Solvable Lie Algebra and Lie’s Theorem

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Solvable Lie Algebra and Lie’s Theorem

Mr. Sarvesh Kumar Mishra1

1Assistant Professor, Deptt of Applied Science

Mangalmay Institute of Engineering & Technology Greater Noida (U.P) Greater Noida (U.P)

Dr. Suneeta Choudhary2 2Assistant Professor, Deptt of Applied Science

Mangalmay Institute Of Engineering & Technology Greater Noida (U.P) Greater Noida (U.P)

3Er. Alok Kumar Maurya

3Assistant Professor

Dept. Deptt of Mechanical Engineering Mangalmay Institute Of Engineering & Technology Greater Noida (U.P) Greater Noida (U.P)

4 Er. Ankit Kumar

4assistant Professor

Dept. Deptt of Mechanical Engineering Mangalmay Institute Of Engineering & Technology Greater Noida (U.P) Greater Noida (U.P)

Abstract :A simply connected Lie group is completely determined by its Lie algebra. By this, of course, we mean that it is determined to within isomorphism. We now discuss the possibilities for Lie groups which are not simply connected. We say a Lie group is connected if every two points of it can be joined by an arc lying in the group. If a Lie group is connected, we call it simply connected when every simple closed curve in the group can be continuously shrunk to a point without any part of it passing outside the group in the process. By a component of a Lie group we mean a maximal connected subset, i.e., all the elements which can be connected to some given element by arcs in the group. In this research we consider the situation where the Lie group is not connected. It can be shown that the component containing the identity is always a closed normal subgroup of the Lie group and that the components are precisely the cosets of this normal subgroup. We can regard this collection of cosets forming the quotient group, as an abstract group. (Indeed, if we take it to be a discrete group the natural mapping is analytic.) The study of the algebraic structure of a Lie group which is not connected can almost be broken into two parts:the structure of the connected subgroup forming the component of the identity and the structure of the discrete quotient group.

INTRODUCTION

The rst step in the classication of Lie algebras was the consideration of classical Lie algebras. They were totally outlined over and , but the problem is not so easy over an algebraically closed eld of characteristic dierent from zero. This new Lie theory emerged around 1935 from the studies by Witt, who dened a simple Lie algebra (now called the Witt algebra W1) whose behavior was totally dierent from the Lie algebras studied till then, over or

. Less than ten years later, Jacobson and Zassenhaus put some order in these new algebras, but it was not until the 21st century when a clear classication came. In fact, is a survey on these specic classications of simple nite- dimensional Lie algebras over algebraically closed elds of characteristic > 0. Roughly speaking, and being a prime greater than 3, the simple Lie algebras are either classical or nite dimensional Cartan Lie type (and their deformations) or Melikyian algebras. If the characteristic is big enough, some other interesting properties hold; for instance, for nite-dimensional Lie algebras over an algebraically closed eld of characteristic = > 7, the existence of non-singular Casimir operators (i.e., dealing

with a restricted Lie algebra) is equivalent to the decomposition of the algebra as a direct sum of classical simple Lie algebras.

Literature review

One of the most fundamental concepts in mathematics is that of a group. Germs of group was present, even in ancient times, in the study of motions in space, in the study of congruences of geometric figures. In the beginning of nineteenth century development of group theory started.

Dedekind [1897] studied about groups all of whose subgroups are normal.

Miller and Moreno [1903] studied groups all of whose proper subgroups are abelian. They studied that all such groups are solvable. Their orders cannot be divided by more than two distinct primes.

Schmidt [1924]worked on groups every proper subgroups of which is special.

Malcev [1945]determined the classication of complex solvable Lie algebras.

Golfond [1948] determined groups all of whose proper subgroups are special.

Chandra Harish [1955]representation of semi simple lie group.

METHODOLOGY

Definitions Solvable group:A group is called solvable if it has a subnormal series whose factor group are all abelian, that is, if there are subgroup {1} = 0 1 . . . =

such that 1 is normal , and 1 is an abelian

group, for = 1,2,3 .

DefinitionsFunction space:A function space ()is the collection of all real-valued continuous functions defined on some interval . () is the collection of all function which belongs () with continuous derivatives. A function space is a topological vector space.

DefinitionsOperator: An operator : () () assigns to every function (). It is therefore a mapping between two function spaces.

Definitions Commutator: Let be a group. An element of the form 11 which is denoted by [, ] is called a commutator. The subgroup of generated by all commutator of is called commutator subgroup or the derived subgroup of .

Or

Let , be operators. Then the commutator of and

is defined as[, ] = .

DefinitionsJacobi identity: The Jacobi identity is the relationship [, [, ]] + [, [, ]] + [, [, ]] = 0 between three elements, and , where [, ] is the commutator. The elements of a lie algebra satisfy this identity.

RESULTS

In this section and are field satisfying , (where is a complex field) and all Lie algebras have the underlying field and are finite dimensional.

THEOREM 1: An -dimensional Lie algebra is solvable if and only if there exists a sequence of subalgebras

= 0 1 = 0

Such that, for each , +1 is ideal in and dim

(+1) = 1.

