PBW theorem for affine super Yangian

Abstract

We prove a Poincar\'e--Birkhoff--Witt theorem for the distinguished affine super Yangian $Y=Y_{1/2,1/2}(\widehat{\mathfrak{sl}}(m|n))$, where $m,n\geq2$ and $m\neq n$. With respect to the current filtration, there is a canonical isomorphism \[ \operatorname{gr}_F Y\cong U\!\left(\operatorname{uce}\!\left( \mathfrak{sl}(m|n)\otimes\mathbb C[z^{\pm1},u] \right)\right), \] where $\operatorname{uce}$ denotes the universal central extension. For any totally ordered classical basis homogeneous in parity and current degree, ordered supermonomials in arbitrary fixed filtered lifts form a vector-space basis of $Y$, with odd exponents restricted to zero or one. The proof identifies the classical presentation through Steinberg Lie superalgebras and establishes injectivity using evaluation homomorphisms, iterated coproducts in degreewise completions, and coefficientwise tensor separation.