Strict quantization of coadjoint orbits

Abstract

A strict quantization of a compact symplectic manifold $S$ on a subset $I\subseteq \mathbb{R}$, containing 0 as an accumulation point, is defined as a continuous field of $C^{\ast }$-algebras $\{A_{\hslash }{\}}_{\hslash \in I}$, with $A_0=C_0(S)$, and a set of continuous cross-sections $\{Q(f){\}}_{f\in C^{\infty }(S)}$ for which $Q_0(f)=f$. Here $Q_{\hslash }(f^{\ast })=Q_{\hslash }(f{)}^{\ast }$ for all $\hslash \in I$, whereas for $\hslash \to 0$ one requires that $i[Q_{\hslash }(f),Q_{\hslash }(g)]/\hslash \to Q_{\hslash }(\{f,g\})$ in norm. We discuss general conditions which guarantee that a (deformation) quantization in a more physical sense leads to one in the above sense. Using ideas of Berezin, Lieb, Simon, and others, we construct a strict quantization of an arbitrary integral coadjoint orbit $O_{\lm}$ of a compact connected Lie group $G$, associated to a highest weight $\lm$. Here $I=0\cup 1/\mathbb{N}$, so that $\hslash =1/k$, $k\in \mathbb{N}$, and $A_{1/k}$ is defined as the $C^{\ast }$-algebra of all matrices on the finite-dimensional Hilbert space $V_{k\lm}$ carrying the irreducible representation $U_{k\lm}(G)$ with highest weight $k\lm$. The quantization maps $Q_{1/k}(f)$ are constructed from coherent states in $V_{k\lm}$, and have the special feature of being positive maps.

13 pages

References in corpus (1)

• Quantization of Equivariant Vector Bundles

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