Characteristic identities for Lie (super)algebras

We present an overview of characteristic identities for Lie algebras and superalgebras. We outline methods that employ these characteristic identities to deduce matrix elements of finite dimensional representations. To demonstrate the theory, we look at the examples of the general linear Lie algebras and Lie superalgebras.


Introduction
Characteristic identities have proven to be a useful tool for revealing important and explicit information about representations of Lie algebras and superalgebras. The aim of this article is to give an overview of how characteristic identities may be used for such purposes.
It was Dirac in 1936 [1] who first employed polynomial identities related to sl (2) in the context of relativistically invariant wave equations. Later, Lehrer-Ilamed [2] noted more generally that n 2 elements of the enveloping algebra U (g) of a Lie algebra g satisfy n 2 identities, which in some cases may be expressed as a single polynomial identity of degree n for an n × n matrix with entries from U (g). There were subsequent works, for example, by Louck [3], Mukunda [4] and Louck and Galbraith [5] in which polynomial identities were encountered for a variety of Lie algebras. In [6][7][8], Bracken and Green established a general theory of characteristic identities for classical Lie algebras, which was soon followed by the work of O'Brien, Cant and Carey [9] in which the authors instituted a suitable algebraic formalism. In a series of papers [10][11][12][13][14][15][16], Gould made use of the characteristic identities to deduce information such as invariants, Wigner coefficients and matrix elements pertaining to the irreducible representations of the classical Lie algebras. It is also worth noting that these polynomial identities also extend to finite groups [17] and quantum groups [18,19].
Following the work of Kac [20,21], Jarvis and Green [22,23] investigated characteristic identities and invariants related to the vector representation of the general linear, special linear and orthosymplectic Lie superalgebras. Other works soon followed that were related to other Lie superalgebras (e.g. [24]) and more general representations (e.g. [25]). It was not until the more recent work of the current authors [26,27], however, in which the approach of Gould for Lie algebras was generalised to the case of Lie superalgebras, particularly for the general linear case.

Characteristic identities for Lie algebras
To demonstrate how the characteristic identities may be utilised, we exhibit the theory in the context of the Lie algebra gl(n).

Characteristic matrix
We denote the generators of gl(n) by a ij , i, j = 1, . . . , n which satisfy the defining relations We may assemble the generators into a square n × n matrix A with (i, j) entry a ij , i.e.
and define powers of A recursively by
To assist the reader in understanding the notation used throughout, consider the following straightforward examples that were originally presented in [7]. For gl(1), (trivial) direct calculation yields which is interpreted to mean A ij − σ 1 δ ij = 0 for each i, j. For gl(2), we may similarly calculate This gives two examples of polynomial identities satisfied by the matrix A with coefficients in the centre of the enveloping algebra of gl(1) and gl(2) respectively. It is worth noting that in these cases σ 2 = σ 2 1 , σ 3 = σ 3 1 for gl(1) and σ 3 = 3 2 σ 1 σ 2 − 1 2 σ 3 1 + σ 2 − 1 2 σ 2 1 for gl (2), demonstrating that the Casimir invariants so defined are not generally independent.

The characteristic identity
The characteristic identity of A for gl(n) on a finite dimensional irreducible representation V (λ) is constructed in the following way.
In the previous section we saw examples of how the matrix A satisfies polynomial identities with coefficients expressible in terms of the Casimir invariants. If we now look at such polynomial identities on a finite dimensional irreducible representation V (λ), with highest weight highest weight λ = (λ 1 , λ 2 , . . . , λ n ), we then have from Schur's Lemma that the Casimir invariants take on constant values on V (λ). For example, for gl(n), Bracken and Green [6,7] showed that the identity n j=1 (A − α j ) = 0 holds on a finite dimensional irreducible representation of gl(n), where α j = λ j + n − r are referred to as the characteristic roots. We highlight the fact that the entries of A are now representation matrices.

The adjoint matrix
The matrix A is not the only such matrix one may define. Consider taking the negative transpose of A as One may also show that holds on a finite dimensional irreducible representation of gl(n), where the characteristic roots are in this case given by

General characteristic matrix
The construction of these characteristic matrices at this stage may seem ad hoc, but there is in fact a general formalism which relies on the quadratic Casimir element [9,16]. Such a matrix can then be constructed for an arbitrary semisimple Lie algebra. Let ∆ be the co-product of the enveloping algebra U (g) of a semisimple Lie algebra g, and let π µ denote any irreducible representation of g corresponding to module V (µ). We may define the matrix with algebraic entries Considering its action on an arbitrary finite dimensional irreducible representation π λ gives In the case V (µ) is the vector representation of g = gl(n) (i.e. highest weight µ = (1, 0, . . . , 0)) we obtain A, and for the case we take the dual vector representation of g = gl(n), we obtain A.

Projection operators
We may use the characteristic identities to define projection operators, which, as we shall see, turn out to be a crucial ingredient for determining matrix elements. We modify our notation slightly, so that A n is understood to be the characteristic matrix associated with gl(n). Explicitly, we have from the characteristic identity If we set V to be the vector representation of gl(n), we have the decomposition is a projection. Here, ∆ k is simply an n-tuple with a 1 in the kth entry and zeroes elsewhere. By contrast, denoting by V * the dual vector representation of gl(n), we have the decomposition is a projection. These projections will be used in what follows.

