# $T$-duality on nilmanifolds

## TL;DR

This paper explores the mathematical framework of $T$-duality on nilmanifolds, focusing on Lie algebras, generalized complex structures, and criteria for extending infinitesimal duality to topological cases.

## Contribution

It introduces the concept of infinitesimal $T$-duality on Lie algebras and provides criteria for its realization as topological $T$-duality on nilmanifolds.

## Key findings

- Constructed the Cavalcanti-Gualtieri map for Lie algebras.
- Provided criteria for integrability of infinitesimal $T$-duality.
- Analyzed symplectic structures on 2-step nilpotent Lie algebras.

## Abstract

We study generalized complex structures and $T$-duality (in the sense of Bouwknegt, Evslin, Hannabuss and Mathai) on Lie algebras and construct the corresponding Cavalcanti and Gualtieri map. Such a construction is called "Infinitesimal $T$-duality". As an application we deal with the problem of finding symplectic structures on 2-step nilpotent Lie algebras. We also give a criteria for the intregability of the infinitesimal $T$-duality of Lie algebras to topological $T$-duality of the associated nilmanifolds.