Structure of the Enveloping Algebras

The adjoint representations of several small dimensional Lie algebras  on their universal enveloping algebras  are explicitly decomposed. It is shown that commutants of raising operators are generated as polynomials in several basic elements. The explicit form of these elements is given and the general method for obtaining  these elements is described.


Introduction
Let g be a finite dimensional complex semisimple Lie algebra and U g U ( ) = its corresponding enveloping algebra.If we want to study all two-sided ideals of the associative algebra U (see [1] chapters 6-10), i.e. to find all the vector spaces I U Ì which satisfy for all a b U aIb I , : Î Î we can easily see that such a two-sided ideal is invariant with respect to adjoint action of U: if we take adjoint action r: , ,K for the simplest cases of Lie algebras and to show that even in more complicated examples the decomposition does not look too weird and is relatively easyli obtained by a generally described procedure.

The simplest example -algebra sl(2)
Let us have algebra sl (2) which is a complex linear span of three vectors where by E ij we denote the matrix having 1 in position (i, j) and 0s elsewhere.If we compute the commutation relations of these elements we obtain The enveloping algebra U U sl = ( ( )) 2 fulfils the Poincare--Birkhoff-Witt theorem and consists of all complex linear combinations of monomials It has a natural filtration U n , n ³ 0 given by the degree n of the elements in U n .It is easily seen that the adjoint action has U n as its invariant subspace (by applying a commutator we cannot obtain an element of higher degree).Therefore it is completely reducible on each U n , i.e. we can see U n as a direct sum of invariant subspaces generated by certain highest weight vectors.
The highest weight vector v of weight m satisfies the relations From this we can directly find that all highest weight vectors of small degree: Here is a Casimir element from the center of U. By commuting these vectors we see that , K From this we can claim that for any n ³ 0 The proof of this general formula is based on a dimensional check: First we see that the sum r r ( )

= +lower terms
where "lower terms" contain monomials with H k -and lower.
Next it is well known from representation theory that the dimension of the irreducible representation with highest weight m is 2 1 m + .
On the other hand the dimension of U n is also easy to determine, because it is well known that it is isomorphic (as a vector space) to the direct sum Å = k n k S g 0 ( ) of the k-th symmetric powers of Lie algebra g.We can construct the symmetric algebra S(g) from the tensor algebra T(g) by taking the quotient algebra of T(g) by the ideal generated by all differences of products v w w v Ä -Ä for v w g , Î ; then there is a direct sum decomposition of S(g) as a graded algebra into the summands S k (g) which consist of the linear span of the monomials in vectors of g of degree k.The symmetric algebra S(g) is in effect the same as the polynomial ring in the indeterminates that are basis vectors for g.
Therefore we see that and from this we have We have

Moving to the commutative case
There is an alternative approach to obtain this decomposition -move to the commutative case: It is a well-known fact (see [2]) that the mapping This system can be easily solved, and the general solution is .We see that when F x x k ( ) = , the solution is polynomial and we can transfer back to the enveloping algebra using the canonical isomorphism S g U g ( ) ( ) (we sum over all permutations of the indices 1, …, k).In our example Doing the dimension check as before the we see that decomposition is complete.

