On the (3, N) Maurer-Cartan equation

Abstract

Deformations of the 3-differential of 3-differential graded algebras are controlled by the (3,N) Maurer-Cartan equation. We find explicit formulae for the coefficients appearing in that equation, introduce new geometric examples of N-differential graded algebras, and use these results to study N Lie algebroids.Comment: 21 page

On N-differential graded algebras

“…This implies that d N A (a) ∈ Ker( f ) = 0, and therefore d N A (a) = 0 which is in contradiction with the fact that (A • , d A ) is a proper M-complex. The proof of (2) is analogous to (1), (3) follows from(1)and(2).…”

“…Examples of N -dga's coming from differential geometry are developed in [1]. A q-analogue, for q a primitive N -th root of unity, of our main result Theorem 19 is provided in [2]. In [3] we state an N -generalized Deligne's principle and use the constructions of this paper to study A ∞ -algebras of depth N .…”

On the (3, N ) Maurer-Cartan equation

“…A choice, introduced first by Kerner in [20,21] and further studied by Dubois-Violette [13,14] and Kapranov [18], is to fix a primitive N -th root of unity q and define a q-differential graded algebra A to be a Zgraded associative algebra together with a linear operator d : A −→ A of degree one such that d(ab) = d(a)b + qāad(b) and d N = 0. There are several interesting examples and constructions of q-differential graded algebras [1,2,6,8,9,15,16,19,21].…”