LEMMA ON

Internat. J. Math. & Math. Sci.

Vol. 5 No. 3 (1982) 621-623

621

ON WHITEHEAD’S SECOND LEMMA FOR LIE ALGEBRAS

WILLIAM T. FLETCHER

Department of Mathematics North Carolina Central University

Durham, North Carolina 27707

(Received December 4, 1980 and in Revised form September 15, 1981)

ABSTRACT. A proof of Whitehead’s second lemma which is independent of the first Whitehead lemma is given.

KEY WORDS AND PHRASES. Whehead’s second lemma, Casimir

operor,

1980 MATHEMATICS SUBJECT CLASSIFICATION CODE. 17B99.

1. INTRODUCT ION.

THE SECOND WHITEHEAD LEMMA FOR LIE ALBEBRAS. Let L be a finite-dimensional semi-simple Lie algebra of characteristic O, M a finite-dimensional L-module and (x,y) g(x,y) a bilinear mapping of L L into M such that

g(x,x) O, (1.1)

g([xy],z) + g(x,y)z + g([yz],x) + g(y,z)x + g([zx],y) + g(z,x)y Oo (1.2) Then there exists a linear mapping x- x of L into M such that

]

g(x,y) x y y x- [xy (1.3)

The standard proof of the second Whitehead lenrna for Lie algebras of charact- eristic 0 [i, p. 89] considers the three cases: F, the Casimir operator, is non- singular; F is nilpotent; and F is neither non-singular nor nilpotent and; ⁱⁿ the case where F is nilpotent, the proof depends on the first Whitehead lemma. The purpose of this note is to give a simplification of the proof for the case F is nilpotent which is independent of the first Whitehead lemma. This, ^of course, will simplify the proof of the Levi theorem which is one of the main uses of the lemma.

622 W.T. FLETCHER

For our purposes here we will need the following lemma, the proof of which follows easily from Theorem 2 of [2] and the properties of the complementary basis.

LEMMA i. If A is a flnite-dimensional Lie algebra which has non-degenerate (eI

Killing formwith basis e2, ^e and complementary basis (e e 2 n

e),

f([ex], [Yek] = f([ekx], [ye]).

k k

2. PROOF OF THE MAIN RESULT.

We introduce the following notation: R is the kernel of the representation of L determined by M, L

I ^{is an ideal such that L R}

LI, (ui)

(ul);

i i, 2, m are dual bases of L

I ^{relative to the trace} form of the given representation, F is the Casimir operator determined by the dual bases and is the

in M Suppose that F is nilpotent, then m 0, R--L, and mapping x

(xui)u

i

the representation determined by M is the zero representation. Then (1.2) reduces to:

g([xy],z) + g([yz],x) + g([zx],y) O, (2.1) Let (e

i)

i’);

and sum over i to get z

[wei]

Since L is non-degenerate in the sense of [2], Theorem 2 of [2] implies that [we ]] -w and (2 2) reduces to

[el

i{g([xe] ^[wei])-

^g([eix], [ew]))=

We can verify that

([xel], [wei])+ g([xei], [we]))

g([xe], [wei]),

ON WHITEHEAD’S SECOND LEMMA FOR LIE ALGEBRAS 623

Now if we substitute the above result in (2.3), we get

g(w,x) _wg([e_i [x]],

.) ^o.

Thus, we obtain a linear mapping e of L into M such that

g(w,x) [wx] g([e

i [wx]],

el).

REFERENCES

i. JACOBSON, N. Lie Algebras, Interscience Publ., No. i0, New York, 1962.

2. CAMPBELL, H.E. On the Casimlr Operator, Pacific J. of Math. 7, No. 3 (1957), 1325- 1331.

Mathematical Problems in Engineering

Special Issue on

Time-Dependent Billiards

Call for Papers

This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi accelera- tion (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.

This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).

We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.

To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.

Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System athttp://

mts.hindawi.com/according to the following timetable:

Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009

Guest Editors

Edson Denis Leonel,Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil ; [email protected]

Alexander Loskutov,Physics Faculty, Moscow State University, Vorob’evy Gory, Moscow 119992, Russia;

[email protected]

Hindawi Publishing Corporation http://www.hindawi.com