LI Xiao-chao,JIN Quan-qin

(1.Department of Mathematics,Huanghuai University,Zhumadian 463000,China;2.Department of Mathematics,Tongji University,Shanghai 200092,China)

Solvable Lie Algebras with NilradicalTheir Casimir Invariants

LI Xiao-chao1,JIN Quan-qin2

(1.Department of Mathematics,Huanghuai University,Zhumadian 463000,China;2.Department of Mathematics,Tongji University,Shanghai 200092,China)

solvable Lie algebra;nilradical;Casimir invariant

§1.Introduction

Lie algebras and their invariants play a relevant role in physical models,such as the standard model,nuclear collective motions and rotational states in particle and nuclear physics.Although semisimple Lie algebras occupy a central position within the Lie algebras appearing in physical models,the class of solvable algebras has shown to be of considerable interest,as follows from their applicability to the theory of completely integrable Hamiltonian systems.Levi’s theorem[5]tells us that any finite-dimensional Lie algebra is isomorphic to a direct sum of a semisimple Lie algebra and a maximal solvable ideal.

§2.Preliminaries

Any solvable Lie algebra s contains a unique maximal nilpotent ideal,i.e.,the nilradical n.We will assume that n is known,that is,in some basis{X1,X2,···,Xn}we know the Lie brackets

We consider the problem of the classification of all the solvable Lie algebras s with nilradical n.This can be achieved by adding further elements Y1,···,Ymto the basis{X1,X2,···,Xn} to form a basis of s.Since the derived algebra of a solvable Lie algebra is contained in the nilradical[4],i.e.,[s,s]?n,we have

Now,we will consider the adjoint representation of s,restrict it to the nilradical n and find ad|n(Yi).It follows from the Jacobi identities that ad|n(Yi)is a derivation of n.In other words(see e.g.,[10,12]), finding all sets of matrices Biin(2.2)satisfying the Jacobi identities is equivalent to finding all sets of outer nil-independent derivations of n:

Different sets of derivations may correspond to isomorphic Lie algebras,so redundancies must be eliminated,see e.g.,[10,12].The equivalence is generated by the following transformations

where(Sij)is an invertible m×m matrix,(Tib)is a m×n matrix and the invertible n×n matrix(Rab)must be chosen so that the Lie brackets(1.1)are preserved.

§3.Classi fication of Solvable Lie Algebras with the Nilradical

3.1 Nilpotent Algebra

shows that dki=0(3≤i≤2n,k＜i).Consider[X2,X3]=0,we have d12=0.Similarly,the condition

implies D(X2n+1)=d2n+1,2n+1X2n+1.Hence,D=(dij)are lower triangular matrices. Consider the derivations on basis elements x=Xi,y=Xj,we get

From the third equation of case(C1),we by induction obtain

From case(C3),we obtain

From equations(3.1)and(3.2),we have(not include d2n,2,since i≥2 in(3.2))

The equations(3.1)and(3.3)are consistent.Finally,we can easily get

we can transform all Dkinto the form

3.2 Solvable Lie Algebras with the Nilradical

From Subsection 3.1,we know that the diagonal elements of the matrix Dkcan be completely determined by αk,βk.According to the theory of linearly algebra,the number of nil-independent elements m can be at most two since a set of three or more derivations of the form(3.4)cannot be linearly nil-independent.

In the theorems,“solvable”will always mean solvable,indecomposable,non-nilpotent.

Theorem 3.2Five types of solvable Lie algebras of dimension dim s=2n+2 exist for any n≥3.They are represented by the following

Whereμ1,μ2,μ3andμ4can only be 0 or 1,[]denotes the integer part.

Proof(1)α1/=0.A scaling change allows us to suppose that α1=1.Change the basis

first we put d2n?1,2to zero,then d2n?3,2etc up to d52.

turns d21,d2n+1,1,d2n+1,2to zero.

If β=1,a further change of basis

allow us to put d2n+1,1,d2n+1,2to zero.But d21cannot be removed,the only possibility is to consider scaling transformations.It is clear that we can scale nonzero d21to 1.

If β=2-n,a further change of basis

turns d2,1,d2n+1,2to zero.But d2n+1,1cannot be removed,the only possibility is to consider scaling transformations.It is clear that we can scale nonzero d2n+1,1to 1.

