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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 …

Let me say that since you are interested in square-free elements where the weight is equal to the number of generators, you actually are asking questions about multilinear elements, that is elements o …

Even though one may argue that it is not stepping too much outside the area, in a sense, you might want to look at the paper of Xiaojun Chen, Wee Liang Gan, https://arxiv.org/abs/0804.4748.

For the algebro-geometric setting, that is polynomial automorphisms (and conjugacy in the group of polynomial automorphisms), I recall a talk and several surveys by Hanspeter Kraft where it was stated …

The structure is just that of a graded Lie ring (homologically graded - to create the correct Koszul signs), once you shift degrees by 1. This structure is not at all exotic, you see it in Gerstenhabe …

The answer to this question may be well-known, but I failed to locate it in any obvious source. From the results of Bott (Ann. Math. 66 (1957), 203-248) and Kostant (Ann. Math. 74 (1961), 329-387), it …

I would suggest the two following references: C.-F. Bödigheimer, F.R. Cohen, Rational cohomology of configuration spaces of surfaces. Algebraic Topology and Transformation Groups, Springer LNM 1361 ( …

This is not true. Consider the algebra $A=T(V)/V^{\otimes 2}$, it is a commutative algebra whose augmentation ideal has zero multiplication. We have $\mathrm{Tor}_A(k,k)\cong T(V[1])$ with the shuffle …

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\ …

To prove that the Hilbert series (the generating function of the sequence of dimensions of homogeneous components) of a finitely generated commutative graded algebra is a rational function, the easies …

If you require just "not homeomeorphic", then there are very silly examples of all sorts. What you want to ask is "not homotopic", I suppose. For that, I know some useful references in "Which H-spac …

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 …

Superalgebras have been used in various questions of algebra in a very striking way. To give some instances: Kemer's proof of the fact that, over a field of zero characteristic, every system of ident …