The Kantor-Koecher-Tits Construction for Jordan Coalgebras

TL;DR

This paper explores the connection between Jordan and Lie coalgebras, demonstrating how to construct a Lie coalgebra from a Jordan coalgebra and analyzing its subcoalgebra and coideal structures.

Contribution

It introduces a method to derive Lie coalgebras from Jordan coalgebras using the Kantor--Koecher--Tits process and characterizes their subcoalgebra and coideal structures.

Findings

Constructed Lie coalgebras from Jordan coalgebras.

Established correspondence between dual algebras and Lie algebras.

Characterized subcoalgebras and coideals of the constructed coalgebras.

Abstract

The relationship between Jordan and Lie coalgebras is established. We prove that from any Jordan coalgebra $\langle A, \Delta\rangle$, it is possible to construct a Lie coalgebra $\langle L(A), \Delta_{L}\rangle$. Moreover, any dual algebra of the coalgebra $\langle L(A), \Delta_{L}\rangle$ corresponds to a Lie algebra that can be determined from the dual algebra for $\langle A,\Delta\rangle$, following the Kantor--Koecher--Tits process. The structure of subcoalgebras and coideals of the coalgebra $\langle L(A), \Delta_{L}\rangle$ is characterized.