2025/05/15 by Andrew Riesen, Riesen, Andrew · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2505.10439
openalex publication_date 2025/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we extend Feigin-Frenkel duality at the critical level to complex rank by identifying two seemingly unrelated constructions in complex rank. On the affine side, we interpolate Molev's construction of higher Segal-Sugawara vectors and thereby describe the centers of universal affine vertex algebras at the critical level in Deligne's interpolating categories. On the W-side, we construct the classical W-algebras associated with Feigin's Lie algebras of complex rank \mathfrakglλ and \mathfrakpoλ as Poisson vertex algebras, realizing their Drinfeld-Sokolov reduction via an interpolated Adler-Gelfand-Dickey bracket. Upon specialization to positive integer rank in types A, B, and C, this recovers the usual Feigin-Frenkel duality at the critical level. As applications, we obtain a uniform construction of several families of higher Segal-Sugawara vectors for Lie superalgebras and recover a complex-rank analogue of the universal Bethe algebra.