hilbert space. Through a similarity transformation, this model is transformed into the one involving hyper geometric functions gl. By carrying out exponentiation, the required model of irreducible unitary representation of SIX 2) is established. This model, in turn, gives rise to interesting identities.

CONTENTS

1. Introduction 1

1.1 Lie Groups and Lie Algebras 1

1.2 Exponential of a Matrix 8

1.3 Some Examples of Local Lie Groups and

Their Lie Algebras 9

1.4 Representations of Lie Groups and Lie Algebras 11 1.5 Local Transformation Groups 11 1.6 Hypergeometric Functions of One Variable 15 1.7 Functions of Two and Several Variables 17 1.8 Basic Hypergeometric Functions 21

1.9 Generating Functions 23

2. Models Based on Fractional Differentiation 26 2.1 Operators Involving Fractional Derivatives 27

2.2 Representations of siC2,C) 33

2.3 Representation DCu,m0) 36

2.4 Type B Representation Iu 40

2.5 Type A Representation iu 52

3. Models Based on Mellin Transformation 55 3.1 Representations of _{sLC2,C3 56}

3.2 Representations of SC0,13 58

3.3 Mellin Transforms of Functions 62 3.4 Model of Representation

-C1+a3 /2 of s/C2,C) 64 3.5 Model of Representation To of gco,i) 71 3.6 Model of Representation

1+a,k of gco,l) 75

4. Models Based on Euler Transformation 80

4.1 Symmetry Algebra of 2F1 80

4.2 Models of Irreducible Representations of sLC2,C) 88 4.3 New Models of Irreducible Representations of sLC2,C) 93

4.4 Identities 103

5. Models of q-Representations ₁₁₀

5.1 The 'Basic' Concept 111

5.2 The Lie Algebras C a, b) and Their q-Representations 114 5.3 q-Representation Q PC w

' ^m0:) of go,o) q 0

5.4 q-Representations of VC0,1)

5.5 q-Representations of gcl,o) 126

6. Models of Irreducible Unitary Representations of Compact

Lie Group SUC2) 132

6.1 Finite Dimensional Irreducible Representation

of slC2,C) 133

6.2 One Variable Model of DC2v) 136 6.3 Two Variable Model of DC2v) 138

6.4 The Group SUC23 141

6.5 Models of Irreducible Representation of suC2) 145

6.6 Conclusion 150

117 120

BIBLIOGRAPHY 151