×

On the exceptional series, and its descendants. (English) Zbl 1017.22008

Summary: Many of the striking similarities which occur for the adjoint representation of groups in the exceptional series also occur for certain representations of specific reductive subgroups. The tensor algebras on these representations are easier to describe and may offer clues to the original situation. The subgroups which occur form a magic triangle which extends Freudenthal’s magic square of Lie algebras. We describe these groups from the perspective of dual pairs, and their representations from the action of the dual pair on an exceptional Lie algebra.

MSC:

22E60 Lie algebras of Lie groups
20G05 Representation theory for linear algebraic groups
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E10 General properties and structure of complex Lie groups

Software:

LiE
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Cohen, A. M.; de Man, R., Computational evidence for Deligne’s conjecture regarding exceptional Lie groups, C. R. Acad. Sci. Paris, Série I, 322, 427-432 (1996) · Zbl 0849.22017
[2] Deligne, P., La série exceptionnelle de groupes de Lie, C. R. Acad. Sci. Paris, Série I, 322, 321-326 (1996) · Zbl 0910.22008
[3] Deligne, P.; de Man, R., La série exceptionnelle de groupes de Lie II, C. R. Acad. Sci. Paris, Série I, 323, 577-582 (1996) · Zbl 0910.22009
[4] Gross, B.; Wallach, N., A distinguished family of unitary representations for the exceptional groups of real rank =4, (Lie Theory and Geometry (1994), Birkhäuser), 289-304 · Zbl 0839.22006
[5] Schwarz, G. W., Invariant theory of \(G_2\) and \(Spin_7\), Comment. Math. Helv., 63, 624-663 (1988) · Zbl 0664.14006
[6] M.A.A. van Leeuwen, A.M. Cohen, B. Lisser, LiE, a package for Lie group computations, CAN, Amsterdam, 1992; M.A.A. van Leeuwen, A.M. Cohen, B. Lisser, LiE, a package for Lie group computations, CAN, Amsterdam, 1992
[7] Wenzl, H., Tensor categories of Lie type \(E_N\), Adv. Math. (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.