Abstract
The concept of extended affine root systems was introduced by K. Saito [6] to construct a flat structure for the space of the universal deformation of a simple elliptic singularity. An extended affine root system is by definition an extension of an affine root system by one dimensional radical (see Definition 1.2). It is a natural problem to construct a Lie algebra associated with the root system. In [7], P. Slodowy constructed a Lie algebra for an arbitray extended affine root system in such a way that the set of its real roots coincides with the root system. For instance, in the case of A[' Dfor E{- they may be expressed in the form g^C^?, ^E]Here g is a finite dimensional simple Lie algebra of type At, Dt or Et and the commutation relations are defined by the formula
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