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Ideals of the enveloping algebra U(osp(1, 2)) 1 | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Ideals of the enveloping algebra U(osp(1, 2)) 1

C. BURD´IK 1, M. HAVL´I ˇCEK 1, O. NAVR´ ATIL 2*, and S. POˇSTA 1

1Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Trojanova 13, 120 00 Prague 2, Czech Republic E-mail: [email protected]

2Department of Mathematics, Faculty of Transportation Sciences, Czech Technical University, Na Florenci 25, 110 00 Prague, Czech Republic

*Corresponding Author:
O. NAVR´ ATIL
Department of Mathematics, Faculty of Transportation Sciences
Czech Technical University, Na Florenci 25
110 00 Prague, Czech Republic
E-mail: [email protected]

Received date: December 12, 2007; Revised date: March 08, 2008

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Abstract

We explore the general form of two sided ideals of the enveloping algebra of the Lie superalgebra osp(1, 2). We begin by disclosing the internal structure of U(osp(1, 2)) computing the decomposition of adjoint representation. The classification of the ideals we reach is done via presenting generators for the each ideal and by showing that each ideal is generated uniquely.

Introduction

Until today there exists no example of enveloping algebra with complete classification of all twosided ideals with the exception of U(sl2) [1]. In this article we present this complete classification for the enveloping algebra of Lie superalgebra osp(1, 2). This Lie superalgebra contains Lie algebra sl2 as subalgebra, but the classification of all both-sided ideals of U(osp(1, 2)) is richer and is not simply related to the classification of ideals of U(sl2). The classification of all primitive ideals was done in the paper [2] and is contained in our more general result.

Structure of U(osp(1, 2))

The five dimensional Lie superalgebra g = osp(1, 2) has enveloping algebra U = U(g) which is complex associative algebra generated by elements E±, H and F± satisfying the following commutation relations:

image

Here [x, y] = xyyx denotes the commutator and {x, y} = xy + yx the anti-commutator. It is well known that the Poincar´e-Birkhoff-Witt theorem holds in this algebra and the basis of U can be taken as ordered monomials

image

The adjoint representation of g, i.e the mapping ad :image defined for imageimage

image(2.1)

where the degree is defined by the formulas

image

acts as a (super)derivation. It means that if we want to define adjoint representation on the whole space U we can take imageand proceed by linearity.

Now we compute the decomposition of U similarly as in the case of U(sl2). The enveloping algebra U possess a natural filtration imagegiven by degree n of elements in Un. It is easily seen that adjoint representation has Un as its invariant subspace (by applying supercommutator we can not obtain element of higher degree). It is completely reducible on each Un [3] i. e. we can see Un as a direct sum of invariant subspaces generated by certain highest weight vectors. The highest weight vector v of weight m satisfies relations

image

From these relations we can directly find that all highest weight vectors of small degree. Let us denoteimage is any subset; for set containing only one element we shorten image. Commuting highest weight vectors of small degree we see that

image

where

image

is Casimir element which generates the center of the algebra [3]; element

image

has the weight 1/2 . Note that the element imagehas the weight image and dimension of the space generated by the highest weight element with the weight m is 4m + 1. Because the elements F+ and F satisfy the relations

image(2.2)

we deduce that the highest weight element A can’t be presented in the decomposition in the power greater than one. Thus we can claim that for any n ≥ 0

image(2.3)

The proof of this claim is based on dimensional check. It’s not difficult to see that the sum imageimagewhere v1, v2 are highest weight vectors, is direct if and only if v1, v2 are linear independent. The vectors image are linear independent for differentimage The representation generated by the highest weight vectorimagehas the dimensionimage

On the other hand the dimension of Un is also easy to determine. The dimension of vector space of homogeneous polynomials of degree d in k variables isimageDue to the relation (2.2) we must consider only monomials which have zero or one factor equal to F±. For the dimension of Un we have the following recurrence relation:

image

(if we want to construct monomial of degree n we take into account monomials from three elements E±, H of degree n−2 to which we append FF+, monomials of degree n−1 to which we append F orF+ and finally monomials of degree n). Simplifying we get

image

For the dimension of the space on the right hand side of (2.3) we get the same result:

image

The following decomposition of adjoint action therefore takes place:

image(2.4)

Important relations in U(osp(1, 2))

By direct calculation, we see that if I is any both-sided ideal of U, the following important implications hold in U:

image(3.1)

image(3.2)

image(3.3)

Using these relations, the structure of ideals generated by the highest weight elements E+n and E+n A can be obtained. Let image and denote

image

Then

image

where (x) means ideal generated by element x.

Let’s now have any both-sided idealimageBecause U is Noetherian ring, I is finitely generated, so we can write

imagee(3.4)

for someimageThanks to decomposition (2.4) we can replace xi’s by certain highest weight vectors. There exist numbers image and complex polynomialsimage

image(3.5)

Now we reduce the number of generators used in (3.5) by successive replacing the generators by more suitable ones.

First, if there are two generators having the same n’s and same image say image andimagewe can replace them by one generator image

Further simplification is possible due to the relations (3.1) and (3.2). Let there be, say, two generators, imagewhere n1 < n2. First, we may assume without loss of generality that P2,P1. Next, we can replace the two generators by the suitable couple P(C)Q(C)E+n1 and P(C)E+n2, where P(C) = gcd{P1(C), P2(C)} and Q(C)|fn2,n1 . (The similar simplification applies to the generators having image= 1.)

And thirdly, assume there are two generators in the list of the form Q(C)E+n1 and E+n2 , n1 < n2, Q(C)|fn2,n1 . Then it is possible to replace them by new couple Q(C)fn1,0 and E+n2 . (Again, the similar simplification applies to the generators havingimage = 1.)

After finite number of steps, we are able to get the following: Every ideal I can be written of the form

IMAGE(3.6)

where n,m ¸ 0 and P1(C), Q1(C), P2(C) and Q2(C) are four polynomials such that

image

The form (3.6) can still be simplified using the relation (3.3). Finally we can reach the form

image(3.7)

where image is some polynomial and Q is such thatimage(3.8)

where image

image

Using (3.8) we can state that the form (3.7) is unique, i. e. for different numbers image and polynomials P, Q we get different ideals.

Conclusion

We have shown that the most general form of every two sided ideal of the enveloping algebra U(osp(1, 2)) is given by formula (3.7). We have found that this form is unique for each ideal and thanks to this uniqueness we have obtained complete classification of both sided ideals of U. It would be nice to explore the origin of the fact that the surprisingly richer structure arises when we compare to the case of U(sl2).

Acknowledgement

This work was partially supported by GACR 201/05/0857.

References

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