The Classification of Almost Affine (Hyperbolic) Lie Superalgebras

Abstract

We say that an indecomposable Cartan matrix A with entries in the ground field of characteristic 0 is almost affine if the Lie sub(super)algebra determined by it is not finite dimensional or affine (Kac-Moody) but the Lie (super)algebra determined by any submatrix of A, obtained by striking out any row and any column intersecting on the main diagonal, is the sum of finite dimensional or affine Lie (super)algebras. A Lie (super)algebra with Cartan matrix is said to be almost affine if it is not finite dimensional or affine (Kac-Moody), and all of its Cartan matrices are almost affine.We list all almost affine Lie superalgebras over complex numbers correcting two earlier claims of classification and make available the list of almost affine Lie algebras obtained by Li Wang Lai. 1991 Mathematics Subject Classification. 2000 Mathematics Subject Classification: 17B65. Key words and phrases. Hyperbolic Lie superalgebra. DL is thankful toÉ. Vinberg and O. Shwartsman for helpful comments and to A. Protopopov for his help with our graphics, see [Pro]. 1 2 DANIL CHAPOVALOV 1 , MAXIM CHAPOVALOV 2 , ALEXEI LEBEDEV 3 , DIMITRY LEITES 2There is another type of Lie (super)algebras that used to go under the same name "hyperbolic", but are defined differently and currently are referred to as Lorentzian Lie algebras; for their precise definition, see [RU, GN]).The study of Lorentzian Lie algebras makes superization not just natural, but rather i ne v i t a b l e we'd say, see [RU, GN]: Borcherds, and later Gritsenko and Nikulin found various applications of simple Lorentzian Lie algebras and superalgebras (for one of these applications Borcherds was awarded with a Fields medal). This certainly justifies the quest for the simple Lorentzian Lie algebras and superalgebras.Whereas the almost affine Lie algebras are useful, e.g., for cosmologic billiards, at the moment the only applications of almost affine Lie superalgebras we know of are due to the fact that some of them (those of rank 3) coincide with the known Lorentzian Lie superalgebras. This is already good, but the notion of almost affine Lie (super)algebras seems to be most natural even without such coincidence, and hence worth investigating. Besides, any almost affine Lie superalgebra whose even part is an almost affine Lie algebra might hint at a hidden supersymmetry of the problem related with the latter.Although the classification problem of the almost affine Lie superalgebras (twice claimed to be done) should be (and is) much simpler than the classification problem of Lorentzian Lie (super)algebras (still an open problem, and it is not even clear if it is tame even if there are finitely many of them), it was not solved in one go.This paper was written after we failed to understand even the definitions given in [FS] to say nothing of results (which, in turn, were supposed to correct the results of [TDP]); several counterexamples to the claims of [FS] immediately spring to mind. Here we rectify the results of [TDP] and [FS]; in particular, we give precise definitions and an ...

The Poincaré Series of the Hyperbolic Coxeter Groups with Finite Volume of Fundamental Domains

“…affine Kac-Moody) Lie algebra g(A), where A is a Cartan matrix. The hyperbolic groups of (quasi-)Lannér type serve as the Weyl groups of what we suggest to call almost affine Lie algebra a g(A); for the list of almost affine Lie algebras, see the arXiv:0906.1860 version of [11]. We assume that all Cartan and Coxeter matrices are indecomposable, unless otherwise stated.…”

“…hyperbolic) Lie algebras due to [41,51]; for the list of such diagrams/matrices, see also [11]. f Recall that a subgraph is complete if each of its nodes is connected to every other of its nodes.…”

“…the case of A (1) 1 (the answer for which coincides with that for A (2) 2 , see Table 13). Unless the authors of [18], where the multiparameter growth functions are applied, or somebody else, will ask us to do the job, we intend to imitate the behavior of Prof. Macdonald, and redirect the reader: For the classification of Li and Saçlioglu, see more accessible list in [11]; the code is available at [10] and how to proceed with the code is described in the last section of this work. The task is now routine while to list the results will double the length of the tables.…”

Defining Relations of Almost Affine (Hyperbolic) Lie Superalgebras

“…For the definition of Lie superalgebras with Cartan matrix, in particular, of the almost affine Lie superalgebras, see [2]. We recall the definition of Chevalley generators and the defining relations expressed in terms of these generators, see [5] and a review [1] which also contains the modular case.…”

“…Here we list defining relations of almost affine Lie superalgebras with indecomposable Cartan matrices classified in [2]. For the Lie superalgebras whose Cartan matrices are symmetrizable and without zeros on the main diagonal, the relations were known; they are described by almost the same rules as for Lie algebras and they are only of Serre type if the off-diagonal elements of such Cartan matrices are non-positive.…”