  In this paper we study a family of commutative algebras generated by two
infinite sets of generators. These algebras are parametrized by Young diagrams.
We explain a connection of these algebras with the fusion product of integrable
irreducible representations of the affine $sl_2$ Lie algebra. As an application
we derive a fermionic formula for the character of the affine fusion product of
two modules. These fusion products can be considered as a simplest example of
the double affine Demazure modules.
