International Journal of Mathematics

This is a sequel paper of [Weak multiplier bialgebras, Trans. Amer. Math. Soc., in press] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals [The Larson–Sweedler theorem for weak multiplier Hopf algebras, in preparation] are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite-dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.

Given a Morse–Smale function on an effective orientable orbifold, we construct its Morse–Smale–Witten complex. We show that critical points of a certain type have to be discarded to build a complex properly, and that gradient flows should be counted with suitable weights. Its homology is proven to be isomorphic to the singular homology of the quotient space under the self-indexing assumption. For a global quotient orbifold [M/G], such a complex can be understood as the G-invariant part of the Morse complex of M, where the G-action on generators of the Morse complex has to be defined including orientation spaces of unstable manifolds at the critical points. Alternatively in the case of global quotients, we introduce the notion of weak group actions on Morse–Smale–Witten complexes for non-invariant Morse–Smale functions on M, which give rise to genuine group actions on the level of homology.

Let E be a real Banach space with dimension at least 2. In this paper, we prove that if D ⊂ E is a John domain which is homeomorphic to an inner uniform domain via a CQH map, then each neargeodesic in D is a cone arc.

In the paper, we prove the existence of solutions of the complex Monge–Ampère type equation -χ(u)(dd^cu)ⁿ = μ in the class if there exist subsolutions in this class. As an application, we prove that the complex Monge–Ampère equation (dd^cu)ⁿ = μ is solvable in the class if there exist subsolutions locally. Moreover, by an example we show that the conditions in our above result are sharp.

This paper is to study further properties of harmonic maps between Finsler manifolds. It is proved that any conformal harmonic map from an n(>2)-dimensional Finsler manifold to a Finsler manifold must be homothetic and some rigidity theorems for harmonic maps between Finsler manifolds are given, which improve some results in earlier papers and generalize Eells–Sampson's theorem and Sealey's theorem in Riemannian Geometry.

We produce new examples supporting the Mond conjecture which can be stated as follows. The number of parameters needed for a miniversal unfolding of a finitely determined map-germ from n-space to (n + 1)-space is less than (or equal to if the map-germ is weighted homogeneous) the rank of the nth homology group of the image of a stable perturbation of the map-germ. We give examples of finitely determined map-germs of corank 2 from 3-space to 4-space satisfying the conjecture. We introduce a new type of augmentations to generate series of finitely determined map-germs in dimensions (n, n + 1) from a given one in dimensions (n - 1, n). We present more examples in dimensions (4, 5) and (5, 6) based on our examples, and verify the conjecture for them.

Over the past 20 years, a great deal of work has been done on the moduli spaces of coherent systems on algebraic curves. Until recently, however, there has been very little work on the fixed determinant case, except for the special case of rank 2 and canonical determinant. This situation has changed due to two papers of Osserman, who has obtained lower bounds for the dimensions of the fixed determinant moduli spaces in some cases. Our object in this paper is to show that some of Osserman's bounds are sharp.

We express the mean Euler characteristic (MEC) of a contact structure in terms of the mean indices of closed Reeb orbits for a broad class of contact manifolds, the so-called asymptotically finite contact manifolds. We show that this class is closed under subcritical contact surgery and examine the behavior of the MEC under such surgery. To this end, we revisit the notion of index-positivity for contact forms. We also obtain an expression for the MEC in the Morse–Bott case.

A symplectic or orthogonal bundle V of rank 2n over a curve has an invariant t(V) which measures the maximal degree of its isotropic subbundles of rank n. This invariant t defines stratifications on moduli spaces of symplectic and orthogonal bundles. We study this stratification by relating it to another one given by secant varieties in certain extension spaces. We give a sharp upper bound on t(V), which generalizes the classical Nagata bound for ruled surfaces and the Hirschowitz bound for vector bundles, and study the structure of the stratifications on the moduli spaces. In particular, we compute the dimension of each stratum. We give a geometric interpretation of the number of maximal Lagrangian subbundles of a general symplectic bundle, when this is finite. We also observe some interesting features of orthogonal bundles which do not arise for symplectic bundles, essentially due to the richer topological structure of the moduli space in the orthogonal case.

In this paper, the Lie bialgebra structures on a class of not-finitely graded Lie algebras related to generalized Virasoro algebras Ŵ are considered. We prove that all Lie bialgebras on Ŵ are coboundary triangular.

A new surgery method for contact manifolds is introduced in this paper. The surgeries are defined by using round handles with symplectic structures which are also introduced in this paper. These symplectic round handles have the global Liouville vector fields. Then the symplectic structures are exact. In the 3-dimensional case, the surgery is described only in terms of 3-dimensional contact topology.

For a locally compact quantum group 𝔾, we generalize some notions of amenability such as amenability of locally compact quantum groups and inner amenability of locally compact groups to the case of right Banach L¹(𝔾)-modules. Also, we investigate the concept of harmonic functionals over right Banach L¹(𝔾)-modules and use these devices to study, among others, amenability of 𝔾.