Medical, Pharma, Engineering, Science, Technology and Business

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

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

^{2}Department 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

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

Until today there exists no example of enveloping algebra with complete classification of all twosided
ideals with the exception of *U*(sl_{2}) [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 *sl _{2}* as subalgebra, but the classification of all both-sided ideals of

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:

Here [*x, y*] = *xy* − *yx* 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

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

(2.1)

where the degree is defined by the formulas

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

Now we compute the decomposition of *U* similarly as in the case of *U(sl _{2})*. The enveloping
algebra

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

where

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

has the weight 1/2 . Note that the element has the weight and dimension of
the space generated by the highest weight element with the weight *m* is 4*m + 1*. Because the
elements *F ^{+} and F^{−}* satisfy the relations

(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

(2.3)

The proof of this claim is based on dimensional check. It’s not difficult to see that the sum where *v _{1}, v_{2}* are highest weight vectors, is direct if and only if

On the other hand the dimension of *U _{n}* is also easy to determine. The dimension of vector
space of homogeneous polynomials of degree

(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* F*^{−}*F*^{+}, monomials of degree *n−1* to which
we append *F*^{−} or^{}*F*^{+} and finally monomials of degree n). Simplifying we get

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

The following decomposition of adjoint action therefore takes place:

(2.4)

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

(3.1)

(3.2)

(3.3)

Using these relations, the structure of ideals generated by the highest weight elements* E ^{+n }and
E^{+n}*

Then

where (*x*) means ideal generated by element *x*.

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

(3.4)

for someThanks to decomposition (2.4) we can replace *x _{i}*’s by certain highest weight
vectors. There exist numbers and complex polynomials

(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 say andwe can replace them by one generator

Further simplification is possible due to the relations (3.1) and (3.2). Let there be, say, two generators, where *n1 < n2*. First, we may assume without loss of generality that *P _{2},P_{1}*. Next, we can replace the two generators by the suitable couple

And thirdly, assume there are two generators in the list of the form *Q(C)E ^{+n1}* and

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

(3.6)

where *n,m ¸ 0* and *P _{1}(C), Q_{1}(C), P_{2}(C)* and

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

(3.7)

where is some polynomial and *Q* is such that(3.8)

where

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

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(sl _{2})*.

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

- CatoiuS (1998) Ideals of the Enveloping Algebra U (sl2). J Algebra 202: 142-177.
- Pinczon G (1990) The Enveloping Algebra of the Lie Superalgebraosp(1,2). J Algebra 132: 219-242.
- ScheunertM (1979) The Theory of Lie Superalgebras.Springer-Verlag, New York.
- DixmierJ (1974) Algebras Enveloppantes.Gauthier-Villars Editeur, Paris.
- Konstant B (1963) Lie group representations on polynomial rings. Am. J Math 85: 327-404.
- KirillovA (1972) Elements of Representation Theory.Nauka, Moscow.
- FlathD (1990) Decomposition of the enveloping algebra of sl3. J Math Phys31: 1076-1077.

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