Affine modular Lie algebras

The paper is devoted to the construction of affine Kac–Moody algebras via defining relations arising from an integral affine Cartan matrix, in the case where the ground field has positive characteristic $p>7$. Explicit constructions are given for algebras obtained in this way; in distinction with the zero characteristic case they have infinite-dimensional center. A theorem on the universality of this central extension is proved, and a series of finite-dimensional factors of affine modular algebras over the ideals having trivial intersection with Cartan's subalgebra is constructed. Moreover, a construction for the PI-envelopes of affine Kac–Moody algebras is given.