Algebras and modules
Model category presentations
Geometry on formal duals of algebras
The framed little 2-disk operad is the operad in Top whose topological space of -ary operations is the space of maps
from -copies of the 2-ball to itself, which restrict on each component to a map that is a combination of
of the disk (regarded via its standard embedding into the 2-simensional Cartesian space) such that the images of all disks are disjoint.
The homology of the framed little 2-disk operad in chain complexes is the BV-operad the operad for BV-algebras:
This is due to (Getzler).
The framed little disk operad is formal.
This means that there is a zig-zag of quasi-isomorphisms
This is due to (Giansiracusa-Salvatore-Severa 09)). See also (Valette, slide 35).
Accordingly one makes the following definition:
The operad for homotopy BV-algebras is any cofibrant resolution of , or equivalently of .
Write for the ribbon braid group? on elements and for the kernel of the surjection onto the symmetric group.
Say that a ribbon operad? is an -operad if the ribbon braid group?s act freely and properly on and if each topological space is contractible.
If is an -operad, then the sequence of quotient spaces forms a symmetric operad equivalent to the frame little disks operad.
This is (Wahl, lemma 1.5.17).
algebraic deformation quantization
The framed little 2-disk operad was introduced in
- Ezra Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories , Comm. Math. Phys. 159 (1994), no. 2, 265–285. (arXiv)
For the relation to ribbons see
- Nathalie Wahl, Ribbon braids and related operads PhD thesis, Oxford (2001) (pdf).
The formality of was shown in
Discussion of homotopy BV-algebras is in
Slides of a talk summarizing this are at
- Bruno Valette?, Homotopy Batalin-Vilkovisky algebras (pdf)