Highest weights, projective geometry, and the classical limit. I: Geometrical aspects and the classical limit
type: Item
instance of: scholarly article
title: Highest weights, projective geometry, and the classical limit. I: Geometrical aspects and the classical limit
zbMATH Open document ID: 0948.22015
DOI: 10.1016/S0393-0440(99)00002-9
published in: Journal of Geometry and Physics
publication date: 2000-11-12T00:00:00Z
review text: It is a classical result that the orbit \(\Phi\) of any highest weight vector \(v\) of a semi-simple compact Lie group \(G\) is an algebraic variety. In a previous paper [Q. J. Math. Oxford, II. Ser. 33, 91--96 (1982; Zbl 0471.22015)] the author proved the following: Theorem: For any irreducible \(G\)-module \(V\) a vector \(v \in V\) is a highest weight vector for some maximal torus subgroup of \(G\) if and only if the cyclic \(G\)-submodule of \(V \otimes V\) generated by \(v \otimes v\) is irreducible. Another, quite different, proof of this theorem was given by \textit{W. Lichtenstein} [Proc. Am. Math. Soc. 84, 605--608 (1982; Zbl 0501.22017)]. The theorem implies that these orbits \(\Phi\) are even homogeneous quadratic varieties (if \(p\) is the projection onto the irreducible submodule of \(V \otimes V\) whose highest weight vector is double that of \(V\), the vectors of \(\Phi\) satisfy the quadratic equation \(p(v \otimes v) = v \otimes v\)). The paper under review contains yet another proof of the above theorem which is more elementary than the other proofs. Moreover, the automorphism group \(\Gamma\) of \(\Phi\) is calculated. It turns out that in most cases the connected component of \(\Gamma\) is just the complexification \(G_{\mathbb C}\) of \(G\), which obviously leaves invariant \(\Phi\). Many beautiful geometrical examples are included throughout the manuscript and the theorem is extended to certain types of algebras like Clifford algebras and to quantum groups. Other relationships to quantisation and to the classical limit are also exploited [see e.g. \textit{E. Lieb}, Commun. Math. Phys. 31, 327--340 (1973) and \textit{B. Simon}, Commun. Math. Phys. 71, 46--91 (1980; Zbl 0436.22012)].
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 51A50
Mathematics Subject Classification ID: 81S10
Mathematics Subject Classification ID: 15A66
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 1463369
zbMATH Keywords: highest weight
zbMATH Keywords: projective geometry
zbMATH Keywords: flag manifold
zbMATH Keywords: pure spinor
zbMATH Keywords: classical limit
zbMATH Keywords: moment map
zbMATH Keywords: semi-simple compact Lie group
zbMATH Keywords: Clifford algebras
zbMATH Keywords: quantum groups
author: Keith C. Hannabuss
MaRDI profile type: Publication
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