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A non-cofibrant dg operad whose homology is $\mathrm{Pois}$ and which looks like $\mathrm{Com}\circ\mathrm{Lie}_\infty$ appears in Section 4.1 of this paper of Anton Khoroshkin and Pedro Tamaroff, the …

To record my answer in comments properly: brace algebras coming from operads satisfy one obvious constraint: for every $x$ and sufficiently large $n$ we have $x\{x_1,\ldots,x_n\}=0$, since we cannot plug …

I am interested in the dimension, or, even better, in the $S_8$-module structure of the space of degree 8 multilinear operations on Jordan algebras. Recall that a Jordan algebra is a commutative but n …

I managed to run Albert on a very powerful computer at work, and the computation of the desired dimension converged: it seems equal to 19089. I would very much like to confirm that it is correct (I am …

This is a reference request. I came across the class of nonassociative algebras satisfying the following identity: $$ (a,b,c)+\omega(b,c,a)+\omega^2(c,a,b)=0. $$ Here: by an "algebra" I mean a vect …

In other words, the (co)associativity for (co)operads has a sequential axiom and a parallel axiom, and these terms vanish because of the parallel axiom. …

I don't know if you found an answer since you posted the question, but I will write this just in case: there is a "cute" (easy) definition in case of nonsymmetric operads which generalises the A-infinity … story rather trivially (derivation homotopy), while for symmetric operads the definition is more involved. …

Well, for your question 1 you presumably may ask yourself first about a relationship between (shifted) Lie algebras and Lie 2-algebras. Lie 2-algebras of Hanlon and Wachs can be viewed as $L_\infty$-a …

One possible answer is contained in the paper of Victor Ginzburg and Travis Schedler, "Free products, cyclic homology, and Gauss-Manin connection", https://arxiv.org/abs/0803.3655. You will be in part …

For $K=\mathbb{R}$, the positive part of your operad (mentioned in Gabriel's comment), and its algebras have been discussed by Tom Leinster and others in connection to entropy. See, for example, http …

To add a bit to what Damien says, addressing your question on how to generalise the gauge approach (which is equivalent to the approach outlined by Damien, as proved by several people): You can view …

Depending on whether you want it to agree with the symmetric structure or only with monoidal structure, this would be usually referred to, respectively, as twisted commutative algebras or twisted asso …

Consider the unique formal power series $g(z)$ with $g(0)=0$ and $g'(0)=1$ satisfying the equation $$ g(z)-g(z)^8+g(z)^{15}=z, $$ that is the inverse of $$ z-z^8+z^{15} $$ in the group of formal …

Let us denote by $p\colon Y\to X$ and $i\colon X\to Y$ the maps of your SDR. Since $pi=\mathop{\mathrm{id}}\nolimits_X$, the map $i$ is injective, and is an isomorphism with its image. The map $\pi=i\ …

(Alternatively, one can use the language of operads to discuss that; I prefer that latter language but choose to write a more "classical" definition here). …