Re: Lie algebra - real life encounter

From: Bill Dubuque (wgd_at_nestle.csail.mit.edu)
Date: 07/09/04 

Date: 09 Jul 2004 18:04:31 -0400

"legs x'd" <lol@legscrossing.com> wrote:
>
> Hoping to hear some testimonials from some of you
> on where, when, and why you first used a Lie algebra.
> Did it give insight into the problem at hand or was
> it just a nice generalization? Any book recommendations?

One nice application is to symmetries of differential equations
and special functions. See my prior post below for references.

Subject: Re: A new algebraic structure?
Newsgroups: sci.math
Date: 1997/04/23

"Etherman" <etherman@mdc.net> writes:
>
> I've been thinking about the problem of solutions to nonlinear DE's.
> With linear eqs. you can multiply the solutions by constants and
> add them and still get solutions. You can't do this with nonlinear
> eqs. So I'm considering the an algebraic structure that is a
> generalization of multiplication by scalars and addition. ...

Have a look at the literature on symmetry groups of differential
equations (which is closely related to the group-theoretic approach
to special functions, esp. via separation of variables and
representation theory, e.g. see the book of Willard Miller cited below).

Following are some pointers into the literature.

-Bill Dubuque

Survey papers
=============

Schwarz, Fritz.
Symmetries of differential equations: from Sophus Lie to computer algebra.
SIAM Review 30 (1988), no. 3, 450--481.
MR 89i:22013 (Reviewer: W. F. Ames) 22E05 (01A60 35A30 58G35 68Q40)

Vinogradov, A. M.
Local symmetries and conservation laws.
Acta Appl. Math. 2 (1984), no. 1, 21--78.
MR 85m:58192 (Reviewer: Toru Tsujishita) 58G37 (35A30 58A17 58G35)

Winternitz, Pavel
Lie groups and solutions of nonlinear differential equations.
Nonlinear phenomena (Oaxtepec, 1982), 263--331,
Lecture Notes in Phys., 189, Springer, Berlin-New York, 1983.
MR 85k:58090 (Reviewer: Ernest G. Kalnins) 58G35 (22E05 34A99 35A30 35C99)

Textbooks
=========

Ames, W. F.; Anderson, R. L.; Dorodnitsyn, V. A.; Ferapontov, E. V.;
Gazizov, R. K.; Ibragimov, N. H.; Svirshchevski\u\i, S. R.
CRC handbook of Lie group analysis of differential equations.
Vol. 1. Symmetries, exact solutions and conservation laws.
CRC Press, Boca Raton, FL, 1994. xiv+429 pp. ISBN: 0-8493-4488-3
MR 95h:58145 (Reviewer: Niky Kamran) 58G35 (00A20 22E05 35A30 58-02)

Olver, Peter J.
Applications of Lie groups to differential equations. Second edition.
Graduate Texts in Mathematics, 107.
Springer-Verlag, New York, 1993. xxviii+513 pp. ISBN: 0-387-94007-3
MR 94g:58260 58G35 (22E60 35A30 58F05 58F07)

Stephani, Hans
Differential equations. Their solution using symmetries.
Cambridge University Press,
Cambridge-New York, 1989. xii+260 pp. ISBN: 0-521-35531-1; 0-521-36689-5
MR 91a:58001 (Reviewer: L. M. Berkovich)
58-01 (34-01 35-01 35A30 58F35 58F37 58G35 58G37)

Edelen, Dominic G. B.
Applied exterior calculus.
A Wiley-Interscience Publication.
John Wiley & Sons, Inc., New York, 1985. xix+471 pp. ISBN: 0-471-80773-7
MR 87i:58006 (Reviewer: Ian M. Anderson) 58A15 (49F05 58-01 58G35 80-02)

Ibragimov, Nail H.
Transformation groups applied to mathematical physics.
Translated from the Russian.
Mathematics and its Applications (Soviet Series).
D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1985.
xv+394 pp. ISBN: 90-277-1847-4
MR 86e:58001 58-02 (58F35 58G35)

Ovsiannikov, L. V.
Group analysis of differential equations.
Translated from the Russian by Y. Chapovsky.
Translation edited by William F. Ames.
Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers],
New York-London, 1982. xvi+416 pp. ISBN: 0-12-531680-1
MR 83m:58082 (Reviewer: J. S. Joel) 58G35 (35A30 58H15)

Miller, Willard, Jr.
Symmetry and separation of variables.
With a foreword by Richard Askey.
Encyclopedia of Mathematics and its Applications, Vol. 4.
Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1977.
xxx+285 pp. ISBN: 0-201-13503-5
MR 57 #744 (Reviewer: Ernest G. Kalnins) 33A75

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