Department Mathematik



Kac-Moody algebras

Yorck Sommerhäuser

Presented at the Ring Theory Conference, Miskolc, Hungary, July 21-26, 1996
  • Preprint: Reihe "Graduiertenkolleg Mathematik im Bereich ihrer Wechselwirkung mit der Physik": gk-mp-9611/44 (dvi, ps)
  • Zeitschrift: J. Algebra (angenommen)


We construct symmetrizable Kac-Moody algebras via a generalized semidirect product construction.


Kac-Moody algebras are a special kind of Lie algebras that generalize the class of finite dimensional semisimple Lie algebras. They were introduced by V. G. Kac and R. V. Moody in 1967 (cf. [8], [10]). They are defined via generators and relations and some additional construction. It was shown much later by O. Gabber and V. G. Kac that Kac-Moody algebras arising from symmetrizable generalized Cartan matrices can be defined entirely via generators and relations (cf. [6]).

In the present article, we give an alternative construction of symmetrizable Kac-Moody algebras which is not based on generators and relations at all. The starting point of the present construction is the so-called triangular decomposition, which describes Kac-Moody algebras as the vector space direct sum of three Lie subalgebras which should be viewed as generalizations of the strictly upper triangular matrices, the diagonal matrices and the strictly lower triangular matrices respectively. Here, we derive a general construction for gluing three Lie algebras to a single one in such a way that the original three Lie algebras appear as subalgebras of the new one, and apply this construction to symmetrizable Kac-Moody algebras. The spirit of the construction is very close to the well-known semidirect product of Lie algebras. It is an analogue of a similar construction for the deformed enveloping algebras of Kac-Moody algebras (cf. [15], [16]). The article is organized as follows: In the preliminary Section 2, we review some background material and introduce some basic notions that will be used in the sequel. In particular, we introduce the notion of a Yetter-Drinfel'd Lie algebra. This kind of Lie algebras will be the main object in Section 4. The notion of a Yetter-Drinfel'd Lie algebra introduced here is not interrelated with the notion of a Lie algebra in the category of Yetter-Drinfel'd modules considered in [12], [13]. In Section 3, we carry out a general framework construction which determines completely the Lie algebras which can be decomposed into three Lie subalgebras in a certain way. Here, we proceed in a completely analogous fashion as in the first construction in [15], and at the end of the section we briefly explain the connection of the two constructions. We also look at a special case of the first construction for Lie coalgebras, a construction that we call the two-sided cosemidirect product, and explain how the so-called bismash product of Lie algebras arises as a special case of the first construction. In Section 4, we discuss a special case of this construction which can be used if two of the three Lie algebras are Yetter-Drinfel'd Lie algebras that are dual in a certain sense. Although conceptually very similar, this construction can not be considered as a special case of the second construction in [15], because the applicability of this construction would require that the enveloping algebras of the two dual Lie algebras are dual too, which never happens unless both Lie algebras are abelian. We discuss further special cases of this second construction, which arise if the Yetter-Drinfel'd Lie algebras involved are symmetric or skew-symmetric, as well as the cases where the background Lie bialgebra is quasitriangular or coquasitriangular. We also explain how the classical double construction of V. G. Drinfel'd arises as a special case of the second construction. In the last section, we describe a cohomological method to construct a bilinear form between two free Lie algebras. We then show that by factoring out the radical of the bilinear form and then applying the second construction one can construct symmetrizable Kac-Moody algebras.

All vector spaces occurring in this article are defined over a base field that is denoted by K, except for the last section, where we are working over the complex numbers as in [9]. The term `linear map' will always mean `K-linear map'.