nLab
Koszul duality

Koszul duality is a duality and phenomenon generalizing the duality between the symmetric and exterior algebra of a vector space to so-called quadratic differential graded algebras (which can be obtained as a free dga module an ideal of relations which live in degree 2). For a pair of Koszul dual algebras, there is a correspondence between certain parts of their derived categories (precise formulation involves some finiteness conditions). In a setup in which one of the algebras is replaced by a cocomplete dg coalgebra, there is a formulation free of finiteness conditions, but involving twisting cochain (see that entry).

Informal discussion

There are a lot of algebras whose derived categories are equivalent in surprising ways. Morita equivalences are pretty simple, especially for finite dimensional algebras; essentially the only free parameter is the dimension of the object.

Namely, if $A\mathrm{Mod}$ and $B\mathrm{Mod}$ are equivalent, then the image of $A$ as a module over itself is a projective generator of $B\mathrm{Mod}$, and for a finite-dimensional algebra, essentially the only thing you can do is take several copies of the indecomposible? projectives of $B$.

On the other hand, if you take the derived category of dg-modules over $A$ (the dg part of this is not a huge deal; it’s just that they’re very close to, but a bit better behaved than, actual derived/triangulated categories, which are just crude truncations of truly funtorial dg/$A_{\mathrm{\infty }}$ versions), this is equivalent to the category of dg-modules over the endomorphism algebra? (this is in the dg sense, so it’s a dg-algebra whose cohomology is the $\mathrm{Ext}$ algebra) of any generating object. There are a lot more generating objects than projective generators, so there are a lot of derived equivalences.

In particular, let $A$ be a finite dimensional algebra $A$, than an obvious not-very-projective generating object is the sum of all the simple modules, say $L$. As mentioned above, there’s an equivalence $A\mathrm{dg}\mathrm{Mod}=\mathrm{Ext}(L,L)\mathrm{dg}\mathrm{Mod}$, just given by taking $\mathrm{Ext}(L,-)$.

In general, $\mathrm{Ext}(L,L)$ is in a very complicated object (for example, often for group algebras over finite field?s), but sometimes it turns out to be nice. For example, if $A$ is an exterior algebra, you’ll get a polynomial ring on the dual vector space. Another (closely related) example is that the cohomology of a reductive group? (over $ℂ$) is Koszul dual to the cohomology of its classifying space.

One way to ensure that $\mathrm{Ext}(L,L)$ is nice is if the algebra $A$ is graded. Then $\mathrm{Ext}(L,L)$ inherits an “internal” grading in addition to its homological one. If these coincide, then $A$ is called Koszul.

In this case, $B=\mathrm{Ext}(L,L)$ is forced to be formal (if it had any interesting $A_{\mathrm{\infty }}$ operations, they would break the grading), so you’re dealing with a derived equivalence between actual algebras, though you have to be a bit careful about the dg-issues. Thus the derived category of usual modules over $A$ is equivalent to dg-modules over $B$ (with its unique grading) and vice versa. This can be fixed by taking graded modules on both sides.

Examples

• The most famous example of Koszul dual algebras are the exterior algebra ${\mathrm{Alt}}^{•}V[-1]$ and the polynomial algebra ${\mathrm{Sym}}^{•}{V}^{*}[-2]$.

• A regular block of category O? for any semisimple Lie algebra is a self-Koszul dual.

  • More generally, a singular block of parabolic category O? is dual to a different singular block of parabolic category O where the combinatorial data determining the central character and finiteness conditions switch.

• Braden, Licata, Proudfoot and Webster gave a combinatorial construction of a large family of Koszul dual algebras in Gale duality and Koszul duality.

Generalization to operads

There is a further generalization to quadratic operads, predicted in

• M. Kontsevich, Formal (non)commutative symplectic geometry. The Gelʹfand Mathematical Seminars, 1990–1992, 173–187, Birkhäuser Boston, Boston, MA, 1993.

and developed in

• V. Ginzburg, M. Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), no. 1, 203–272 (arXiv)

Other references

Other historical references on Koszul duality include

• A. A. Beĭlinson, V. A. Ginsburg, V. V. Schechtman, Koszul duality. J. Geom. Phys. 5 (1988), no. 3, 317–350.

• A. Beilinson, V. Ginzburg, W. Soergel, Koszul duality patterns in representation theory. J. Amer. Math. Soc. 9 (1996), no. 2, 473–527.

Koszul duality is also discussed in John Baez’ This Week’s Finds in Mathematical Physics:

• Week 238
• Week 239