# hqsmcf — Result In Brief

Project ID:
274032

Funded under:
FP7-PEOPLE

Country:
Switzerland

Domain:
Industry

## Simultaneous deformations in algebra and geometry

EU-funded mathematicians have demonstrated how simultaneous deformation problems are naturally governed by strongly homotopy Lie algebras and how such algebras can be constructed.

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The concept of deformation is pervasive in mathematics. It is used to study objects of a particular type by organising them into families and determining how they are related. In essence, this is the approach used for determining the moduli of an algebraic or a more general structure.

The deformation of various kinds of mathematical structures can be encapsulated in the language of strongly homotopy Lie algebras. Most of the time, obtaining the Lie algebra governing the given deformation problem can be difficult. There are known techniques to solve this problem based on operads.

Within the EU-funded project HQSMCF (Homotopy quantum symmetries, monoidal categories and formality), mathematicians developed further a technique proposed to produce Lie algebras out of simple concepts of graded linear algebra. This was adapted to the study of simultaneous deformations.

The HQSMCF team applied this relatively easier technique to old problems in algebra, recovering known results and also showed geometrical applications, which would not be possible otherwise. Specifically, they used it successfully on simultaneous deformations of various algebraic structures and their morphisms.

Strongly homotopy Lie algebras are associated with manifolds equipped with a closed form. To study Lie group actions on these manifolds, mathematicians introduced a theory of homotopy moment maps. Such maps are essentially morphisms from the Lie algebra of the group into higher Poisson Lie algebra.

Relationships were established with previous work in classical field theory, algebroid theory and differential geometry. Lastly, mathematicians used the theory developed to construct various strongly homotopy Lie algebras as higher central extensions of Lie algebras geometrically.

Along with progress in proving the formality of the strongly homotopy Lie algebras, HQSMCF established close collaborations between leading mathematicians. It is hoped that this network of European and overseas academic institutions will evolve beyond the end of the project.

The deformation of various kinds of mathematical structures can be encapsulated in the language of strongly homotopy Lie algebras. Most of the time, obtaining the Lie algebra governing the given deformation problem can be difficult. There are known techniques to solve this problem based on operads.

Within the EU-funded project HQSMCF (Homotopy quantum symmetries, monoidal categories and formality), mathematicians developed further a technique proposed to produce Lie algebras out of simple concepts of graded linear algebra. This was adapted to the study of simultaneous deformations.

The HQSMCF team applied this relatively easier technique to old problems in algebra, recovering known results and also showed geometrical applications, which would not be possible otherwise. Specifically, they used it successfully on simultaneous deformations of various algebraic structures and their morphisms.

Strongly homotopy Lie algebras are associated with manifolds equipped with a closed form. To study Lie group actions on these manifolds, mathematicians introduced a theory of homotopy moment maps. Such maps are essentially morphisms from the Lie algebra of the group into higher Poisson Lie algebra.

Relationships were established with previous work in classical field theory, algebroid theory and differential geometry. Lastly, mathematicians used the theory developed to construct various strongly homotopy Lie algebras as higher central extensions of Lie algebras geometrically.

Along with progress in proving the formality of the strongly homotopy Lie algebras, HQSMCF established close collaborations between leading mathematicians. It is hoped that this network of European and overseas academic institutions will evolve beyond the end of the project.