Derivations and invariant forms of Jordan and alternative tori

Neher, Erhard; Yoshii, Yōji

doi:10.1090/s0002-9947-02-03013-1

Cited by 19 publications

(8 citation statements)

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“…Lie tori with this root-grading type are classified in [BGK,BGKN,Yo1]. It follows from this classification together with [NY,4.9] that L ≃ sl l (Q) for Q a quantum torus in n variables and structure matrix q = (q ij ) an n × n quantum matrix with at least one q ij not a root of unity (3.7).…”

Section: An Interlaced Extension and Letmentioning

confidence: 99%

Conjugacy of Cartan subalgebras in EALAs with a non-fgc centreless core

Chernousov

Neher

Pianzola

2017

Trans. Moscow Math. Soc.

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Section: An Interlaced Extension and Letmentioning

confidence: 99%

Conjugacy of Cartan subalgebras in EALAs with a non-fgc centreless core

Chernousov

Neher

Pianzola

2017

Trans. Moscow Math. Soc.

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“…Lie tori with this root-grading type are classified in [BGK,BGKN,Yo1]. It follows from this classification together with [NY,4.9] that L ≃ sl l (k q ) for k q a quantum torus in n variables and q = (q ij ) an n × n quantum matrix with at least one q ij not a root of unity. Ne1,8]) that D induces the Λ-grading of L in the sense that…”

Section: Review: Lie Tori and Ealasmentioning

confidence: 99%

On conjugacy of Cartan subalgebras in extended affine Lie algebras

Chernousov

Neher

Pianzola

et al. 2016

Advances in Mathematics

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That finite-dimensional simple Lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as Coxeter-Dynkin diagrams and indecomposable irreducible root systems, is arguably one of the most elegant results in mathematics. The definition of the root system is done by fixing a Cartan subalgebra of the given Lie algebra. The remarkable fact is that (up to isomorphism) this construction is independent of the choice of the Cartan subalgebra. The modern way of establishing this fact is by showing that all Cartan subalgebras are conjugate.For symmetrizable Kac-Moody Lie algebras, with the appropriate definition of Cartan subalgebra, conjugacy has been established by Peterson and Kac. An immediate consequence of this result is that the root systems and generalized Cartan matrices are invariants of the Kac-Moody Lie algebras. The purpose of this paper is to establish conjugacy of Cartan subalgebras for extended affine Lie algebras; a natural class of Lie algebras that generalizes the finite-dimensional simple Lie algebra and affine Kac-Moody Lie algebras.Part of the properties of an EALA (E, H) is a root space decomposition: E = α∈Ψ E α with E 0 = H. The "root system" Ψ is an example of an extended affine root system. The main question, of course, is whether Ψ is an invariant of E. In other words, if H ′ is a subalgebra of E for which the pair (E, H ′ ) is given an EALA structure, is the resulting root system Ψ ′ isomorphic (in the sense of [extended affine] root systems) to Ψ? That this is true follows immediately from the main result of our paper. 0.1. Theorem (Theorem 7.6). Let (E, H) be an extended affine Lie algebra of fgc type. Assume E admits the second structure (E, H ′ ) of an extended affine Lie algebra. Then H and H ′ are conjugate, i.e., there exists a k-linear automorphism f of the Lie algebra E such that f (H) = H ′ .

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“…For the next lemma we recall that a subset S ⊂ Λ is called a pointed reflection subspace if 0 ∈ S and 2S − S ⊂ S, see for example [NY,2.1], where it is also shown that any pointed reflection subspace is a union of cosets modulo 2Z[S], including the trivial coset 2Z[S]. Here Z[S] denotes the Z-span of S. In particular, a pointed reflection subspace is in general not a subgroup.…”

Section: (Xmentioning

confidence: 99%

“…If A = Z(A) ⊕ [A, A], e.g. if A is a torus ([NY, Prop. 3.3]), then sp 2 (A, π) is centreless and hence is (isomorphic to) the TKK-algebra of the Jordan algebra J = H 2 (A, π) and the Jordan (J, J).Example 7.11.…”

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Lie tori of type B_2 and graded-simple Jordan structures covered by a triangle

Neher¹,

Tocón²

2011

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We classify two classes of B2-graded Lie algebras which have a second compatible grading by an abelian group Λ: (a) Λ-gradedsimple, Λ torsion-free and (b) division-Λ-graded. Our results describe the centreless cores of a class of affine reflection Lie algebras, hence apply in particular to the centreless cores of extended affine Lie algebras, the so-called Lie tori, for which we recover results of Allison-Gao and Faulkner. Our classification (b) extends a recent result of Benkart-Yoshii.Both classifications are consequences of a new description of Jordan algebras covered by a triangle, which correspond to these Lie algebras via the Tits-Kantor-Koecher construction. The Jordan algebra classifications follow from our results on graded-triangulated Jordan triple systems. They generalize work of McCrimmon and the first author as well as the Osborn-McCrimmon-Capacity-2-Theorem in the ungraded case.

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Derivations and invariant forms of Jordan and alternative tori

Cited by 19 publications

References 30 publications

Conjugacy of Cartan subalgebras in EALAs with a non-fgc centreless core

Conjugacy of Cartan subalgebras in EALAs with a non-fgc centreless core

On conjugacy of Cartan subalgebras in extended affine Lie algebras

Lie tori of type B_2 and graded-simple Jordan structures covered by a triangle

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