All Questions
Tagged with operads qa.quantum-algebra
19 questions
30
votes
7
answers
5k
views
Shuffle Hopf algebra: how to prove its properties in a slick way?
Let $k$ be a commutative ring with $1$, and let $V$ be a $k$-module. Let $TV$ be the $k$-module $\bigoplus\limits_{n\in\mathbb N}V^{\otimes n}$, where all tensor products are over $k$.
We define a $k$...
- 33.8k
21
votes
1
answer
2k
views
On Tamarkin's proof of Etingof-Kazhdan quantization of Lie bialgebra
This question is motivated by the desire to understand better the interplay between Drinfeld associators, algebraic structures up to homotopy and related results.
Recall that a Lie bialgebra is a Lie ...
- 8,524
20
votes
1
answer
1k
views
Are G_infinity algebras B_infinity? Vice versa?
What is the relationship between $G_\infty$ (homotopy Gerstenhaber) and $B_\infty$ algebras?
In Getzler & Jones "Operads, homotopy algebra, and iterated integrals for double loop spaces" (a paper ...
- 1,038
14
votes
0
answers
792
views
Splitting of homomorphism from cactus group to permutation group
We all learned in kindergarten that the category of finite-dimensional (type I, say) $U_q(\mathfrak{g})$-modules is braided monoidal for $\mathfrak{g}$ a complex semisimple Lie algebra. This gives an ...
- 8,559
12
votes
1
answer
946
views
Compatibility of the KZ connection with operadic composition
In what sense is the Knizhnik-Zamolodchikov connection compatible with the operadic composition in the little discs operad and/or the operad of $\overline{M}_{0,n}$'s?
Here are (some) details, ...
- 407
12
votes
0
answers
333
views
Does the $(\mathbb Z/2)$-graded isomorphism $E_n \cong E_{n+2}$ have any nice properties?
This question assumes everything is dg. Let's decide to work over the "field" $\mathbb Q[\mu,\mu^{-1}]$ where $\mu$ has homological degree $+2$. Then chain complexes are just $\mathbb Z/2$-graded. ...
- 54.6k
11
votes
1
answer
356
views
What is the relation between 2-Gerstenhaber, CohFT, and Gerstenhaber geometrically?
Background. As we know from Fred Cohen's Thesis, taking homology of the little 2-discs operad $\mathcal{D}_2$ with coefficients in a field of characteristic zero produces the Gerstenhaber operad $\...
- 1,981
8
votes
1
answer
578
views
Transmutation versus operads
A while ago, I was reading Majid's book Foundations of quantum group theory, and Section 9.4 has a rather fascinating description of a Tannaka-Krein reconstruction result for quantum groups. In ...
- 45.7k
8
votes
1
answer
561
views
Identifying the little disk operad with parenthesized braids
Let $D_2$ be the topological operad of little disks. This operad can be modelled "combinatorially" in terms of an operad of groupoids called $\newcommand{\PaB}{\mathbf{PaB}}\PaB$, the operad of ...
- 40.2k
8
votes
0
answers
199
views
When are the categories of algebras over props (co)complete?
Suppose P is a (colored) prop in a closed symmetric monoidal locally presentable category C. Is the category Alg_P of algebras over P in C locally presentable?
It seems that one can relatively easy ...
- 37.8k
7
votes
0
answers
432
views
What is the endomorphism cooperad?
In Loday and Vallette's book on algebraic operads, they use the "Endomorphism cooperad $End^c_{s\mathbb{K}}$", where $s\mathbb{K}$ is the base field, shifted into (homological) degree one. This is an ...
- 2,074
6
votes
1
answer
302
views
Knot Factorization Homology inputs
Following the paper by Ayala, Francis, and Tanaka: https://arxiv.org/pdf/1409.0848.pdf
If we are talking about knots we are talking about framed 3-manifolds with a framed 1-dimensional sub-manifold ...
- 527
6
votes
0
answers
353
views
Homotopy transfer of cyclic L-infinity algebras
Suppose $W$ is a cyclic $L_\infty$ algebra, i.e. $W$ has a non-degenerate, symmetric, invariant pairing $\langle\cdot,\cdot\rangle_W$. Let $V$ be a cochain complex, and suppose given the data of a ...
4
votes
1
answer
423
views
Formality of the little $n$-disks operad and deformation theory
In [Another proof of M. Kontsevich formality theorem], Tamarkin provides a proof of the formality of the differential graded Lie algebra controlling the deformation of a polynomial associative algebra....
4
votes
0
answers
205
views
Where is the Courant operad discussed?
Where is the Courant operad discussed? And hopefully defined precisely.
By the Courant operad or rather a suitable generalization of operad to accommodate the inner product, the operad whose ...
- 3,880
3
votes
1
answer
365
views
Brace algebra structure on the Hochschild complex of an associative algebra
As shown by Gerstenhaber and Voronov [Higher operations on the Hochschild complex], the Hochschild complex of an associative algebra is endowed with a natural structure of brace algebra. The first ...
3
votes
1
answer
277
views
What is the relation between cobar duality and Feynman transform
If $O$ is a cyclic operad, it can be regared as a modular operad $P$ with $P(g,n)=0$, for $g >0$. So we have cobar dual $BO$ and Feynman transform $FP$(with trivial cocycle). Is there any ...
- 31
3
votes
1
answer
272
views
Embedding e_n -> e_m
Let $e_n$ be the operad of (say, rational) chains of the operad of little n-disks. Consider the natural embedding $e_n\to e_m$, where $n<m$ and $n,m>1$ induced by the embedding $\mathbb{R}^n\to \...
- 743
1
vote
0
answers
185
views
Exact sequence of L-infinity-algebras
We call a sequence of $L_\infty$-algebras (weak) maps
$$0\to L\xrightarrow{f} M\xrightarrow{g} N\to 0$$
is exact if it is exact on the the underlying chain complexes level.
Thought I don't know ...
- 1,271