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Highest weights, projective geometry, and the classical limit. I: Geometrical aspects and the classical limit. (English) Zbl 0948.22015
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 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. E. Lieb, Commun. Math. Phys. 31, 327–340 (1973) and B. Simon, Commun. Math. Phys. 71, 46–91 (1980; Zbl 0436.22012)].

MSC:
22E46 Semisimple Lie groups and their representations
51A50 Polar geometry, symplectic spaces, orthogonal spaces
81S10 Geometry and quantization, symplectic methods
15A66 Clifford algebras, spinors
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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