1-2-3 Seminar: Koszul Duality, Operads, and Young Tableaux

Cody Tipton, University of Washington
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PDL C-401

One of the first examples of Koszul duality in the world of quadratic algebras is the duality between the symmetric algebras S(V) and the exterior algebra \Lambda(V^*).  This arises from taking relations in S(V) and finding the orthogonal complement with respect to some non-degenerate bilinear form.  Since these algebras are Koszul (having some minimal graded resolution) then there is some equivalence between some subcategories of their derived categories.  We will first introduce the conept of Koszul duality in the realm of algebras and describe some important examples.
Next, we will introduce the extension of Koszul duality in the world of operads and give a few implications of the relationships between their respective algebras, specifically to the deformation theories of them.  The most important examples of Koszul duality will come from the operads Ass, Com, and Lie, the operads encoding associative, commutative associative, and Lie algebras respectively. In particular, the algebra Ass is Koszul dual to itself and Com is Koszul dual to Lie.
Finally, we will introduce the two operads I have been working with, which are proved to be Koszul dual.  Specifically, we will talk about the operad n-Lie_d, the operad encoding n-Lie algebras of degree d, and its Koszul dual n-Com_{-d+n-2}, the operad encoding algebras with n-arity operations of degree -d+n-2 that satisfy relations coming from a Young tableaux of shape (n,n-1).