International Journal of Mathematics

We describe all the holomorphic Hermitian vector bundles (E, h) over the upper half plane in ℂ with the property that (f* E, f* h) is holomorphically isometric to (E, h) for any holomorphic automorphism f of the upper half plane. We give an explicit construction of all such holomorphic Hermitian vector bundles using some linear algebraic data.

A CW-complex X is called a [G,m]-complex if X is an m-dimensional complex with π₁(X) ≅ G and the universal cover is (m - 1)-connected. We show that if G has an infinite amenable normal subgroup, then the asphericity of a [G,m]-complex X is equivalent to the vanishing of L²-Euler characteristic of . This result corresponds to a generalization and a variation of earlier several works. Also, we show that the L²-Betti numbers of a group which belongs to the class of groups K𝔉 eventually vanish. As a byproduct, we give an example of a group which belongs to the class of groups H𝔉 but does not belong to the class of groups K𝔉.

We prove transformation formulae for generating functions of Gromov–Witten invariants on general toric Calabi–Yau threefolds under flops. Our proof is based on a combinatorial identity on the topological vertex and analysis of fans of toric Calabi–Yau threefolds.

By using C*-correspondences and Cuntz–Pimsner algebras, we associate to every subshift (also called a shift space) 𝖷 a C*-algebra , which is a generalization of the Cuntz–Krieger algebras. We show that is the universal C*-algebra generated by partial isometries satisfying relations given by 𝖷. We also show that is a one-sided conjugacy invariant of 𝖷.

This is the first in a series of papers in which we study vertex-algebraic structure of Feigin–Stoyanovsky's principal subspaces associated to standard modules for both untwisted and twisted affine Lie algebras. A key idea is to prove suitable presentations of principal subspaces, without using bases or even "small" spanning sets of these spaces. In this paper, we prove presentations of the principal subspaces of the basic -modules. These convenient presentations were previously used in work of Capparelli–Lepowsky–Milas for the purpose of obtaining the classical Rogers–Ramanujan recursion for the graded dimensions of the principal subspaces.

We show that the left regular representation π_l of a discrete quantum group (A, Δ) has the absorbing property and forms a monoid in the representation category Rep(A, Δ).

Next we show that an absorbing monoid in an abstract tensor *-category gives rise to an embedding functor (or fiber functor) , and we identify conditions on the monoid, satisfied by , implying that E is *-preserving.

As is well-known, from an embedding functor the generalized Tannaka theorem produces a discrete quantum group (A, Δ) such that . Thus, for a C*-tensor category with conjugates and irreducible unit the following are equivalent: (1) is equivalent to the representation category of a discrete quantum group (A, Δ), (2) admits an absorbing monoid, (3) there exists a *-preserving embedding functor .