Algebraic deformation program on minimal nilpotent orbit
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 \(\rho: \mathbb{C}[{\mathcal D}]\to E\) of algebras such that \(\rho(X)= 2v\pi(X)\) for \(X\) in the Lie algebra. (Here a value \(v\) of the parameter has been selected.) The authors show that the irreducible module associated to the functional on the Cartan subalgebra coming from \({\mathcal D}\) is connected to \({\mathcal D}\) and show it explicitly in their \(C_ n\) example.
scientific article; zbMATH DE number 540167
scientific article