Nonhomogeneous quadratic duality and curvature

By L. Positselski. 2014

50 citations recorded by the source corpus.

A quadratic algebra is a graded algebra with generators of degree 1 and relations of degree 2. Let A be a quadratic algebra with the space of generators V and the space of relations I ⊂ V ⊗ V . The classical quadratic duality assigns the quadratic algebra A with generators from V ∗ and the relations I ⊂ V ∗ ⊗ V ∗ to the algebra A. According to the classical results of Priddy and Löfwall [1, 3], A is isomorphic to the subalgebra of Ext∗A(k, k) generated by Ext 1 A(k, k). Priddy called an algebra A a Koszul algebra if this subalgebra coincides with the whole of Ext∗A(k, k). Koszul algebras constitute a wonderful class of quadratic algebras, which is closed under a large set of operations, contains the main examples, and perhaps admits a finite classification. In this paper, we propose an extension of the quadratic duality to the nonhomogeneous case. Roughly speaking, a nonhomogeneous quadratic algebra (or a quadratic-linear-scalar algebra, a QLS-algebra) is an algebra defined by (generators and) nonhomogeneous relations of degree 2. A quadratic-linear algebra (QL-algebra) is an algebra defined by nonhomogeneous quadratic relations without the scalar parts; in other words, it is an augmented QLS-algebra. The precise definition takes into account the fact that a collection of nonhomogeneous relations does not necessarily “make sense” (its coefficients must satisfy some equations; the Jacobi identity is a classical example). The dual object for a QL-algebra is [6] a quadratic DG-algebra [7]. The dual object for a QLS-algebra is a set of data which we call a quadratic CDG-algebra (“curved”), defined up to an equivalence. The classical Poincaré–Birkhoff–Witt theorem on the universal enveloping algebra structure [9] finds its natural place in this context as a particular case of the fact that every Koszul CDG-algebra corresponds to a QLS-algebra. Remarkable examples of nonhomogeneous quadratic duality are provided by Differential Geometry. The algebra of differential operators on a manifold may be considered as a QL-algebra defined by the commutation relations for the vector fields. The dual object for this algebra is the de Rham complex; the corresponding equivalence of the categories of modules is constructed in [4]. The algebra of differential operators in a vector bundle is a QLS-algebra. The dual object is the algebra of differential forms with coefficients in linear operators in this bundle and with the exterior differential defined by means of a connection. Its square is not zero—it is equal to the commutator with the curvature; the equivalence relation mentioned in the previous paragraph corresponds to changing the connection. Thus, the curvature corresponds to the scalar part of the relations. A question arises about the obstructions to existence of a QL-algebra structure (i.e., of “a flat connection”) on a given QLS-algebra. In the present paper we construct obstructions of this kind generalizing the Chern classes of vector bundles [8] and (which is less evident) the Chern–Weil classes of principal G-bundles. Our analogues of the secondary characteristic classes [5] form the Chern–Simons functor on the category of CDG-algebras.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e81ee996d4e0ac2ceee5dd412149c3154c114713

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Curved Koszul duality theory (2010, background): “Monoids Relations Homogeneous quadratic Quadratic and linear Quadratic, linear and constant Associative [33] [30,32] algebras Operads [6,7] Section 4 of this paper [8] Properads [35]”
Hochschild (co)homology of the second kind I (2010, background): “Expositions in the generality of CDG-rings, which might be somewhat more accessible to an inexperienced reader, can be found in [13, 15, 19].”