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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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