On regular subalgebras of symmetrizable Kac-Moody algebras and related combinatorics

Abstract

In this thesis, we study two aspects of Kac-Moody algebras. One is to understand the subalgebras that can be embedded inside a Kac-Moody algebra as subalgebras generated by real root vectors. The other one is to explicitly classify the regular subalgebras and the maximal regular subalgebras of an untwisted affine Kac-Moody algebra. Dynkin classified the semisimple regular subalgebras of a finite-dimensional semisimple Lie algebra back in 1949. One of the key tools he used for the classification is π-systems. For non-finite Kac-Moody algebras, π-system became an integral part of understanding the embedding of different types of algebras in a Kac-Moody algebra. Till now all the articles existing in the literature, which study π-systems, assume that the π-systems are either linearly independent or finite. It seems that our article is the first one to address infinite π-systems in the context of the embedding problem. This paves a way for us to understand the infinite (linearly independent) π-systems for Borcherds Kac-Moody algebras and understand the embedding problem in that setting. We used Deodhar's preorder to prove that every closed subroot system in a Kac-Moody root system admits a π-system and this π-system need not be finite in general. Moreover, for any closed subroot system ψ of Δ we proved that there exists a unique π-system Π(ψ), which is contained in the set of positive roots. Since the subroot systems of a root system are not very `well behaved', this is quite surprising and it generalizes the previously well-known fact that they simple systems and positive systems determine each other in the level of the subroot system. Using this unique π-system Π(ψ), we prove that for a real closed subroot system ψ, the real roots of a root generated subalgebra g(ψ) is equal to ψ. This result was a much-awaited one in the literature because almost after 70 years of Dynkin's result, Roy and Venkatesh [Trans. Grp. 2019] proved that the same is true for an affine root system. These two results provide a bridge between the algebraic and combinatorial side which shows that the root-generated subalgebras are in bijection with the real closed subroot systems which are in turn in one-to-one correspondence with the π-systems contained in the positive roots of a Kac-Moody algebra. In the last part of our analysis of regular subalgebras generated by root vectors, we prove that for any closed subroot system ψ, the root generated subalgebra is isomorphic to a quotient of the derived subalgebra of the Kac-Moody algebra corresponding to the (infinite) Cartan matrix defined by the unique π-system of the closed subroot system ψ by an ideal contained in the centre of the algebra. This result is a generalization of the existing results when the π-system is linearly independent and the ideal is zero when the π-system is linearly independent also follows from our result. In particular, as long as the roots are concerned, to understand the root generated subalgebras, it is enough to consider the derived algebras of a Kac-Moody algebra g'(A) corresponding to a (n infinite) GCM A. Classification of regular subalgebras of an affine Kac-Moody Lie algebra is an interesting problem in its own right. Barnea et. al. started such classification in 1998. Later Felikson et al. used combinatorics of root systems to classify the regular subalgebras in 2008, more precisely the root generated subalgebras of an affine Kac-Moody algebra. We took a completely different approach, namely, using the classification of the closed subroot system of a real affine root system given by Roy and Venkatesh, we classify the regular subalgebras of affine Kac-Moody Lie algebras with a symmetric set of roots and we get the classification of root generated subalgebras as a Corollary. Moreover, we also classify the maximal symmetric regular subalgebras we show a bijective correspondence between the maximal real closed subroot systems of the affine Lie algebra and the maximal symmetric regular subalgebras different from [g,g]. Which also shows in affine case, given a maximal closed subroot system ψ of Δ, the poset (with set inclusion as the partial order) A_ψ:={s:Δ(s)^{re}=ψ} contains a unique maximal element.