Explicit L∞L_\infty-operations on Hochschild cochains of A∞A_\infty-algebra

It is well-known that the Hochschild cochain complex CC∗(A) of an associative algebra A carries a lot of structure. In particular: a differential, a cup product, and a bracket, which make the Hochschild cohomology HH∗(A) into a Gerstenhaber algebra. For concretenes, I have in mind that we’re using the standard bar complex model for CC∗ … Read more

Homotopy transfer of cyclic L-infinity algebras

Suppose W is a cyclic L∞ algebra, i.e. W has a non-degenerate, symmetric, invariant pairing ⟨⋅,⋅⟩W. Let V be a cochain complex, and suppose given the data of a strong deformation retraction (i,p,k) of W onto V, i.e. p is a cochain map W→V, i is a cochain map V→W, pi is the identity on … Read more

What is the correct definition of weak map between 2-term $L_\infty$-algebras?

The definition of $L_\infty$-algebra is by now pretty standard. I gather that the sign conventions given in Lada–Markl’s paper Strongly homotopy Lie algebras, Communications in Algebra 23 Issue 6 (1995) (arXiv:hep-th/9406095) are widely used, and I will keep to them here. I will not rehash the definition of $L_\infty$-algebra, because I’m sure the people who … Read more

What is a homotopy between L∞L_\infty-algebra morphisms

A L∞-algebra can be defined in many different ways. One common way, that gives the ‘right’ kind of morphisms, is that a L∞-algebra is a graded cocommutative and coassociative coalgebra, cofree in the category of locally nilpotent differential graded coalgebras and their morphisms are coalgebra morphisms that commute with the codifferential. Breaking this compact definition … Read more