PROOF: Let g be solvable. Form the commutator series

and interpolate subspace in the sequence so that dim

(+1)=1 for all i. We have

= 0 1 = 0.

REMARK

1. When g is a solvable Lie algebra and is representation,

() is solvable. This follows immediately from

2. The theorem is the base step in an induction that will show that has a basis in all the matrices of () are triangular. This conclusion appears as theorem 3 below. If is solvable lie algebra of matrices and is the identity and one of the conditions on is satisfied, then g can be conjugated so as to be triangular.

PROOF: We induct on dim = 1, then () consist of the multiples of a single transformation, and the results follows.

Assume the theorem for all solvable Lie algebras of dimension less than dim g satisfying the eigenvalue condition. Since g is solvable, [, ] . choose a subspace of codimension 1 in g with [, ] . then [, ] [, ] , and is an ideal. So is solvable. (also the eigenvalue condition holds for h if it holds for .) By inductive hypothesis we can choose with ()

For any , we can find such that

+1.

= () for all , where () is a scalar valued

Then

[ , ] [, ] = +1 .

+1

function defined for .

1 = 0, 0 = = ()1

+1

and let = { . . . . .

. }. Then () . Let

Hene is a subalgebra for each , and +1 is an ideal . Conversely let the sequence exist. Choose so that =

+ +1 .

We show by induction that , so that = 0 . In fact, 0 = 0 .If , then

+1 = [, ] [ + +1, + +1] [, +1] + [+1, +1] +1 ,

and the induction is complete. Hence is solvable

The kind of sequence in the theorem is called an elementary sequence. The existence of such a sequence has the following implication. Write = +1 . Then is a 1-demensional subspace of , hence a subalgebra. Also +1 is ideal . In a view of proposition 1.22

..

is exhibited as a semidirect product of a 1-dimensional Lie algebra and +1 . The theorem says that solvable Lie algebra are exactly those that can be obtain from semidirectproduct , starting from 0 and adding one dimension at a time .

Let be vector space over , and be a Lie algebra. A

0

be an eigenvalue for () in . We show that v is an eigenvalue for each () , .

First we show that

() () {0, . . , 1}

(1)

For all . We do so by induction on . Formula (1) is valid for = 0 by definition of 0. Assume (1) for . then

()+1 = ()()

= ([, ]) + ()()

([, ])

+ ()() {0, . . , 1}

(By induction)

([, ])

+ ()() {0, . . , 1, ()0, . . , ()1

(By induction)

()() {0, . . , }

representation of on is homomorphism of Lie algebra

: () , which we often write simply as

= ()+1

{0, . . , }

: . Because of the definition of bracket in

, the conditions on are that it be linear and satisfy

([, ]) = ()() ()() for all , . (1)

THEOREM 2: Let g be solvable, let 0 be a fintedimensional vector space over , and let

be a representation. If is algebraically closed,

This proves (1) for p+1 and completes the induction. Next we show that

([, }) = 0 .

(2)

In fact, (1) says that () and that, relative to the basis 0, 1, ., the linear transformation () has matrix

()

() = ( )

then there is a simultaneous eigenvector

0

for all the

0 ()

Thus Tr () = () dim E, and we obtain

members () . More generally (for ).there is a

simultaneous eigenvector if all the eigenvalues of all

(), , lie in .

([, ]) dim E = Tr ([, }) =Tr[(), ()] = 0.

Since our fields have characteristic 0, (2) follow, Now we can sharpen (1) to

() = () for all . (3)

To prove (3), we induct on . for = 0, the formula is the definition of 0. Assume (3) for p. then

()+1 = ()()

= ([, ]) + ()()

= ([, ]) + ()()

(by

induction)

= 0 + ()+1 by (2)

= ()+1.

This completes the induction and proves (3). Because of (3), () = () for all and in particular for =

. Hence the eigenvector of () is also an eigenvector of (). The theorem follows.

Before carrying out the induction indicated in Remark 2, we observe something about eigenvalues in connection with representations. Let be a representation of on a finite dimensional , and let be an invariant subspace: () . Then the formula ()( + ) =

() + defines a quotient representation of on /. The characteristic polynomial of () on is the product of the characteristic polynomial on and that on /, and hence the eigenvalues for / are a subset of those for .

CONCLUSION

THEOREM 1: An -dimensional Lie algebra is solvable if and only if there exists a sequence of subalgebras

= 0 1 = 0

Such that, for each , +1 is ideal in and dim

(+1) = 1.

THEOREM 2 : Let g be solvable, let 0 be a finte

dimensional vector space over , and let be a representation. If is algebraically closed, then there is a simultaneous eigenvector 0 for all the members() . More generally (for ).there is a simultaneous eigenvector if all the eigenvalues of all

(), , lie in .

REFERENCES

3. Artin, M. [2000]: Algebra, Prentice Hall of India.

4. Besche, H.-U. and Eick, B [1999]: "The Groups of Order at Most 1000 Except 512 and 768." J. Symb. Comput.27, 405- 413.

5. Bourbaki N. [1961]:Elements deMathematique Vol. 26, Chap. on Algebres de Lie.,

6. Bredon, G. E [2005]: Topology & Geometry. New York: Springer-Verlag, p. 69.