  We use the Hecke algebras of affine symmetric groups and their associated
Schur algebras to construct a new algebra through a basis, and a set of
generators and explicit multiplication formulas of basis elements by
generators. We prove that this algebra is isomorphic to the quantum enveloping
algebra of the loop algebra of $\mathfrak {gl}_n$. Though this construction is
motivated by the work \cite{BLM} by Beilinson--Lusztig--MacPherson for quantum
$\frak{gl}_n$, our approach is purely algebraic and combinatorial, independent
of the geometric method which seems to work only for quantum $\mathfrak{gl}_n$
and quantum affine $\mathfrak{sl}_n$. As an application, we discover a
presentation of the Ringel--Hall algebra of a cyclic quiver by semisimple
generators and their multiplications by the defining basis elements.
