Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits

Carbone, Lisa; Chung, Sjuvon; Cobbs, Leigh; McRae, Robert; Nandi, Debajyoti; Naqvi, Yusra; Penta, Diego

doi:10.1088/1751-8113/43/15/155209

Cited by 37 publications

(58 citation statements)

References 17 publications

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“…We have composed computer program to do the checking. The computation result shows except for the number 131,132,133,137,139,141 hyperbolic Lie algebras in the list of Carbone [1], all the subalgebras we constructed are non symmetrizable.…”

Section: (B)mentioning

confidence: 98%

Polynomial invariants of Weyl groups for Kac–Moody groups

Zhao¹,

Chang²

2014

Pacific J. Math.

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In this paper, we prove that the ring of polynomial invariants of the Weyl group for an indecomposable and indefinite Kac-Moody Lie algebra is generated by invariant symmetric bilinear form or is trivial depending on A is symmetrizable or not. The result was conjectured by Moody[24] and assumed by Kac [18]. As applications we discuss the rational homotopy types of Kac-Moody groups and their flag manifolds.

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Section: (B)mentioning

confidence: 98%

Polynomial invariants of Weyl groups for Kac–Moody groups

Zhao¹,

Chang²

2014

Pacific J. Math.

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“…s ∈ S m , for fields from (16), (24)- (27), when Restrictions (31) and (32) are imposed, are equivalent to the equations of motion for the σ -model governed by the action:…”

Section: The Sigma Modelmentioning

confidence: 99%

On Brane Solutions with Intersection Rules Related to Lie Algebras

Ivashchuk

2017

Symmetry

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Abstract:The review is devoted to exact solutions with hidden symmetries arising in a multidimensional gravitational model containing scalar fields and antisymmetric forms. These solutions are defined on a manifold of the form M = M 0 × M 1 × . . . × M n , where all M i with i ≥ 1 are fixed Einstein (e.g., Ricci-flat) spaces. We consider a warped product metric on M . Here, M 0 is a base manifold, and all scale factors (of the warped product), scalar fields and potentials for monomial forms are functions on M 0 . The monomial forms (of the electric or magnetic type) appear in the so-called composite brane ansatz for fields of forms. Under certain restrictions on branes, the sigma-model approach for the solutions to field equations was derived in earlier publications with V.N.Melnikov. The sigma model is defined on the manifold M 0 of dimension d 0 = 2 . By using the sigma-model approach, several classes of exact solutions, e.g., solutions with harmonic functions, S-brane, black brane and fluxbrane solutions, are obtained. For d 0 = 1 , the solutions are governed by moduli functions that obey Toda-like equations. For certain brane intersections related to Lie algebras of finite rank-non-singular Kac-Moody (KM) algebras-the moduli functions are governed by Toda equations corresponding to these algebras. For finite-dimensional semi-simple Lie algebras, the Toda equations are integrable, and for black brane and fluxbrane configurations, they give rise to polynomial moduli functions. Some examples of solutions, e.g., corresponding to finite dimensional semi-simple Lie algebras, hyperbolic KM algebras: H 2 (q, q) , AE 3 , H A (1) 2 , E 10 and Lorentzian KM algebra P 10 , are presented.

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“…Similarly, the proof of Theorem 3.9 does not use any assumption that the Kac-Moody algebra be of finite type, so our result extends to arbitrary simply-laced types. (see [5] for the notation and list of Dynkin diagrams), whose Dynkin diagram is the complete graph on four vertices. Then the partitions are enumerated as (ν (1) , ν (2) , ν (3) , ν (4) ) and Table 5.1 Well-known embeddings g −→ g of affine Kac-Moody algebras by type as given in [11] (n = 1).…”

Section: Proof Consider Any (B J) ∈ H and Definementioning

confidence: 99%