×

Representations of Lie algebras arising from polytopes. (English) Zbl 1234.17005

Summary: We present an extremely elementary construction of the simple Lie algebras over \(\mathbb C\) in all of their minuscule representations, using the vertices of various polytopes. The construction itself requires no complicated combinatorics and essentially no Lie theory other than the definition of a Lie algebra; in fact, the Lie algebras themselves appear as by-products of the construction.

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
PDFBibTeX XMLCite
Full Text: arXiv