On the hyperalgebra of the loop algebra glₙ

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Let UZ(glₙ) be the Garland integral form of U(glₙ) introduced by Garland Ga, where U(glₙ) is the universal enveloping algebra of glₙ. Using Ringel--Hall algebras, a certain integral form UZ(glₙ) of U(glₙ) was constructed in Fu13. We prove that the Garland integral form UZ(glₙ) coincides with UZ(glₙ). Let k be a commutative ring with unity and let U k(glₙ)=UZ(glₙ)⊗ k. For h≥ 1, we use Ringel--Hall algebras to construct a certain subalgebra, denoted by u(n)ₕ, of U k(glₙ). The algebra u(n)ₕ is the affine analogue of u(glₙ)ₕ, where u(glₙ)ₕ is a certain subalgebra of the hyperalgebra associated with glₙ introduced by Humhpreys Hum. The algebra u(glₙ)ₕ plays an important role in the modular representation theory of glₙ. In this paper we give a realization of u(n)ₕ using affine Schur algebras.