The algebras sl(3) and gl(3)
The case of algebra sl( 3) is more complicated.The decomposition of this case is done in [3].Let us show where the complications in the decomposition arise, using the similar case of Lie algebra gl(3) with the basis Eij, i j , , , -1 2 3 and commutation relations We turn directly into the commutative case.First we set up, as before, the system of differential equations for the highest weight vectors of weight (N 1 , N 2 , N 3 ): From this immediately follows N N N then any solution f can be written as where the new unknown function F satisfies the following system of differential equations: 13 13 21 21 31 31 Solving this system, we obtain that the general solution of the system has the form: (we omit the summation of repeated indices going over 1, 2, 3).
We now encounter the first problem which arises in the process of decomposition.Although we know the general solution of this system of differential equations, an important question is how to obtain enough polynomial solutions in variables x ij from this general solution.This can be generally difficult.For example, the expression ( ) although rational in variables x j , c j is polynomial in variables The second question which arises is, in some sense, the opposite kind of problem.We can generate too many polynomial solutions and their polynomial combinations will no longer be linear independent (or, equivalently, the polynomial solutions will be functionally dependent).For example, in our case of algebra gl(3) we have This dependency signals that it is forbidden to have product x 3 x 4 in the decomposition.
Luckily, both above-mentioned problems are easily found out by a dimension check.If there are not enough solutions we will not have enough linear independent vectors to construct the decomposition, so that the number of linear independent vectors we generate is less than the dimension of the corresponding filtration part U n .In the opposite case, if the number is greater, it clearly indicates that there is some linear dependency between the considered vectors.
we can construct vectors and compute their weights.The weight of the product of any elements is obtained by the sum of the weights of the corresponding elements; we have element weight The degree of the general element is Now we can perform a dimension check to see if our hypothesis is correct: from representation theory, the dimension of the representation with highest weight (N 1 , N 2 , N 3 ) is hence the dimension of the representation generated by our general element is ~( , , , It follows that we must prove that ~( , , , ) , , , , , , 1 2 3 1 2 3 x x The second term is obviously equal to the first term.For the third sum we have and sum each of the summands separately.For the first one we have ~( , , , ) , , , , and the dimension check succeeds.

Lie algebra gl(4)
Finally let us demonstrate how the procedure goes in the case of Lie algebra gl (4).For the vectors with highest weights (N 1 , N 2 , N 3 , N 4 ) we obtain the following equations

Now let us denote
The highest weights of the corresponding elements of the enveloping algebra are element weight The dimension of the representation with highest weight Now we start as in the case of gl(3) to count the dimensions of the filtration subspaces to see the conditions which arise as the degree of the elements grows.
Elements of degree 1 are element dimension Elements of degree 2 are element dimension Y 1 20 total = 136 … etc. Analyzing degree 6, we encounter the first linear dependency which signals some polynomial relations (syzygies) between the generators.Namely we have Higher degrees give us many other relations of this type which lead to the system of conditions that must be fulfilled to keep the set of highest weight vectors linear independent.Going up to degree 12 finishes the analysis, and we no longer encounter new relations.Thus we are ready to formulate the hypothesis and after a successful dimension check we can state the final result, which is contained in the following theorem: form a desired decomposition of the enveloping algebra U(gl(4)).

Conclusion
We have studied the structure of enveloping algebras U(g) where g sl gl gl = ( ), ( ), ( ) 2 3 4 and we have decomposed the adjoint representation into its irreducible components.We have found explicitly the highest weight vector in any such component.Our result can be useful for a further study of the tensor products of representations and ideals of enveloping

Y 0 Y
It is well known that such adjoint action r is completely reducible for g semisimple: it is possible to cut the enveloping algebra U into pieces U k is an invariant subspace (corresponding to an irreducible representation (component) of r) generated by its highest weight vector Y k : .we know a large set of vectors which must be contained also in the ideal I.This can help considerably in the classification of all ideals of U.The main aim of this paper is to give an explicit decompo-, i.e. to give a list of highest weight vectors Y k , k = 0 1 Now it is sufficent to prove that the dimensions of two spaces on both sides (1) are equal, i.e. that This is easily done with the help of the sum of the geometric series

1 = 2 = ( ) , c e kk 3 3 =
these elements we define the following solutions of the system (summation of repeated indices going over 1… 4): c e kk 1 ( ) , c e kk 2 acting on the space of polynomials in three variables e 12 , e 21 , h 1 , we can find all highest weight vectors by solving this system of differential equations: x Acta Polytechnica Vol.47 No.2-3/2007Summing up these facts, if we now anti-symmetrize x j , c j into elements X j , C j of the enveloping algebra U, i.e. f x x