Matrix elements of irreducible representations
We may utilise the characteristic identity to determine the matrix elements of finite dimensional irreducible representations. Here we demonstrate the procedure, at the same time highlighting the power of this approach. Of particular note is the fact that we may deduce, up to a phase factor, not only the matrix elements of the elementary generators of gl(n), i.e. those of the form a i i+1 , but also the nonelementary generators.

Vector operators
Before presenting the details of the matrix element formulae, we first need to describe vector operators. Define a gl(n) vector operator as a collection of n operators ψ i satisfying [a ij , ψ k ] = δ jk ψ i .
Related to the dual vector representation, we also define a gl(n) contragredient vector operator as a collection of n operators φ j satisfying From the work of Green [7], we have where ψ n r j = ψ i P n r ij = P n r ji ψ i are linearly independent and change that value of the highest weight label λ r,n by one, leaving the other λ k,n unchanged, i.e.
Also, in the case of the contragredient vector operator, we have

Construction of invariants
Following the work of Gould [13], we are now in a position to construct eigenvalues of invariants C k,n+1 (defined below). Consider the gl(n + 1) identity expressed in the form Looking at the (j, n + 1) entry, j = 1, 2, . . . , n gives is a gl(n) invariant. After some manipulation, we arrive at It is known [8,10] that for any polynomial p(x), since the ψ n r j are linearly independent. Also, n+1 k=1 C k,n+1 = 1, since the C k,n+1 are defined in terms of projections. This gives a set of linear equations that can be solved for C k,n+1 , giving Similarly, for we have

Generalisation to Lie superalgebras
For the Lie superalgebra gl(m|n), we now have generators a pq , p, q = 1, 2, . . . , m + n satisfying Analogous to the Lie algebra case, we define the characteristic matrices by setting On the irreducible representation V (Λ), with Λ = (Λ 1 , Λ 2 , . . . , Λ m |Λ m+1 , . . . , Λ m+n ), it is also possible to show [25] that the following characteristic identities hold with associated characteristic roots By analogy with the methods outlined for gl(n), we may construct similar objects for gl(m|n) as listed below.
There is, however, one major complication in the context of Lie superalgebras. The representation theory is not as well-understood as that of Lie algebras, and in particular the branching rules for an arbitrary irreducible representation are not known. In fact, in general it is not even known if an irreducible representation is completely reducible, i.e. there is no analogue of Weyl's Theorem in general. To avoid this problem, we look at a special subset of the irreducible representations first considered by Scheunert, Nahm and Rittenberg [29].
For the case ǫ = 0, V (Λ) is said to be a type 1 star irrep. Otherwise, for ǫ = 1, V (Λ) is called a type 2 star irrep. We have complete reducibility in these cases, and so determine branching rules, which is the one complication in the above procedure.

Classification of unitary (star) irreducible representations
The star representations, which we now refer to as "unitary" due to the existence of the positivedefinite form (i.e. an inner product), were classified by Gould and Zhang [30,31]. Here we only focus on the type 1 case.
There are similar conditions for type 2 star irreps which may be found in [30], but we will not include the details here.

Form of the matrix elements
The main outcome of our research so far is that for type 1 star irreps., generators of gl(m|n + 1) have action of the form a ℓ p+1 |Λ q,s = u N [u p , u p−1 , . . . u ℓ+1 , u ℓ ] λ q,s + ∆ up,p + · · · + ∆ u ℓ ,ℓ , ℓ < p + 1, and similarly for a p+1 ℓ . The precise form of the coefficients N [u p , u p−1 , . . . u ℓ+1 , u ℓ ] (i.e. matrix elements) will not be given here, but are similar in form to the matrix elements of equation (2) above which are related to the Lie algebra gl(n). In fact, for the cases under consideration, we make the point that the procedure closely follows that already presented for gl(n). For the fully explicit details of matrix element formulae and other features such as branching rules related to the type 1 unitary irreps of gl(m|n), see [27]. Based on the convenient form of the highest weight given by equation (3), we see that our results agree with those presented in the work of Stoilova and van der Jeugt [32], Molev [33] and also the essentially typical representations considered by Palev [34,35].

Future work
Currently work is in preparation for providing the explicit details of the type 2 unitary irreps. It is also of interest to consider how to treat the cases of representations that do not admit an invariant inner product, and hence are not guaranteed to be completely reducible. One possibility is to investigate mixed tensor representations. That is, those representations that occur within the tensor product of a type 1 unitary irrep. with a type 2 unitary irrep.
Having made some progress for the classical Lie superalgebras, it is also of interest to determine the analogous matrix element formulae for the quantum group counterparts.
Finally, our procedure may be considered beyond the vector representation (a type 1 unitary irrep.) and the dual vector representation (a type 2 unitary irrep.), leading to a more general pattern calculus. This remains the topic of future work.