Geometry and affine Lie algebras. (English) Zbl 0737.17008
Generalizing work of J. R. Faulkner [J. Algebra 26, 1–9 (1973; Zbl 0285.17004)], the author investigates the inner ideal geometry of a faithful irreducible highest weight module \(M\) over an affine Lie algebra \(L\). First the inner ideals of \(M\) are determined using a certain inner product on \(M\). The objects in the geometry are found using equivalence classes determined by roots of \(L\) and certain automorphisms of \(M\). The incidence relations are realized as inclusion relations among the inner ideals. Which incidences are allowed can be determined from the Dynkin diagram of \(L\) and information about the highest weight of \(M\). This is related to the work of J. Tits [J. Math. Pures Appl. (9) 36, 17–38 (1957; Zbl 0079.36203)].
Reviewer: Ernest L.Stitzinger (Raleigh)
MSC:
| 17B65 | Infinite-dimensional Lie (super)algebras |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
References:
| [1] | M. I. Dillon, Weight strings in nonstandard representations of Kac-Moody algebras, Trans. Amer. Math. Soc., to appear.; M. I. Dillon, Weight strings in nonstandard representations of Kac-Moody algebras, Trans. Amer. Math. Soc., to appear. · Zbl 0741.17010 |
| [2] | Faulkner, J. R., On the geometry of inner ideals, J. Algebra, 26, 1-9 (1973) · Zbl 0285.17004 |
| [3] | Kac, V. G., Infinite Dimensional Lie Algebras (1983), Birkhäuser: Birkhäuser Boston · Zbl 0574.17002 |
| [4] | Kac, V. G.; Peterson, D. H., Infinite dimensional Lie algebras, theta functions and modular forms, Adv. in Math., 53, 125-264 (1984) · Zbl 0584.17007 |
| [5] | Steinberg, R., Lectures on Chevalley Groups, Yale University lecture notes (1967), New Haven, CT |
| [6] | Tits, J., Sur la géométrie des R-espaces, J. Math. Pures Appl., 36, 17-38 (1957) · Zbl 0079.36203 |
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