Quivers, Symmetrizable Cartan Matrices and Representation Theory

Christof Geiss (Instituto de Matemáticas-UNAM, )

Abstract: We briefly review the representation theory of complex semisimple Lie algebras, including Lusztig’s semicanonical basis for the symmetric cases. We note that the basic theory of semismiple Lie algebras is completely uniform for all Dynkin types. However, some more advanced geometric constructions, like the semicanonical basis work well only for the symmetric cases due to their close link to quiver representations. The same phenomenon occurs for Kac-Moody Lie algebras.

In joint work with B. Leclerc and J. Schröer we proposed a 1-Iwanaga Gorenstein H, defined over an arbitrary field K, associated to the datum of a symmetrizable Cartan Matrix C, a symmetrizer D of C and an orientation $Omega$. The H-modules of finite projective dimension behave in many aspects like the modules over a hereditary , and we can associate to H a kind of preprojective $$.

If we look, for K algebraically closed, at the varieties of representations of $Pi$ which admit a filtration by generalized simples, we find that the components of maximal dimension provide a realization of the crystal $B(-infty)$ corresponding to C. For K being the complex numbers we can construct,following ideas of Lusztig, an algebra of constructible functions which contains a family of “semicanonical functions”. Those are naturally parametrized by the above mentioned components of maximal dimension.
Modulo a about the support of the functions in the “Serre ideal” the semicanonical functions yield a basis of the enveloping algebra U(n) of the positive part of the Kac-Moody Lie algebra g(C).

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https://arxiv.org/find/math/1/au:+Geiss_C/0/1/0/all/0/1

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