Polynomial Realisations of Lie (Super)Algebras and Bessel Operators

Abstract

We study realisations of Lie (super)algebras in Weyl (super)algebras and connections with minimal representations. The main result is the construction of small realisations of Lie superalgebras, which we apply for two distinct purposes. Firstly it naturally introduces, and generalises, the Bessel operators for Jordan algebras in the study of minimal representations of simple Lie groups. Secondly, we work out the theoretical realisation concretely for the exceptional Lie superalgebra D(2, 1; α), giving a useful hands-on realisation.

A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra osp(m,2|2n)

“…The Bessel operators map R 2 into R 2 if and only if λ = 2 − M , where M = m − 2n is the superdimension of R m|2n and R 2 is the ideal in P R m|2n generated by R 2 .Therefore we will only use the Bessel operator with the parameter 2 − M in this paper and we set B(x i ) := B 2−M (x i ). We obtain the following two properties of the Bessel operator from Proposition 4.2 in[4].…”

The Joseph Ideal for $$\mathfrak {sl}(m|n)$$

“…The procedure in [BC,Section 3] then gives realisations of g as (complex) polynomial differential operators on a real flat supermanifold with same dimensions as g -, so on R m−1|n . We choose coordinates x i with corresponding partial differential operators ∂ i , for 2 ≤ i ≤ m + n, both are even for i ≤ m and odd otherwise.…”

“…As g 0 ∼ = gl(m − 1|n), the space of characters g 0 → C is in bijection with C. If we apply the construction in [BC,Section 3] to the character corresponding to µ ∈ C, we find a realisation π µ satisfying (8) π µ (X ε j −ε 1 ) = x j and π µ (X ε 1 −ε j ) = (µ − E)∂ j for 2 ≤ j ≤ m + n, with E = ∑ m+n i=2 x i ∂ i . The other expressions for π µ follow from the above and the fact that, since π µ is a realisation, we have for all X ,Y in g π µ (X )π µ (Y ) − (−1) |X||Y | π µ (Y )π µ (X ) = π µ ([X ,Y ]).…”

Lie Theory and Its Applications in Physics

“…As g 0 ∼ = gl(m − 1|n), the space of characters g 0 → C is in bijection with C. If we apply the construction in [BC,Section 3] to the character corresponding to µ ∈ C, we find a realisation π µ satisfying (8) π µ (X ε j −ε 1 ) = x j and π µ (X ε 1 −ε j ) = (µ − E)∂ j for 2 ≤ j ≤ m + n, with E = ∑ m+n i=2 x i ∂ i . The other expressions for π µ follow from the above and the fact that, since π µ is a realisation, we have for all X,Y in g Now we interpret π µ as a representation of g on the space of polynomials, i.e.…”