Reflectable bases for affine reflection systems

Locally finite root supersystems

“…Set V Q := span Q A ⊆ V := F ⊗ Z A. One can see that the form (·, ·) F restricted to V Q is nondegenerate (see [1,Lemma 1.6…”

Extended affine root supersystems

“…The following example shows that the condition X = A(ℓ, ℓ) is necessary in Lemma 1.13. This is a phenomena occurring in the super-version of root systems; more precisely, one knows that for an affine reflection system (A, (·, ·), R) i.e., an extended affine root supersystem with no nonsingular root, R 0 = {0} if and only if A 0 = {0}; see [3]. Example 1.14.…”

“…In what follows by a reflectable set for a locally finite root system S, we mean a subset Π of S \ {0} such that W Π (Π) coincides with the set of nonzero reduced roots S × red = S \ {2α | α ∈ S}, in which W Π , is the subgroup of the Weyl group generated by r α for all α ∈ Π; see [3]. We also recall from [7] that a symmetric reflection subspace (or s.r.s for short) of an additive abelian group A is a nonempty subset X of A satisfying X − 2X ⊆ X; we mention that a symmetric reflection subspace satisfies X = −X.…”

“…More precisely, reflectable bases are important in the study of the structure of locally extended affine root systems [18] and integral bases are important in the theory of locally finite Lie algebras [11]. A linearly independent subset Π of the set of real roots of an affine reflection system is called a reflectable base if all nonzero reduced real roots can be obtained from the iterated action of reflections based on the elements of Π. Reflectable bases for affine reflection systems have been studied in [3]. A linearly independent subset Π of a locally finite root supersystem R is called an integral base if each element of R can be written as a Z-linear combination of the elements of Π.…”

A Length Function for Weyl Groups of Extended Affine Root Systems of Type A1

“…The image of R under¯is shown byR. By [AYY,Corollary 1.9],R inĀ is a locally finite root system. The type and the rank of R are defined to be the type and the rank ofR, respectively.…”