18
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Does Koszul duality between $Comm$ and $Lie$ imply the power series identity $\exp(\ln(1-z))-1 = -z$?
Let me flesh out the answer a little. The general statement is given by Theorem 7.5.1 in the book Algebraic Operads by Loday and Vallette.
First a definition. Let $P = P(E,R)$ be a quadratic operad, ...
- 5,040
13
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Does Koszul duality between $Comm$ and $Lie$ imply the power series identity $\exp(\ln(1-z))-1 = -z$?
Yes. (Mathoverflow won't let me make this my total answer, so ...)
Koszul duality says that a certain chain complex of graded vector spaces is acyclic. Thus the alternating sum of the Poincare ...
- 11.1k
11
votes
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Koszulness of the cohomology ring of moduli of stable genus zero curves
It is: https://arxiv.org/abs/1902.06318 - this paper also explains how to use the Koszul dual algebra for something, where something is estimating Betti numbers of the free loop spaces of $\overline{M}...
- 16.9k
10
votes
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Is algebra: ac=ca, bd = db , ad - da = cb - bc ("Manin matrix algebra") - a Koszul algebra?
If I understand correctly Theorem 4.1.1 of https://www.math.univ-paris13.fr/~vallette/Operads.pdf then the answer is "yes".
I'll take the order $a<b<c<d$ and use lex ordering like ...
- 10.7k
10
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Which finite posets are Koszul self-dual?
Elaborating on Richard's suggestion, I think that it is necessary (but don't know about sufficiency) for the poset $P$ to be Gorenstein* over the field $k$ used in defining the incidence algebra $A_P$....
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7
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Does Manin's construction of non-commutative endomorphism algebra $\mathrm{End}(A)$ produce Koszul algebra, if $A$ is Koszul?
Q1: This algebra is just the Manin black product of $A$ and $A^!$ (in other words, the Koszul dual of the Segre product of $A$ and $A^!$), and hence it is Koszul. (As requested, the Segre product of ...
- 16.9k
5
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Strange formulas that gave rise to Koszul duality
No. If the left-hand side is in the singular block, the right hand-side should be in the regular parabolic block, i.e. its Koszul dual.
(The regular block for the Borel subalgebra is Koszul-self-dual....
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5
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What are Koszul dualities?
I have finally found a source which puts together the pieces in a satisfactory way, at least in the stable setting, here:
Amabel, Araminta. "Poincaré/Koszul Duality for General Operads." ...
5
votes
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Infinity-homotopies
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 ...
- 16.9k
4
votes
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Augmented algebras over semisimple ring
This is the set-up of Beilinson, Alexander; Ginzburg, Victor; Soergel, Wolfgang, Koszul duality patterns in representation theory, J. Am. Math. Soc. 9, No. 2, 473-527 (1996). ZBL0864.17006.
It has ...
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4
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Strange formulas that gave rise to Koszul duality
This is a more concrete explanation for what Rafael Mrđen had said.
Let $x,w\in W^\Sigma$ and $\ell(x,w)=\ell(w)-\ell(x)$.
It is well-known that $\mathrm{ch}L(w\cdot\mu)=\sum_{x\in W^\Sigma}(-1)^{\...
- 1,875
3
votes
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Derived Koszul complex
A slight generalization of "usual" Koszul complex is summarized in Illusie's thesis Complexe Cotangent et Deformations I, 4.3.1.2 & 4.3.1.3 & 4.3.1.6 (where the result is attributed ...
- 2,806
3
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What bigrading is used in this spectral sequence?
This is simply a summary of the comments. The problem was I did not raise the index $p$ throughout.
Raise the index in $F_p\Omega^n$ to get a filtration $F^p\Omega^n$. The spectral sequence is now ...
- 1,554
3
votes
Tensor products of $\infty$-algebras over operads
If you know the statement for strict algebras, then you "know" the statement for $\infty$-algebras by general nonsense. For example, the fact that a Lie algebra tensored with a commutative algebra ...
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3
votes
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Differential of the Twisted complex for algebraic operads
Since $\alpha$ is of degree $-1$, these terms come in pairs appearing with opposite signs that cancel each other. In other words, the (co)associativity for (co)operads has a sequential axiom and a ...
- 16.9k
3
votes
Koszul duality and $\operatorname{H}^*(BG)$ — concise proof?
I am posting this as an answer because it is a bit long for a comment. Questions close to this one have appeared before on MathOverflow. You ask specifically about Koszul duality, but for $k=\mathbb{...
3
votes
Is there something "Koszul dual" to formal groups?
I'm not sure what you mean by "elaboration" since the category of formal groups is equivalent to that of Lie algebras. There is however a way Koszul duality can enter this story (I'm not an expert so ...
- 8,524
2
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An exercise from Loday and Vallette about Koszul morphism
Recall: $\alpha$ being Koszul means that $C\otimes_\alpha A$ is acyclic, menaning that the augmentation map $C\otimes_\alpha A\overset{\epsilon\otimes\epsilon}{\longrightarrow}\mathbb{K}$ is a quasi-...
- 8,385
1
vote
Homology and cohomology of free loop spaces
I'll address your question for a reason why the Hochschild homology $HH_*(C^*(M),C^*(M))$ should have a cup product, without resorting to the fact that it is the cohomology of a space. We assume ...
- 5,829
1
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Koszul duality between Weyl and Clifford algebras?
If Koszul dual is used for computing resolutions (for $k$ as $A$-module or for $A$ as $A$-bimodule), you can use this idea to find a small resolution for the Weyl algebra. In this way we computed ...
- 1,028
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