If β=3-2n,a further change of basis

turns d21,d2n+1,1to zero.But d2n+1,2cannot be removed,we can scale nonzero d2n+1,2to 1.

allow us to put d21,d2n+1,1,d2n+1,2to zero.Now the parameters d2s+1,2(2≤s≤n-1)and d2n,2cannot be removed,so that unless all vanish,the derivation D is not diagonal.We can scale nonzero d2n,2to 1.Summarizing the above discussion,we complete the proof of the theorem.

ProofBy taking linear combinations of D1,D2we obtain α1=1,β1=0 and α2= 0,β2=1.By the prove of Theorem 3.2,we can take D1to its canonical form

Let D2be the form of(3.4).Computing the commutator for the above given forms of D1,D2we immediately find

Therefore there is a single canonical form of D1,D2

The transformation

takes ν1,ν2in equation(3.5)into ν1=ν2=0 while leaving all other Lie brackets invariant. We conclude that in this case the solvable Lie algebra with?Q2n+1is unique.

§4.Generalized Casimir Invariants

4.1 General Method

The term Casimir operator,or Casimir invariant,is usually reserved for elements of the center of the enveloping algebra of a Lie algebra g.These operators are in one-to-one correspondence with polynomial invariants characterizing orbits of the coadjoint representation of g. The search for invariants of the coadjoint representation is algorithmic and amounts to solving a system of linear first-order partial differential equations.Alternatively,global properties of the coadjoint representation can be used.In general,solutions are not necessarily polynomials and we shall call the nonpolynomial solutions generalized Casimir invariants.For certain classes of Lie algebras,including semisimple Lie algebras,perfect Lie algebras,nilpotent Lie algebras and more generally algebraic Lie algebras,all invariants of the coadjoint representation are functions of polynomial ones.

In equation(4.1)the quantities xaare commuting independent variables which can be identified with coordinates in the basis of the space g?dual to the basis{x1,···,xn}of the algebra g.

The invariants of the coadjoint representation,i.e.,the generalized Casimir invariants,are solutions of the following system of partial differential equations

The number of functionally independent solutions of system(4.2)is

where C is the antisymmetric matrix

With respect to the number of independent Casimir operators of g,formula(4.3)is merely an upper bound.Since the method of computation is generally known(see e.g.,[2-3]),we shall not present details and just give the results in the form of theorems.In all cases proofs consist of a direct calculation,i.e.,solving equations(4.2).

4.2 The Generalized Casimir Invariants

Obviously,r(C)=2n-2,by equation(4.3),we get nI=3.Clearly the generators of the center are Casimir operators of the algebra,that is,ξ1=X2n,ξ2=X2n+1.In order to determine another independent invariant,we have to solve the system(4.2)

where 2≤i≤2n-1.For any fixed 2≤i≤2n-1,the function X1X2n+1+(-1)iXi+1X2n+1?iis a solution of equation(4.7).

Therefore that ξ3is an invariant of the algebra.

Theorem 4.2The algebras s2n+2,1,s2n+2,2,s2n+2,3,s2n+2,4,s2n+2,5have two Casimir invariants each.Their forms are

ProofBy equation(4.3),we get nI=2.The Lie algebra s2n+2has two Casimir invariants, dependent only on ξ1,ξ2,ξ3.We have additional truncated differential operator?F,respectively.

It can be easily verified that

It can be easily verified that

We see that

Theorem 4.3The Lie algebra s2n+3of Theorem 3.3 has one Casimir invariant that can be chosen to be

ProofBy equation(4.3),we get nI=1.The Lie algebra s2n+3has one Casimir invariant, again dependent only on ξ1,ξ2,ξ3.We have two additional truncated differential operators,namely

we have

Hence,we get

AcknowledgementThe authors are grateful to the referee for his or her valuable comments and suggestions.

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tion:17B05,17B30,22E70,81R05

:A

1002–0462(2017)01–0099–12

date:2015-12-29

Supported by the National Natural Science Foundation of China(11071187);Supported by the Basic and Advanced Technology Research Project of Henan Province(142300410449);Supported by the Natural Science Foundation of Education Department of Henan Province(16A110035)

Biographies:LI Xiao-chao(1981-),male,native of Zhumadian,Henan,an associate professor of Huanghuai University,Ph.D.,engages in Lie algebra;JIN Quan-qin(1965-),male,native of Raoyang,Hebei,a professor of Tongji University,Ph.D.,engages in Lie algebra.

CLC number:O152.5