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Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium

- *Corresponding Author:
- Stijn LIEVENS
[email protected], [email protected], [email protected]

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**Received date:** December 14, 2007; **Revised date:** March 08, 2008

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

Using the equivalence of the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) to the defining triple relations of n pairs of parabose operators b± i we construct a class of unitary irreducible (infinite-dimensional) lowest weight representations V (p) of osp(1|2n). We introduce an orthogonal basis of V (p) in terms of Gelfand-Zetlin patterns, where the subalgebra u(n) of osp(1|2n) plays a crucial role and we present explicit actions of the osp(1|2n) generators.

Following some physical ideas we construct a class of infinite dimensional unitary irreducible representations of the Lie superalgebra [1] in an explicit form. In 1953 Green [2] introduced the so called parabose operators

satisfying

(1)

(to be interpreted as +1 and −1 in the algebraic expressions as a generalization of the ordinary Bose operators. The Fock space V (p) of n pairs of PBOs is a Hilbert space with a vacuum |0>, defined by means of (j, k = 1, 2, . . . , n)

(2)

and by irreducibility under the action of the algebra spanned by the elements 1, . . . , n), subject to (1). The parameter p is referred to as the order of the paraboson system. However the structure of the parabose Fock space is not known, also a proper basis has not been introduced. We solve these problems using the relation between n pairs of PBOs and the defining relations of the LS osp(1|2n), discovered by Ganchev and Palev [3]. The orthosymplectic superalgebra osp(1|2n) [1] consists of matrices of the form

(3)

where a and a1 are (1 × n)-matrices, b is any (n × n)-matrix, and c and d are symmetric
(n × n)-matrices. The even elements have a = a1 = 0 and the odd elements are those with
b = c = d = 0. Denote the row and column indices running from 0 to 2n and by e_{ij} the matrix
with zeros everywhere except a 1 on position (i, j). Then as a basis in the Cartan subalgebra h
of osp(1|2n) consider

(4)

In terms of the dual basis ±j of h*, the root vectors and corresponding roots of osp(1|2n) are given by:

Introduce the following multiples of the odd root vectors

(5)

Then the following holds [3]:

**Theorem 1** (Ganchev and Palev). As a Lie superalgebra defined by generators and relations, osp(1|2n) is generated by 2n odd elements subject to the parabose relations (1).

From (4) and (5) it follows that and using (2) we have:

**Corollary 2. **The parabose Fock space V (p) is the unitary irreducible representation of osp(1|2n)
with lowest weight

We can construct the representation V (p) [4] using an induced module construction with an appropriate chain of subalgebras.

**Proposition 3.** A basis for the even subalgebra sp(2n) of osp(1|2n) is given by the elements The n^{2} elements 1, . . . , n are a basis for the sp(2n) subalgebra u(n).

The subalgebra u(n) can be extended to a parabolic subalgebra j, k = 1, . . . , n} [4] of osp(1|2n). Recall that with Then the space spanned by |0i is a trivial one-dimensional u(n) module of weight Since the module can be extended to a one-dimensional module. Now we define the induced osp(1|2n) module with lowest weight By the Poincar´e-Birkhoff-Witt theorem [1,4], it is easy to give a basis for .

where However in general is not a simple module and let M(p) be the maximal nontrivial submodule of . Then the simple module (irreducible module), corresponding to the paraboson Fock space, is The purpose is now to determine the vectors belonging to M(p) and also to find explicit matrix elements of the osp(1|2n) generators in an appropriate basis of V (p).

From the basis in it is easy to write down the character of :

(6) .

Such expressions have an interesting expansion in terms of Schur functions.

**Proposition 4** (Cauchy, Littlewood). Let *x _{1}, . . . , x_{n}* be a set of n variables. Then [5]

(7)

Herein the sum is over all partitions ¸ and sλ(x) is the Schur symmetric function [6].

The characters of finite dimensional u(n) representations are given by such Schur functions s¸(x). For such finite dimensional u(n) representations labelled by a partition ¸, there is a known basis: the Gelfand-Zetlin basis (GZ) [7]. We shall use the u(n) GZ basis vectors as our new basis for . Thus the new basis of consists of vectors of the form

(8)

where the top line of the pattern, also denoted by the n-tuple [m]^{n} , is any partition ¸ (consisting
of non increasing nonnegative numbers) with The label p itself is dropped in the
notation of |m). The remaining n−1 lines of the pattern will sometimes be denoted by |m)^{n−1}. So all m_{ij} in the above GZ-pattern are nonnegative integers, satisfying the betweenness conditions Note that, since the weight of |0> is , the
weight of the above vector is determined by

(9)

The triple relations (1), imply that is a standard u(n) tensor of rank (1, 0, . . . , 0). Therefore we can attach a unique GZ-pattern with top line 10 · · · 0 to every corresponding to the weight . Explicitly:

(10)

where the pattern consists of j − 1 zero rows at the bottom, and the first n − j + 1 rows are of the form 10 · · · 0. The tensor product rule in u(n) reads where and a subscript ±k indicates an increment of the kth label by ±1: A general matrix element of can now be written as follows:

The first factor in the right hand side is a u(n) Clebsch-Gordan coefficient [8], the second factor is a reduced matrix element. By the tensor product rule, the first line of has to be for some k-value.

The special u(n) CGCs appearing here are well known, and have fairly simple expressions. They can be found, e.g. in [8,9]. The actual problem is now converted into finding expressions for the reduced matrix elements, i.e. for the functions for arbitrary n-tuples of non increasing nonnegative integers

(11)

So one can write:

(12)

(13)

For j = n, the CGCs in (12)-(13) take a simple form [8], and one has

(14)

(15)

In order to determine the n unknown functions F_{k}, one can start from the following action:

(16)

Expressing the left hand side by means of (14)–(15), one finds a system of coupled recurrence
relations for the functions F_{k}. Taking the appropriate boundary conditions into account, we
have been able to solve this system of relations [9].

**Proposition 5.** *The reduced matrix elements F _{k} appearing in the actions of on vectors |m)
of V (p) are given by:*

(17)

*where ε and O* are the even and odd functions defined by *ε*_{j} = 1 if j is even and 0 otherwise,
O_{j} = 1 if j is odd and 0 otherwise.

The proof consists of verifying that all triple relations (1) hold when acting on any vector
|m). Each such verification leads to an algebraic identity in n variables m_{1n, . . . },m_{nn}. In these
computations, there are some intermediate verifications: e.g the action should leave
the top row of the GZ-pattern |m) invariant (since belongs to u(n)). Furthermore, it
must yield (up to a factor 2) the known action of the standard u(n) matrix elements *E _{jk}* in the
classical GZ-basis. Consider now the factor in the expression
of This is the only factor in the right hand side of (17) that may become zero. If this
factor is zero or negative, the assigned vector belongs to

**Theorem 6. **The osp(1|2n) representation V (p) with lowest weight is a unirrep if
and only if

The structure of V (p) is determined by

where `(¸) is the length of the partition λ.

The explicit action of the osp(1|2n) generators in V (p) is given by (12)-(13), and the basis is orthogonal and normalized. For p 2 {1, 2, . . . , n − 1} this action remains valid, provided one keeps in mind that all vectors with must vanish.

Note that the first line of Theorem 6 can also be deduced from [10,11], where all unirreps of osp(1|2n) are classified.

N. I. Stoilova was supported by a project from the Fund for Scientific Research – Flanders (Belgium) and by project P6/02 of the Interuniversity Attraction Poles Programme (Belgian State – Belgian Science Policy). S. Lievens was also supported by project P6/02.

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