Lie groups : a problem-oriented introduction via matrix groups / Harriet Pollatsek.

Format:

Book

Author/Creator:

Pollatsek, Harriet Suzanne Katcher.

Contributor:

Hazel M. Hussong Fund.

Series:

MAA textbooks

Language:

English

Subjects (All):

Lie groups.

Lie groups--Problems, exercises, etc.

Matrix groups.

Genre:

Problems and exercises.

Physical Description:

xii, 177 pages : illustrations ; 26 cm.

Place of Publication:

[Washington, D.C.] : Mathematical Association of America, [2009]

Summary:

Can be used as supplementary reading in a linear algebra course, or as a primary text in a "bridge" course that helps students make the transition to courses that emphasize definition and proofs, as well as for an upper level elective.

The work of the Norwegian mathematician Sophus Lie extends ideas of symmetry and leads to many applications in mathematics and physics. Ordinarily, the study of the "objects" in Lie's theory (Lie groups and Lie algebras) requires extensive mathematical prerequisites beyond the reach of the typical undergraduate. By restricting to the special case of matrix Lie groups and relying on ideas from multivariable calculus and linear algebra, this lovely and important material becomes accessible even to college sophomores. Working with Lie's ideas fosters an appreciation of the unity of mathematics and the sometimes surprising ways in which mathematics provides a language to describe and understand the physical world.

Lie Groups is an active learning text that can be used by students with a range of backgrounds and interests. The material is developed through 200 carefully chosen problems. This is the only book in the undergraduate curriculum to bring this material to students so early in their mathematical careers. Book jacket.

Contents:

1 Symmetries of vector spaces 1

1.1 What is a symmetry? 1

1.2 Distance is fundamental 5

1.3 Groups of symmetries 8

1.4 Bilinear forms and symmetries of spacetime 14

1.5 Putting the pieces together 20

1.6 A broader view: Lie groups 26

2 Complex numbers, quaternions and geometry 29

2.1 Complex numbers 29

2.2 Quaternions 33

2.3 The geometry of rotations of R³ 35

2.4 Putting the pieces together 39

2.5 A broader view: octonions 41

3 Linearization 45

3.1 Tangent spaces 45

3.2 Group homomorphisms 51

3.3 Differentials 55

3.4 Putting the pieces together 63

3.5 A broader view: Hilbert's fifth problem 66

4 One-parameter subgroups and the exponential map 69

4.1 One-parameter subgroups 69

4.2 The exponential map in dimension 1 70

4.3 Calculating the matrix exponential 72

4.4 Properties of the matrix exponential 76

4.5 Using exp to determine L(G) 78

4.6 Differential equations 81

4.7 Putting the pieces together 87

4.8 A broader view: Lie and differential equations 91

4.9 Appendix on convergence 93

5 Lie algebras 99

5.1 Lie algebras 99

5.2 Adjoint maps-big 'a' and small 104

5.3 Putting the pieces together 108

5.4 A broader view: Lie theory 111

6 Matrix groups over other fields 115

6.1 What is a field'? 115

6.2 The unitary group 116

6.3 Matrix groups over finite fields 121

6.4 Putting the pieces together 127

6.5 A broader view: finite groups of Lie type and simple groups 132.

Notes:

Includes bibliographical references (pages 169-170) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Hazel M. Hussong Fund.

ISBN:

9780883857595

0883857596

OCLC:

434563150

Publisher Number:

99937171102