Dynkin diagrams of hyperbolic Kac Moody superalgebras

Tripathy, L K; Das, Balaram; Pati, K. C.

doi:10.1088/0305-4470/36/8/307

Cited by 3 publications

(2 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The constructs in this case so generated are hyperbolic Kac-Moody superalgebras. [4][5][6][7][8][9][10][11] Through the results of three papers, 1,3,12 it is now believed that the hyperbolic Kac-Moody superalgebras have also been completely classified. In all these papers it is also shown that similar to hyperbolic Kac-Moody algebras, the hyperbolic Kac-Moody superalgebras are also finite in number for rank >2 with maximum rank being 6.…”

Section: Introductionmentioning

confidence: 99%

Embedding of simply laced hyperbolic Kac-Moody superalgebras

Nayak

Pati

2013

Journal of Mathematical Physics

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

Embedding of simply laced hyperbolic Kac-Moody superalgebras

Nayak

Pati

2013

Journal of Mathematical Physics

View full text Add to dashboard Cite

show abstract

“…From these motivations, the authors of Ref. [11] have recently classified hyperbolic Kac-Moody (HKM) superalgebras by a procedure quite close to that followed to classify the hyperbolic Kac-Moody ones, showing that they are limited in rank, now the maximum rank being 6, and that they are finite in number (for rank > 2). However, as we remarked that many diagrams are missing, while some of the proposed Dynkin-Kac diagrams correspond in fact to diagrams of untwisted or twisted affine Lie superalgebras (sometimes in the not distinguished basis), we present here a, hopefully exhaustive, classification of HKM superalgebras, together with a corresponding simple roots basis and we discuss a class of singular subalgebras.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic Kac–Moody superalgebras

Frappat

Sciarrino

2005

Journal of Mathematical Physics

View full text Add to dashboard Cite

show abstract

The Classification of Almost Affine (Hyperbolic) Lie Superalgebras

Chapovalov¹,

Chapovalov²,

Lebedev³

et al. 2021

JNMP

View full text Add to dashboard Cite

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 ...

show abstract

Dynkin diagrams of hyperbolic Kac Moody superalgebras

Cited by 3 publications

References 15 publications

Embedding of simply laced hyperbolic Kac-Moody superalgebras

Embedding of simply laced hyperbolic Kac-Moody superalgebras

Hyperbolic Kac–Moody superalgebras

The Classification of Almost Affine (Hyperbolic) Lie Superalgebras

Contact Info

Product

Resources

About