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Solutions of the Yang-Baxter equation: groups, algebras and braces

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Arne van Antwerpen
Vrije Universiteit Brussels
Tue, 08/11/2022 - 14:00 - 15:00
MPIM Lecture Hall

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Contact: Gaetan Borot (


The Tate--Hochschild complex is a complex stitched together from Hochschild homology and cohomology of an associative Frobenius algebra.It appears naturally in the study of singularities and in representation theory. There are known operations on this complex which are extensions the cup product,Gerstenhaber bracket and their duals which include the Goresky-Hingston (co)product,the existence of which is already non-trivial. There are also mixed products, whichyield and m_3 multiplication which is part of an A_infty structure with all m_i= 0 for i >4, aswas shown by Rivera and Wang.

Together with Rivera and Wang, we show that these operations are part of a universal family of operations obtained analogously as the operations on the Hochschild complex we previously defined. This allows us to identify a series of higher bracket operations of which the bi-bracket is dual to the m_3 operation and the tri-bracket guarantees the higher associativity. Other operations guarantee the Poisson property of the bi-bracket. We will introduce this formalism and comment on how this leads to a new type of bordification of the Chas-Sullivan string topology space.

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