An algebra associated with a subspace lattice over a finite field and its relation to the quantum affine algebra Uq(slˆ2)

It is known that the incidence algebra of a subspace lattice over a finite field with q elements is a homomorphic image of the quantum algebra Uq1/2(sl2). In this paper, we extend this situation. For a fixed proper subspace (which is an object of the subspace lattice), we define naturally a new algebra H which contains the incidence algebra as a proper subalgebra, and show how it is related to the quantum affine algebra Uq1/2(slˆ2). We show that there is an algebra homomorphism from Uq1/2(slˆ2) to H, and that H is generated by its image together with the center. Moreover, we show that any irreducible H-module is also irreducible as a Uq1/2(slˆ2)-module and is isomorphic to the tensor product of two evaluation modules. We also obtain a small set of generators of the center of H.