A sigma, or skew, derivation is a natural generalisation of the notion of derivation depending on an algebra automorphism $\sigma$ which when equal to $id = \sigma$ reduces to the usual notion of a derivation. For a precise definition see here


Does there exist a notion of skew differential graded algebra in the literature? If so where do these objects arise?

EDIT: To confirm I am asking if there exists a graded analogue of skew derivation algebra. So an $\mathbb{N}_0$-graded algebra $A = \bigoplus_{k \in \mathbb{N}_)0} A_k$, together with a degree $1$ map $d$ satisfying $d^2 = 0$, and a skew analogue of the graded Leibniz rule: $$ d(a \wedge b) = da \wedge \sigma(b) + (-1)^k \sigma(a)db, ~ a \in A_k $$

  • $\begingroup$ @Najib: I have edited the question to include a definition of the object I am wondering about. $\endgroup$ Commented Apr 21, 2020 at 15:18
  • $\begingroup$ Is this the kind of construction you're interested in? perso.univ-rennes1.fr/bernard.le-stum/Publications_files/… $\endgroup$ Commented Apr 21, 2020 at 15:54
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    $\begingroup$ Is your wedge graded (anti)commutative? If so then this $d$ seems poorly defined. $\endgroup$ Commented Apr 21, 2020 at 16:20
  • $\begingroup$ @Alex: Yes, it is assumed to be anti-commutative. I have adjusted the ansatz. $\endgroup$ Commented Apr 22, 2020 at 11:54
  • $\begingroup$ @NicolaCiccoli: The same but with functioning references: arxiv.org/abs/1503.05022v1 $\endgroup$ Commented Apr 22, 2020 at 12:02

1 Answer 1


This edited version of the "skew Leibniz rule" has appeared in geometry: if $\varphi: N\to M$ is a map of (super) manifolds, a section $X$ of the pullback bundle $\varphi^\star TM$ is a linear map $C^\infty(M)\to C^\infty(N)$ satisfying precisely your skew-leibniz rule: $$ X(fg)=X(f)\varphi^\star g +(-1)^{\deg X\deg f} (\varphi^\star f) X(g)$$

(See J. Nestruev, Smooth manifolds and observables, paragraph 9.47.)

So if you put $\mathbb N_0$-gradings on your structure sheaves (what is $\mathbb N_0$? Positive integers?) and pick an $X$ of degree $+1$ which squares to zero you seem to arrive at your setup. (Identifying $\varphi^\star$ with $\sigma$ and $C^\infty(M)$ with $A$.)

On the algebraic side there is the (comparatively more obscure) notion of $({\bf l},{\bf r})$-coderivation of Berglund (Definition 3.2) which should be dual to your proposal in the case ${\bf l}={\bf r}=\sigma$.

  • $\begingroup$ Just like any normal person, I write $\mathbb{N}_0 := \{0,1,2,3,\dots\}$. $\endgroup$ Commented Apr 24, 2020 at 11:51
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    $\begingroup$ @JakeWetlock; Oh I've never seen that before. In that case supermanifolds with N_0 gradings on their structure sheaf are called "N-manifolds", see the original AKSZ paper $\endgroup$ Commented Apr 24, 2020 at 12:41

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