The Weyl realizations of Lie algebras, and left–right duality

Abstract

Abstract. We investigate dual realizations of non-commutative spaces of Lie algebra type in terms of formal power series in the Weyl algebra. To each realization of a Lie algebra g we associate a star-product on the symmetric algebra S(g) and an ordering on the enveloping algebra U (g). Dual realizations of g are defined in terms of left-right duality of the star-products on S(g). It is shown that the dual realizations are related to an extension problem for g by shift operators whose action on U (g) describes left and right shift of the generators of U (g) in a given monomial. Using properties of the extended algebra, in the Weyl symmetric ordering we derive closed form expressions for the dual realizations of g in terms of two generating functions for the Bernoulli numbers.The theory is illustrated by considering the κ-deformed space.

Nonassociative Snyder ϕ4 quantum field theory

“…K µ = k µ . Such a realization is called Weyl realization in the literature, see [45] and references therein. Furthermore, for Hermitian realizations of the Snyder-type spaces we have…”

Generalization of Weyl realization to a class of Lie superalgebras

“…[29] which states that the Weyl symmetric realization of a Lie algebra g 0 can be generalized to Lie superalgebras g = g 0 ⊕ g 1 satisfying [g 1 , g 1 ] = {0} where [· , ·] is the supercommutator in g. For details on the Weyl realization of g 0 see Refs. [30] and [31]. The evidence for this conjecture was found indirectly by using an automorphism of the Weyl superalgebra and comparing the power series expansion of two different realizations of g to second order in the structure constants of g. Here we give an elegant proof of this conjecture using the functional equation satisfied by the generating function for the Bernoulli numbers.…”

“…In Ref. [31] it was shown that the realization (20) can also be found by extending the original Lie algebra g 0 by n 2 generators T µν satisfying the Lie bracket [T µν , T αβ ] = 0 and [T µν , X λ ] = n α=1 C µλα T αν in order to derive the function f (t). Using an action of the Weyl algebra A n on the space of polynomials K[x 1 , .…”

“…, x n ], one can associate an ordering prescription on U (g 0 ) to a given realization (3) (see Ref. [31]). The realization (20) corresponds to the Weyl symmetric ordering on U (g 0 ), hence we refer to Eq.…”

“…As a special case, this also gives a shorter proof of the Weyl symmetric realization of g 0 found in Refs. [30] and [31].…”