Recently, Hausel introduced the notion of a big algebra associated to a representation of a complex reductive Lie algebra. These algebras are commutative and capture a lot of representation-theoretic information, and are also related to the equivariant cohomology of certain singular varieties.
In this work, we consider the case g=gl_n and apply the construction of big algebras to two special gl_n-modules, namely the symmetric and exterior algebras of the affine space of n\times r matrices. We discuss some explicit formulas for generators of the corresponding big algebras in terms of differential operators with polynomial coefficients. Then, we explain how to use these formulas to relate type A big algebras to Bethe subalgebras of the Yangian Y(gl_n), and, as a by-product, reprove the commutativity of big algebras in type A.