# Koszul duality for Operads

@article{Ginzburg1994KoszulDF, title={Koszul duality for Operads}, author={Victor Ginzburg and Mikhail M. Kapranov}, journal={arXiv: Algebraic Geometry}, year={1994} }

(0.1) The purpose of this paper is to relate two seemingly disparate developments. One is the theory of graph cohomology of Kontsevich [Kon 2 3] which arose out of earlier works of Penner [Pe] and Kontsevich [Kon 1] on the cell decomposition and intersection theory on the moduli spaces of curves. The other is the theory of Koszul duality for quadratic associative algebras which was introduced by Priddy [Pr] and has found many applications in homological algebra, algebraic geometry and…

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## 772 Citations

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## References

SHOWING 1-10 OF 58 REFERENCES

The discrete series of GLn over a finite field

- Mathematics
- 1974

In this book Professor Lusztig solves an interesting problem by entirely new methods: specifically, the use of cohomology of buildings and related complexes. The book gives an explicit construction…

Duality for representations of a reductive group over a finite field

- Mathematics
- 1982

The purpose of this paper is to construct a duality operation for representations of a reductuve group over a finite field. Its effect, very roughly speaking, is to interchange irreducible…

Chow quotients of Grassmannian I

- Mathematics
- 1992

We introduce a certain compactification of the space of projective configurations i.e. orbits of the group $PGL(k)$ on the space of $n$ - tuples of points in $P^{k-1}$ in general position. This…

New perspectives on the BRST-algebraic structure of string theory

- Mathematics, Physics
- 1993

Motivated by the descent equation in string theory, we give a new interpretation for the action of the symmetry charges on the BRST cohomology in terms of what we callthe Gerstenhaber bracket. This…

Batalin-Vilkovisky algebras and two-dimensional topological field theories

- Physics, Mathematics
- 1994

By a Batalin-Vilkovisky algebra, we mean a graded commutative algebraA, together with an operator Δ:A⊙→A⊙+1 such that Δ2 = 0, and [Δ,a]−Δa is a graded derivation ofA for alla∈A. In this article, we…

Topological invariants of algebraic functions II

- Mathematics
- 1970

In an earlier paper of the same name (2] the connection between algebraic functions and braids was used in order to calculate the cohomologies of braid groups. In this work the cohomologies of braid…

The Lie algebra structure of tangent cohomology and deformation theory

- Mathematics
- 1985

Abstract Tangent cohomology of a commutative algebra is known to have the structure of a graded Lie algebra; we account for this by exhibiting a differential graded Lie algebra (in fact, two of them)…

Infinite Loop Spaces

- Mathematics
- 1978

The theory of infinite loop spaces has been the center of much recent activity in algebraic topology. Frank Adams surveys this extensive work for researchers and students. Among the major topics…

The geometry of iterated loop spaces

- Mathematics
- 1972

Operads and -spaces.- Operads and monads.- A? and E? operads.- The little cubes operads .- Iterated loop spaces and the .- The approximation theorem.- Cofibrations and quasi-fibrations.- The smash…

Rational homotopy theory

- Mathematics
- 1969

For i ≥ 1 they are indeed groups, for i ≥ 2 even abelian groups, which carry a lot of information about the homotopy type of X. However, even for spaces which are easy to define (like spheres), they…