Classification of simple differential Lie and Jordan (super)coalgebras of finite rank

TL;DR

The paper classifies finite-rank simple differential Lie and Jordan (super)coalgebras by establishing a contravariant equivalence with finite simple Lie and Jordan conformal (super)algebras. Leveraging the DK and FK classifications, it identifies seven families of simple differential Lie (super)coalgebras and seven corresponding Jordan analogues, then provides explicit coproduct structures for their dual bases, yielding concrete OPE-derived coalgebra data for $W_n$, $S_n$, $K_n$, CK$_6$, and CK-type current algebras, as well as the N=2,3,4 cases. The methodology hinges on the duality between differential coalgebras and conformal algebras and a detailed, section-by-section construction of the coalgebra maps. The results deliver explicit, highly structured coalgebra formulas that facilitate the study of OPE encodings in conformal field theory and extend to the Jordan setting, with potential applications in representation theory and mathematical physics. Overall, the work provides a complete, explicit realization of simple differential Lie and Jordan coalgebras of finite rank tied to the established conformal classifications.

Abstract

We classify simple differential Lie and Jordan (super)coalgebras of finite rank. In particular, we provide an explicit description of the Lie supercoalgebras associated with the operator product expansion (OPE) of the n=2,3,4 superconformal Lie algebras and the exceptional Lie conformal superalgebra CK$_6$

Theorems & Definitions (31)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Proposition 2.4
• proof
• Theorem 2.5
• Theorem 2.6
• proof
• Proposition 2.7
• proof