On the computation of polynomial representations of nilpotent Lie groups: a symbolic mathematical approach

Nilpotent Lie algebras and Lie groups are of special interest in a wide variety of applications in such fields as computer science, control theory, robotics, and mathematical physics, since computing with these structures involves only.polynomials. In this paper we show how to compute explicitly polynomial representations and we illustrate with an example. Our approach is based on flags. The method presented here has been developed specifically for doing symbolic computations on nilpotent (and solvable) Lie groups. Properties as special functions, e.g. addition theorems, recurrence formulas, etc., of the polynomials appearing in the representations follow readily from the constructions. Our symbolic approach leads to mathematical computations easy to do with a symbolic manipulator such as Maple. The drastic space limitation does not permit to display a Maple output. The programming code is available by the authors. "Department of Mathematics, Southern Illinois University, Carbonda.l¢, Illinois 62901 USA, e-mail: pfeinsil@cmathl . siu.edu tCRIN-CNRS et D6partement de Math6matiques, Universit6 de Nancy I, B.P. 239, 54506 Vandoeuvre-16s-Nancy, France, e-mail: franzu~loria.fr :CRIN-CNRS, Universit6 de Nancy I, B.P. 239, 54506 Vandoeuvre-16s-Nancy, France, e-mail: schott@loria.fr "Permission to make digital or hard copies of part or all or" this work for ~ersonal or cl~sroom use is granted without lee provided that copies are not nade or distributed for profit or commercial advantage and that copies bear his notice and the full citalion on the first page. Copyrights for components ffthis work owned by others than ACM must be honored. Abstracting with :redit is permitted. To cop:,' otherwise, to republish, to post on servers or to cdistribute to lists, requires prior specific permission and/or a tee. " 2~1997ACM0-89791-850-9 97 0002 3.50 1 I n t r o d u c t i o n The study of nilpotent Lie algebras has ample precedent in the study of formal power series motivated by applications in control theory [6, 7], robotics [8] and computer science [3, 9]. Recent works by the present authors [4, 5] concern Appell systems, special functions, and infinitedimensional representations of Lie groups. We present here an original method based on the notion of flag for computing efficiently representations of nilpotent Lie groups. The formula giving the matr ix elements of the representations is particularly suited for practical computat ion using a symbolic manipulator such as Maple. 2 F l a g s a n d p o l y n o m i a l r e p r e s e n t a t i o n s o f n i l p o t e n t L i e g r o u p s Let (1.(d be a basis for a nilpotent Lie algebra. Let .4 be the corresponding universal enveloping algebra. Denote the d-tuple ( ( 1 , . . . , ~d) by ~. Let n denote a multi-index ( n l . . . . , rid). As basis for . 4 and as basis for polynomials in commuting variables A -( A t , . . . , Ad) we use c . (A) = A? respectively. Note that products involving ~5 are ordered. The elements of the group near the identity may be expressed as products of one-parameter subgroups generated by the (i. I.e., let g ( A , ~ ) = e A'~I e A 2 ~ . e A d ~ This may be expanded in the form and interpreted as as a pairing (C, ko) of the sequences c : { c,, }, q, : { }.