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A geometric realization of \(\mathfrak{sl}(6,\mathbb C)\). (English) Zbl 1210.53085
Summary: Given an orientable weakly self-dual manifold \(X\) of rank two (see the second author, Asian J. Math. 9, No. 1, 79–101 (2005; Zbl 1085.14035)), we build a geometric realization of the Lie algebra \(\mathfrak{sl}(6,\mathbb C)\) as a naturally defined algebra \(L\) of endomorphisms of the space of differential forms of \(X\). We provide an explicit description of Serre generators in terms of natural generators of \(L\). This construction gives a bundle on \(X\) which is related to the search for a natural gauge theory on \(X\). We consider this paper as a first step in the study of a rich and interesting algebraic structure.

MSC:
53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
17B20 Simple, semisimple, reductive (super)algebras
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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