Dual object

Abstract: We construct a duality transform for the coalgebra of the free difference quotient derivation-comultiplication of an operator with respect to a free algebra of scalars. The dual object is realized in an algebra of matricial analytic functions endowed with yet another generalization of the difference quotient derivation.

Abstract: In this work we generalize the entanglement of purification and its conjectured holographic dual to conditional and multipartite versions of the same, where the optimization defining the entanglement of purification is now optimized in either a constrained way or over multiple parties. We separately derive new constraints on both the conditional entanglement of purification and its conjectured holographic dual object that match, further reinforcing the likelihood of this conjecture. We also show that the multipartite objects we define, despite obeying several of the same inequalities, are not holographic duals of each other. Further, we find inequalities that are true only for the bulk objects and thus could provide additional consistency checks for states dual to (semi-)classical bulk geometries.

Abstract: 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 V of generators and the space I C V ® V of relations. The classical quadratic duality assigns a quadratic algebra A ! with generators from V* and relations I ± C V* ® V* to the algebra A. According to the classical results of Priddy and L6fwall [1, 3], A" is isomorphic to the subalgebra of Ext,4 (k, k) generated by Ext'4 (k, k). Priddy called an algebra A a Koszul algebra if that subalgebra coincides with the whole of Ext~ (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 possibly 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-aIgebra) is an algebra defined by nonhomogeneous quadratic relations without scalar parts; in other words, it is ~n augmented QLS-aigebra. In the precise definition we take into account the fact that a collection of nonhomogeneous relations does not necessarily "makes 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 Poincar6-Birkhoff-Witt theorem on the universal enveloping algebra structure [9] attains its natural place in this context as a particular case of the fact that every KoszuI CDG-aIgebra 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 commutation relations for vector fields. The dual object for this algebra is the De Rham complex; in [4] the corresponding equivalence of the categories of modules is constructed. The algebra of differential operators on a vector bundle is a QLS-algebra. The dual object is the algebra of differential forms with coefficients in linear operators on this bundle and with exterior differential defined by means of a connection. Its square is nonzero and it is equal to the commutator with the curvature of the connection; the equivalence mentioned in the previous paragraph corresponds to the change of a connection. Thus, the curvature corresponds to the scalar part of relations. There arises a problem on obstructions to the existence of a QL-algebra s~ructure (i.e., of "a flat connection" ) on a QLS-algebra. In the present paper we construct obstructions of this kind which generalize Chern classes of vector bundles [8] and (this is less evident) Chern-Weil classes of principal G-bundles. Our analogs of the secondary characteristic classes [5] form the Chern-Simons functor on the category of CDG-algebras. The author is grateful to A. B. Astashkevich, R. V. Bezrukavnikov, M. V. Finkel~berg, V. A. Ginzburg, A. E. Polishchuk, and V. S. Retakh for numerous and very useful discussions and to A. A. Kirillov and A. N. Rudakov for their constant attention to the work.

Abstract: The double Schur functions form a distinguished basis of the ring \Lambda(x||a) which is a multiparameter generalization of the ring of symmetric functions \Lambda(x). The canonical comultiplication on \Lambda(x) is extended to \Lambda(x||a) in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood-Richardson coefficients in two different ways thus providing comultiplication rules for the double Schur functions. We also prove multiparameter analogues of the Cauchy identity. A new family of Schur type functions plays the role of a dual object in the identities. We describe some properties of these dual Schur functions including a combinatorial presentation and an expansion formula in terms of the ordinary Schur functions. The dual Littlewood-Richardson coefficients provide a multiplication rule for the dual Schur functions.