Non-Local Equivariant Star Product on the Minimal Nilpotent Orbit

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dc.creatorAstashkevich, Alexander
dc.creatorBrylinski, Ranee
dc.date2000-10-27
dc.date2001-03-20
dc.date.accessioned2026-07-07T04:38:16Z
dc.date.available2026-07-07T04:38:16Z
dc.descriptionWe construct a unique G-equivariant graded star product on the algebra $S(g)/I$ of polynomial functions on the minimal nilpotent coadjoint orbit $\Omin$ of G where G is a complex simple Lie group and $g\neq\sl_2(C)$. This strengthens the result of Arnal, Benamor, and Cahen. Our main result is to compute, for G classical, the star product of a momentum function $μ_x$ with any function f. We find $μ_x\star f=μ_xf+\half\{μ_x,f\}t+Λ^x(f)t^2$. For $\g$ different from $sp_n(\C)$, $Λ^x$ is not a differential operator. Instead $\Lamda^x$ is the left quotient of an explicit order 4 algebraic differential operator $D^x$ by an order 2 invertible diagonalizable operator. Precisely, $Λ^x=-{1/4}\frac{1}{E'(E'+1)}D^x$ where $E'$ is a positive shift of the Euler vector field. Thus $μ_x\star f$ is not local in f. Using $\star$ we construct a positive definite hermitian inner product on $Sg/I$. The Hilbert space completion of $Sg/I$ is then a unitary representation of $G$. This quantizes $\Omin$ in the sense of geometric quantization and the orbit method.
dc.descriptionlatex file, 13 pages. In this new version we use the star product to construct a unitary representation attached to the orbit
dc.identifierhttps://arxiv.org/abs/math/0010257
dc.identifierhttp://arxiv.org/abs/math/0010257
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/60209
dc.subjectQuantum Algebra
dc.subjectRepresentation Theory
dc.subjectSymplectic Geometry
dc.titleNon-Local Equivariant Star Product on the Minimal Nilpotent Orbit
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