Algebraic deformation program on minimal nilpotent orbit
type: Item
instance of: scholarly article
title: Algebraic deformation program on minimal nilpotent orbit
zbMATH Open document ID: 0805.17009
DOI: 10.1007/BF00805856
published in: Letters in Mathematical Physics
publication date: 1995-02-05T00:00:00Z
review text: Let \({\mathfrak g}\) be a finite-dimensional simple complex Lie algebra not of type \(A_ n\); let the adjoint group \(G\) of \({\mathfrak g}\) act on \({\mathfrak g}\) by conjugation and let \({\mathcal D}\) be the unique nontrivial nilpotent orbit of minimal dimension. Elements of the symmetric algebra \(S({\mathfrak g})\) act on \({\mathfrak g}\) via the Killing form; their restriction to \({\mathcal D}\) defines the algebra \(\mathbb{C}[{\mathcal D}]\). The orbit \({\mathcal D}\) has a canonical structure of a symplectic manifold with Poisson bracket \(\{\cdot, \cdot\}\) induced from the natural extension by derivation of the Lie bracket in the Lie algebra to the symmetric algebra. Let \(\psi\) denote the highest-weight root and \(V_ n\) the simple submodule of the \(n\)-th symmetric power of the Lie algebra with highest weight \(n\psi\). A star product on \(\mathbb{C}[{\mathcal D}]\) is a bilinear map \((P,Q)\mapsto P\circ Q= \sum v^ n c_ n (P,Q)\) into the formal power series in \(v\) over \(\mathbb{C}[{\mathcal D}]\) where \(c_ 0\) is multiplication, \(c_ 1\) is Poisson bracket, \(c_ n\) is \((-1)^ n\) anti-symmetric and \(c_ n(1,\cdot) =0\), \(n>0\). The product is covariant if \(P\circ Q- Q\circ P= 2v[P,Q]\) for \(P\), \(Q\) in the Lie algebra, strongly invariant if \(X\circ f- f\circ X= 2v\{X, f\}\) for \(X\) in the Lie algebra and \(f\in \mathbb{C}[{\mathcal D}]\) arbitrary, and graded if \(c_ k(P,Q)\in V_{n+m-k}\) when \(P\in V_ n\) and \(Q\in V_ m\). The authors prove that a graded star product exists, that it is strongly invariant, and that it is unique up to equivalence (an isomorphism of the form \(T=I+ \sum v^ r T_ r\)). They give an example of the construction in the case that the Lie algebra is of type \(C_ n\), \(n\geq 3\). Finally, they carry out the deformation program for \({\mathcal D}\): in the program, an irreducible representation \((\pi, E)\) is said to be connected to the orbit \({\mathcal D}\) if there is an irreducible representation \…
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 540167
zbMATH Keywords: simple complex Lie algebra
zbMATH Keywords: symmetric algebra
zbMATH Keywords: Poisson bracket
zbMATH Keywords: graded star product
zbMATH Keywords: irreducible representation
author: D. Arnal
author: Hédi Ben Amor
author: Benjamin Cahen
reviewed by: Andy R. Magid
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 540167