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 ...
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