Differential Gerstenhaber Algebras Associated to Nilpotent Algebras

This article provides a complete description of the diﬀerential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Kählerian complex structures on six-dimensional nilpotent algebras whose diﬀerential Gerstenhaber algebra is quasi-isomorphic to that of the symplectic structure. In a weak sense of mirror symmetry, this gives a classiﬁcation of pseudo-Kähler structures on six-dimensional nilpotent algebras whose mirror images are themselves.

Keywords

Nilpotent algebra, Gerstenhaber algebra, complex structure, symplectic structure, deformation, mirror symmetry

2010 Mathematics Subject Classification

Primary 32G05. Secondary 13D10, 16E45, 17B30, 32G07, 53D45.

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Published 1 January 2008