The root system and the core of an extended affine Lie algebra

Abstract

Extended affine Lie algebras are higher nullity generalizations of finite dimensional simple Lie algebras and affine Kac Moody Lie algebras. In this paper we completely describe the structure of the core modulo its centre and the root system for extended affine Lie algebras of type B l (l ≥ 3), C l (l ≥ 2), F 4 and G 2 . MathematicsSubject Classification (2000). Primary 17B65; Secondary 17B60, 17B67, 17C50.Extended affine Lie algebras (or EALAs) were introduced in 1990 by Høegh-Krohn and Torresani [HT] as generalizations of finite dimensional simple Lie algebras and affine Kac-Moody Lie algebras. Another class of examples is the class of toroidal Lie algebras (with suitable derivations added). Toroidal Lie algebras are coordinatized by Laurent polynomials in several commuting variables and they are known to possess interesting vertex operator representations [Ya], [MRY].Although the axiom system for EALAs is simple and natural, it is strong enough to allow a detailed and deep structure theory for the algebras themselves and for their root systems.The structure of an EALA is understood by coordinatizing its core. The coordinate algebras that occur include infinite dimensional quantum versions of all noncommutative and nonassociative algebras that are used classically to construct finite dimensional simple Lie algebras. These can be thought of as noncommutative 1 The first author is grateful for the support of NSERC Grant #A8465 and the hospitality of the University of Ottawa during the preparation of this paper. 2 The second author acknowledges support from a research fellowship from the Alexander von Humboldt Foundation and thanks Peter Slodowy for his hospitality.

Reflection systems and partial root systems

“…In particular, it generalizes the structure theorem of extended affine root systems proven in [ (e) Lie algebras whose root systems (in the appropriate sense) are EARS have been studied in [1,2,3,4,50,51,49]. For SEARS see [42,48], for LEARS see [32] and for GRRS see [9,8].…”

“…Indeed, let f : R → S andf : R →S be extensions of R of nondegenerate reflection systems S andS respectively. Then Ker(f ) = Ker(f ) by (4). From f (R) = S and f (R) =S it follows that f : X → Y andf : X →Ỹ are surjective.…”

“…By (5) and (8), a normal subset is in particular closed. Moreover, by (4) and (5), ∅ and A are always normal subsets of A, and by (6) any proper normal subset B of A has 0 / ∈ B c .…”

“…The subset U := C J ∪ B K is a subsystem of R = BC I (J) and hence restriction along U gives rise to the new reflection systemR = BC I (J) U (4) whose reflective and imaginary roots are…”

A Presentation of Lie Tori of Type $B_ℓ$

“…Let L be a centerless ν−torus over C of type B , ≥ 3. Then by [Yo2,Theorem 7.3] and [AG,Theorem 5.29] we may assume that the corresponding pair of L is (S, 2Z ν ) where…”