Double Yangian and reflection algebras of the Lie superalgebra ${\mathfrak{gl}}_{M|N}$, II: Quantum currents

Abstract

In this paper, we continue our research on the double Yangian and reflection algebras of the Lie superalgebra ${\mathfrak{gl}}_{M|N}$. Extending the Etingof-Kazhdan construction, we introduce the structure of $h$-adic quantum vertex superalgebra on the vacuum module $V^c({\mathfrak{gl}}_{M|N})$ over the double Yangian of level $c\in \mathbb{C}$. Next, we construct families of central elements in the completed double Yangian for ${\mathfrak{gl}}_{M|N}$ at the critical level. Finally, we show that the level $c$ restricted modules over the reflection algebra of ${\mathfrak{gl}}_{M|N}$ are naturally equipped with the structure of quasi $V^{2c}({\mathfrak{gl}}_{M|N})$-module. By using this structure, we find explicit formulas for families of central elements in the completed reflection algebra of ${\mathfrak{gl}}_{M|N}$ at the critical level.

25 pages, new results added, comments are welcome