Mathematics

Given a tame differential calculus over a noncommutative algebra A and an A -bilinear pseudo-Riemannian metric g 0 , consider the conformal deformation g=k. g 0 , k being an invertible element of A. We prove that there exists a unique connection ??on the bimodule of one-forms of the differential calculus which is torsionless and compatible with g. We derive a concrete formula connecting ??and the Levi-Civita connection for the pseudo-Riemannian metric g 0 . As an application, we compute the Ricci and scalar curvature for a general conformal perturbation of the canonical metric on the noncommutative 2 -torus as well as for a natural metric on the quantum Heisenberg manifold. For the latter, the scalar curvature turns out to be a negative constant.

In this paper we use Lie conformal algebras to realize some moonshine type VOAs, whose Greiss algebras are Jordan algebras. On the other hand, we consider some free fields which realizes the corresponding simple VOAs. As an application, we can calculate the correlation functions of these VOAs in a relatively easy way.

The first two sections of the paper provide a convenient scheme and additional diagrammatics for working with Frobenius extensions responsible for key flavors of equivariant SL(2) link homology theories. The goal is to clarify some basic structures in the theory and propose a setup to work over sufficiently non-degenerate base rings. The third section works out two related SL(2) evaluations for seamed surfaces.

We describe explicitly the vertex algebra of (twisted) chiral differential operators on certain nilmanifolds and construct their logarithmic modules. This is achieved by generalizing the construction of vertex operators in terms of exponentiated scalar fields to Jacobi theta functions naturally appearing in these nilmanifolds. This provides with a non-trivial example of logarithmic vertex algebra modules, a theory recently developed by Bakalov.

We apply the construction of the universal lower-bounded generalized twisted modules by the author to construct universal lower-bounded and grading-restricted generalized twisted modules for affine vertex (operator) algebras. We prove that these universal twisted modules for affine vertex (operator) algebras are equivalent to suitable induced modules of the corresponding twisted affine Lie algebra or quotients of such induced modules by explicitly given submodules.

We describe the embedding from the crystal of Kashiwara-Nakashima tableaux in type D of an arbitrary shape into that of i -Lusztig data associated to a family of reduced expressions i which are compatible with the maximal Levi subalgebra of type A . The embedding is described explicitly in terms of well-known combinatorics of type A including the Schützenberger's jeu de taquin and an analog of RSK algorithm.

In the formulation of his celebrated Formality conjecture, M. Kontsevich introduced a universal version of the deformation theory for the Schouten algebra of polyvector fields on affine manifolds. This universal deformation complex takes the form of a differential graded Lie algebra of graphs, denoted fGC 2 , together with an injective morphism towards the Chevalley-Eilenberg complex associated with the Schouten algebra. The latter morphism is given by explicit local formulas making implicit use of the supergeometric interpretation of the Schouten algebra as the algebra of functions on a graded symplectic manifold of degree 1 . The ambition of the present series of works is to generalise this construction to graded symplectic manifolds of arbitrary degree n≥1 . The corresponding graph model is given by the full Kontsevich graph complex fGC d where d=n+1 stands for the dimension of the associated AKSZ type σ -model. This generalisation is instrumental to classify universal structures on graded symplectic manifolds. In particular, the zeroth cohomology of the full graph complex fGC d is shown to act via Lie ∞ -automorphisms on the algebra of functions on graded symplectic manifolds of degree n . This generalises the known action of the Grothendieck-Teichmüller algebra grt 1 ≃ H 0 ( fGC 2 ) on the space of polyvector fields. This extended action can in turn be used to generate new universal deformations of Hamiltonian functions, generalising Kontsevich flows on the space of Poisson manifolds to differential graded manifolds of higher degrees. As an application of the general formalism, new universal flows on the space of Courant algebroids are presented.

We give a general definition of Manin matrices for arbitrary quadratic algebras in terms of idempotents. We establish their main properties and give their interpretation in terms of the category theory. The notion of minors is generalised for a general Manin matrix. We give some examples of Manin matrices, their relations with Lax operators and obtain the formulae for some minors. In particular, we consider Manin matrices of the types B, C and D introduced by A. Molev and their relation with Brauer algebras. Infinite-dimensional Manin matrices and their connection with Lax operators are also considered.

We show that any pivotal Hopf monoid H in a symmetric monoidal category C gives rise to actions of mapping class groups of oriented surfaces of genus g≥1 with n≥1 boundary components. These mapping class group actions are given by group homomorphisms into the group of automorphisms of certain Yetter-Drinfeld modules over H . They are associated with edge slides in embedded ribbon graphs that generalise chord slides in chord diagrams. We give a concrete description of these mapping class group actions in terms of generating Dehn twists and defining relations. For the case where C is finitely complete and cocomplete, we also obtain actions of mapping class groups of closed surfaces by imposing invariance and coinvariance under the Yetter-Drinfeld module structure.

We show that the matrix embeddings in Bratteli diagrams are iterated direct sums of Hopf-Galois extensions (quantum principle bundles) for certain abelian groups. The corresponding strong universal connections are computed. We show that M n (C) is a trivial quantum principle bundle for the Hopf algebra C[ Z n × Z n ] . We conclude with an application relating known calculi on groups to calculi on matrices.