E$_{6(6)}$ Exceptional Drinfel'd Algebras

TL;DR

This paper constructs the E6(6) exceptional Drinfel'd algebra, a Leibniz algebra framework for exploring U-duality in M-theory, using geometric realizations and generalised Yang-Baxter equations.

Contribution

It provides the first detailed construction of the E6(6) EDA, linking algebraic structures to geometric models and generalising classical Yang-Baxter equations.

Findings

Realisation of E6(6) EDA as a generalised Leibniz parallelisation.

Identification of a generalised Yang-Baxter equation for coboundary EDAs.

Examples including embeddings of Drinfel'd doubles and novel structures.

Abstract

The exceptional Drinfel'd algebra (EDA) is a Leibniz algebra introduced to provide an algebraic underpinning with which to explore generalised notions of U-duality in M-theory. In essence it provides an M-theoretic analogue of the way a Drinfel'd double encodes generalised T-dualities of strings. In this note we detail the construction of the EDA in the case where the regular U-duality group is $E_{6(6)}$. We show how the EDA can be realised geometrically as a generalised Leibniz parallelisation of the exceptional generalised tangent bundle for a six-dimensional group manifold $G$, endowed with a Nambu-Lie structure. When the EDA is of coboundary type, we show how a natural generalisation of the classical Yang-Baxter equation arises. The construction is illustrated with a selection of examples including some which embed Drinfel'd doubles and others that are not of this type.