REPRESENTATIONS OF LIE ALGEBRAS ARISING FROM POLYTOPES

Yıl 2008, Cilt: 4 Sayı: 4, 27 - 52, 01.12.2008

Richard M. Green

Öz

We present an extremely elementary construction of the simple Lie algebras over 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.

Anahtar Kelimeler

Lie algebra, minuscule representation, polytope

Kaynakça

• S.C. Billey and V. Lakshmibai, Singular Loci of Schubert Varieties, Progr. Math. 182, Birkh¨auser, Boston, 2000.
• R.W. Carter, Lie algebras of finite and affine type, Cambridge University Press, Cambridge, 2005.
• J.H. Conway and N.J.A. Sloane, The cell structures of certain lattices, in Miscellanea Mathematica (editors P. Hilton, F. Hirzebruch and R. Remmert), Springer-Verlag, (New York, 1991), pp. 71–108.
• B.N. Cooperstein, A note on the Weyl group of type E7, Europ. J. Combina- torics, 11 (1990), 415–419.
• H.S.M. Coxeter, Regular Polytopes, Pitman, New York, 1947.
• P. du Val, On the directrices of a set of points in a plane, Proc. Lond. Math. Soc., (2) 35 (1933), 23–74.
• R.M. Green, Full heaps and representations of affine Kac–Moody algebras, Int. Electron. J. Algebra, 2 (2007), 138–188.
• R.M. Green, Full heaps and representations of affine Weyl groups, Int. Elec- tron. J. Algebra, 3 (2008), 1–42.
• R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977.
• J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Univer- sity Press, Cambridge, 1990.
• V.G. Kac, Infinite dimensional Lie algebras (third edition), Cambridge Uni- versity Press, Cambridge, UK, 1990.
• M. Kashiwara, On crystal bases of the q-analogue of universal enveloping al- gebras, Duke Math. J., 63 (1991), 465–516.
• L. Manivel, Configurations of lines and models of Lie algebras, J. Algebra, 304 (2006), 457–486.
• J.R. Stembridge, Minuscule elements of Weyl groups, J. Algebra, 235 (2001), –743.
• N.J. Wildberger, A combinatorial construction for simply-laced Lie algebras, Adv. Appl. Math., 30 (2003), 385–396.
• N.J. Wildberger, A combinatorial construction of G2, J. Lie Theory, 13 (2003), –165. Richard M. Green
• Department of Mathematics University of Colorado Campus Box 395 Boulder, CO 80309-0395 USA e-mail: [e-mail]