Subalgebras of hyperbolic Kac-Moody algebras

Feingold, Alex J.; Nicolai, Hermann

doi:10.1090/conm/343/06185

Cited by 30 publications

(55 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We also state the following lemma (implicit in [8]) that simplifies the task of verifying the hypothesis of theorem 2.…”

Section: Hementioning

confidence: 99%

“…Indefinite type Kac-Moody algebras however remain a notoriously intractable part of the theory. In spite of much work in this direction (see [7,11] and the references in [8]), obtaining detailed information about the structure of these Lie algebras seems out of reach at present. Most of the available results concern the subclass of hyperbolic Kac-Moody algebras.…”

Section: Introductionmentioning

confidence: 99%

“…In [8], Feingold and Nicolai studied subalgebras of hyperbolic Kac-Moody algebras. They showed that the rank 3 hyperbolic Kac-Moody algebra F = HA (1) 1 contains every rank 2 hyperbolic Kac-Moody algebra with symmetric generalized Cartan matrix; in fact F was also shown to contain an infinite series of indefinite type Kac-Moody algebras.…”

Section: Introductionmentioning

confidence: 99%

“…Among these, the algebra E 10 has been singled out for its relevance to string theory and has received much attention in recent times (e.g: [1][2][3][4][12][13][14]). In [8], Feingold and Nicolai studied subalgebras of hyperbolic Kac-Moody algebras. They showed that the rank 3 hyperbolic Kac-Moody algebra F = HA (1) 1 contains every rank 2 hyperbolic Kac-Moody algebra with symmetric generalized Cartan matrix; in fact F was also shown to contain an infinite series of indefinite type Kac-Moody algebras.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Embeddings of Hyperbolic Kac–Moody Algebras into E 10

Viswanath

2008

Lett Math Phys

View full text Add to dashboard Cite

Abstract. We show that the rank 10 hyperbolic Kac-Moody algebra E10 contains every simply laced hyperbolic Kac-Moody algebra as a Lie subalgebra. Our method is based on an extension of earlier work of Feingold and Nicolai. IntroductionSince their discovery [9,15] by Kac and Moody, Kac-Moody algebras have been playing increasingly important roles in diverse subfields of mathematics and physics. The affine Kac-Moody algebras are by now as well understood as the finite dimensional simple Lie algebras classified by Cartan and Killing. Indefinite type Kac-Moody algebras however remain a notoriously intractable part of the theory. In spite of much work in this direction (see [7,11] and the references in [8]), obtaining detailed information about the structure of these Lie algebras seems out of reach at present. Most of the available results concern the subclass of hyperbolic Kac-Moody algebras. Such algebras only exist in ranks 2-10 and can be completely classified [16] (see also [5] for some missing diagrams). Among these, the algebra E 10 has been singled out for its relevance to string theory and has received much attention in recent times (e.g: [1][2][3][4][12][13][14]).In [8], Feingold and Nicolai studied subalgebras of hyperbolic Kac-Moody algebras. They showed that the rank 3 hyperbolic Kac-Moody algebra F = HA (1) 1 contains every rank 2 hyperbolic Kac-Moody algebra with symmetric generalized Cartan matrix; in fact F was also shown to contain an infinite series of indefinite type Kac-Moody algebras. Analogously, it was shown that there are infinitely many inequivalent Kac-Moody algebras of indefinite type that occur as Lie subalgebras of E 10 .This motivates the main question of this letter: which Kac-Moody algebras of hyperbolic type occur as Lie subalgebras of E 10 ? To answer this question, we extend the method of Feingold-Nicolai and formulate some general principles for constructing Lie subalgebras. These principles are then used to prove our main result: Every simply laced hyperbolic Kac-Moody algebra occurs as a Lie subalgebra of E 10 . This statement can be viewed as further evidence of the distinguished role played by E 10 in the family of 2000 Mathematics Subject Classification. 17B67.

show abstract

“…We also state the following lemma (implicit in [8]) that simplifies the task of verifying the hypothesis of theorem 2.…”

Section: Hementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Embeddings of Hyperbolic Kac–Moody Algebras into E 10

Viswanath

2008

Lett Math Phys

View full text Add to dashboard Cite

show abstract

“…Again decomposing with respect to them will soon leave the realm of highest weight representations and is thus subject to the obstructions mentioned above. Some related ideas have recently appeared in [23]. …”

Section: Proposition 1 Letmentioning

confidence: 99%

E11 as E10 representation at low levels

Kleinschmidt

2004

Nuclear Physics B

View full text Add to dashboard Cite

The Lorentzian Kac-Moody algebra E 11 , obtained by doubly overextending the compact E 8 , is decomposed into representations of its canonical hyperbolic E 10 subalgebra. Whereas the appearing representations at levels 0 and 1 are known on general grounds, higher level representations can currently only be obtained by recursive methods. We present the results of such an analysis up to height 120 in E 11 which comprises representations on the first five levels. The algorithms used are a combination of Weyl orbit methods and standard methods based on the Peterson and Freudenthal formulae. In the appendices we give all multiplicities of E 10 occuring up to height 340 and for E 11 up to height 240.

show abstract

Oxidizing Borcherds symmetries

Kleinschmidt

Palmkvist

2013

J. High Energ. Phys.

View full text Add to dashboard Cite

The tensor hierarchy of maximal supergravity in D dimensions is known to be closely related to a Borcherds (super)algebra that is constructed from the global symmetry group E 11−D . We here explain how the Borcherds algebras in different dimensions are embedded into each other and can be constructed from a unifying Borcherds algebra. The construction also has a natural physical explanation in terms of oxidation. We then go on to show that the Hodge duality that is present in the tensor hierarchy has an algebraic counterpart. For D > 8 the Borcherds algebras we find differ from the ones existing in the literature although they generate the same tensor hierarchy.

show abstract

Subalgebras of hyperbolic Kac-Moody algebras

Cited by 30 publications

References 28 publications

Embeddings of Hyperbolic Kac–Moody Algebras into E 10

Embeddings of Hyperbolic Kac–Moody Algebras into E 10

E11 as E10 representation at low levels

Oxidizing Borcherds symmetries

Contact Info

Product

Resources

About