Construction of extended affine lie algebras by the twisting process

Azam, Saeid

doi:10.1080/00927870008826991

Cited by 14 publications

(13 citation statements)

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“…Let K σ c be the subalgebra of K σ consisting of all matrices of the form (3.80) with P = BF −1 . By [4,Lemma 3.17], G σ c = K σ c ⊕ C. It follows that G σ c = G σ ∩ G c if and only if m = 1. Now Lemma 2.56 implies that G σ is tame if m = 1.…”

Section: Also We Havementioning

confidence: 94%

“…In [4] it is shown that many examples of EALA (of types D , A 1 , B , C , and BC ) can be obtained as the fixed points of automorphisms of some other EALA (of types A , D , and C ) which may have a simpler structure. Using Theorem 2.65, we are able to give new proofs of the results obtained in [4]. In Examples 3.78 and 3.79 we have provided the details for two cases which seems to be more delicate.…”

Section: Examplesmentioning

confidence: 98%

“…, G σ k , W and I (see Theorem 2.65) for some particular automorphisms of certain EALA. In 3.78 and 3.79 two examples from [4] are restated in such a way that they fit in the setting presented in Section 2. In 3.81, an example regarding the results in [2] is provided.…”

Section: Introductionmentioning

confidence: 98%

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Fixed point subalgebras of extended affine Lie algebras

2005

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It is a well-known result that the fixed point subalgebra of a finite dimensional complex simple Lie algebra under a finite order automorphism is a reductive Lie algebra so it is a direct sum of finite dimensional simple Lie subalgebras and an abelian subalgebra. We consider this for the class of extended affine Lie algebras and are able to show that the fixed point subalgebra of an extended affine Lie algebra under a finite order automorphism (which satisfies certain natural properties) is a sum of extended affine Lie algebras (up to existence of some isolated root spaces), a subspace of the center and a subspace which is contained in the centralizer of the core. Moreover, we show that the core of the fixed point subalgebra modulo its center is isomorphic to the direct sum of the cores modulo centers of the involved summands.

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Section: Also We Havementioning

confidence: 94%

Section: Examplesmentioning

confidence: 98%

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Fixed point subalgebras of extended affine Lie algebras

2005

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“…In [Yo2], Yoshii considered ∆−graded Lie algebras whose corresponding finite dimensional split simple Lie subalgebras are of type ∆ red . Yoshii's concept comes from the theory of EALAs, namely, the core of an EALA is a ∆−graded Lie algebra in the sense of Yoshii [Az1,AG]. Yoshii also introduced (∆, G)−graded Lie algebras, ∆ an irreducible finite root system and G an abelian group [Yo2].…”

Section: The Structure Of L Cmentioning

confidence: 99%

“…Namely, up to the choice of a certain derivation and a 2-cocycle, an EALA is determined by its core modulo center, called the centerless core [N1,2]. In [BGKN], [BGK], [AG] and [Yo1], the authors give a complete description of the centerless cores of reduced extended affine Lie algebras (see also [Az1]). …”

Section: Introductionmentioning

confidence: 99%

A Generalization of Extended Affine Lie Algebras

Yousofzadeh

2007

Communications in Algebra

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ABSTRACT. We introduce a new class of possibly infinite dimensional Lie algebras and study their structural properties. Examples of this new class of Lie algebras are finite dimensional simple Lie algebras containing a nonzero split torus, affine and extended affine Lie algebras. Our results generalize well-known properties of these examples. IntroductionIn 1985, Saito [Sa] introduced the notion of an extended affine root system which is a nonempty subset R of a finite dimensional real vector space V, equipped with a positive semidefinite bilinear form (·, ·), satisfying certain axioms. Following [Az2], we call them Saito's extended affine root systems or SEARSs for short. In 1990, Hφegh-Krohn and Torresani [H-KT] introduced a new class of Lie algebras over the field of complex numbers. The basic features of these Lie algebras are the existence of a non-degenerate symmetric invariant bilinear form, a finite dimensional Cartan subalgebra, a discrete root system and the ad-nilpotency of the root spaces attached to non-isotropic roots. In 1997, Allison, Azam, Berman, Gao and Pianzola [AABGP] called these Lie algebras extended affine Lie algebras (or EALAs for short). The subalgebra of an EALA generated by its non-isotropic root spaces, called the core, plays a very important role in the study of EALAs. Namely, up to the choice of a certain derivation and a 2-cocycle, an EALA is determined by its core modulo center, called the centerless core In [AABGP], the authors take the properties of the root system of an EALA as axioms for the new notion of an extended affine root systems or EARS for short. The non-isotropic roots of an EARS form a SEARS, but not all SEARS arise in this way. In fact, there is a one to one correspondence between reduced SEARS and non-isotropic parts of EARSs [Az2].

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Extended Affine Lie Algebras and Other Generalizations of Affine Lie Algebras – A Survey

Neher

2010

Progress in Mathematics

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Construction of extended affine lie algebras by the twisting process

Cited by 14 publications

References 12 publications

Fixed point subalgebras of extended affine Lie algebras

Fixed point subalgebras of extended affine Lie algebras

A Generalization of Extended Affine Lie Algebras

Extended Affine Lie Algebras and Other Generalizations of Affine Lie Algebras – A Survey

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