Extended affine Lie algebras and their root systems

Abstract. We classify Jordan G-tori, where G is any torsion-free abelian group. Using the Zelmanov prime structure theorem, such a class divides into three types, namely, the Hermitian type, the Clifford type and the Albert type. We concretely describe Jordan G-tori of each type. IntroductionIt is a well-known fact that the concept of a "Z n -torus", is of great importance in the context of classification of extended affine Lie algebras. This concept was originally defined by Y. Yoshii in [Yo2]. With the appearance of more general extensions of affine Kac-Moody Lie algebras such as, locally extended affine Lie algebras and invariant affine reflection algebras, one naturally extends the concept of a Z n -torus to a G-torus for an abelian group G, where for the algebras under consideration, G is almost always torsion free. In this work we classify, in a descriptive manner, Jordan G-tori, where G is a torsion free abelian group.First we discuss associative G-tori, using the concept of cocycles. Then we show that a Jordan G-torus is strongly prime, and so one can use the Zelmanov prime structure theorem [MZ]. Thus, such a class divides into three types, the Hermitian type, the Clifford type and the Albert type. We classify each type using the result of associative G-tori and similar methods in [Yo1].The paper is organized as follows. In Section 1, we provide preliminary concepts, including direct limits and direct unions, pointed reflection subspaces and (involutorial) associative G-tori. In Section 2, using a direct union approach, we show that a Jordan G-tori J of Hermitian type has one of involution, plus or extension types (see Definition 2.4) and that J is a direct union of Jordan tori of Hermitian type, where J and its direct union components have the same involution, plus or extension type, see Theorem 2.7. In Section 3, we show that a Jordan G-torus J of Clifford type with support S and central grading group Γ, is graded isomorphic to a Clifford G-torus J(S, Γ, {a ǫ } ǫ∈I ), introduced 2010 Mathematics Subject Classification. 17B67, 17C50. Key words and phrases. Jordan tori, extended affine Lie algebras, invariant affine reflection algebras.1 This research was in part supported by a reseach grant from IPM and partially carried out in IPM-Isfahan branch. The author also would like to thank the Center of Excellence for Mathematics, University of Isfahan.2 This research was in part supported by a grant from IPM (No. 91170415) and partially carried out in IPM-Isfahan branch. explicitly in Example 3.3, for some nonempty index set I and choices of a ǫ ∈ F × , ǫ ∈ I, see Theorem 3.5. In Section 4, the final section, we first fully characterize associative G-tori of central degree 3. Then for two subgroups ∆ and Γ of G satisfying 3G Γ ⊆ ∆ ⊆ G, dim Z3 (G/Γ) = 3, and dim(∆/Γ) = 2, we associate to the triple (G, ∆, Γ), a Jordan algebra A t which turns out to be a Jordan G-torus of Albert type, called an Albert G-torus associated to the triple (G, ∆, Γ), see Example 4.16. Then we proceed with showing that given a Jordan...

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“…A non-trivial interesting feature of semilattices is that any semilattice in G contains a Z-basis of G (see [AABGP,Proposition,II.1.11…”

Section: 2mentioning

confidence: 99%

Jordan tori for a torsion free abelian group

2014

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“…The cases wβ ∈ K × α and wβ ∈ K × β can be done similarly. Thus, for the proof of (a) and (b) it remains to establish the claims about the root intervals as well as (1). Because of 2.13.6 it is sufficient to show (1) for all n ∈ N.…”

Section: Proposition With the Assumptions And Notations Ofmentioning

confidence: 99%

“…By applying (3) repeatedly to roots of the form (4) where 1 i m, and vice versa, we see that (1) holds. This together with 3.3.1 implies (2).…”

Section: Then Condition (I) Holds As Well and A C ⊂ σmentioning

confidence: 99%

“…5]. Extended affine Lie algebras, which generalize affine Lie algebras without necessarily being themselves Kac-Moody algebras, require extended affine root systems already in their definition [1], and generalizations of extended affine root systems have come up naturally in the structure theory of extended affine Lie algebras presently developed, see for example [8,5,6], [32], [52] and the closely related root systems appearing in [41] and [42]. An axiom system for the root systems of the basic classical Lie superalgebras is described in [43], and for the root systems of Borcherds' generalization of Kac-Moody algebras in [11,12].…”

Section: Introductionmentioning

confidence: 99%

“…Among partial root systems, the class of extended affine root systems [1] has recently attracted much interest. There are presently many similar approaches with sometimes conflicting terminology.…”

Section: Introductionmentioning

confidence: 99%

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Reflection systems and partial root systems

Loos

Neher

2011

Forum Mathematicum

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Abstract. We develop a general theory of reflection systems and, more specifically, partial root systems which provide a unifying framework for finite root systems, Kac-Moody root systems, extended affine root systems and various generalizations thereof. Nilpotent and prenilpotent subsets are studied in this setting, based on commutator sets and the descending central series. We show that our notion of a prenilpotent pair coincides, for Kac-Moody root systems, with the one defined by Tits in terms of positive systems and the Weyl group.

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On the Relation of Extended Affine Weyl Groups and Indefinite Weyl Groups

Azam

1999

Journal of Algebra

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Extended affine Lie algebras and their root systems

Cited by 159 publications

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Jordan tori for a torsion free abelian group

Jordan tori for a torsion free abelian group

Reflection systems and partial root systems

On the Relation of Extended Affine Weyl Groups and Indefinite Weyl Groups

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