In the theory of inverse systems, in order to study the properties of a space X or a map f : X → Y between spaces, one expands X to an inverse system X or expands f to a map f : X → Y between the inverse systems, and then work on X or f . In this paper, we define approximate injectivity (resp., surjectivity) for approximate maps, and show that a map f : X → Y between compact metric spaces is in...

In this paper, we investigate the structure and representations of the quantum group U(∞) = Uυ(gl∞). We will present a realization for U(∞), following Beilinson–Lusztig–MacPherson (BLM) [1], and show that the natural algebra homomorphism ζr from U(∞) to the infinite q-Schur algebra S(∞, r) is not surjective for any r > 1. We will give a BLM type realization for the image U(∞, r) := ζr(U(∞)) and...

Let H,K be two finitely generated subgroups of a free group, let 〈H,K〉 denote the subgroup generated by H,K, called the join of H,K, and let neither of H, K have finite index in 〈H,K〉. We prove the existence of an epimorphism ζ : 〈H,K〉 → F2, where F2 is a free group of rank 2, such that the restriction of ζ on both H and K is injective and the restriction ζ0 : H ∩K → ζ(H)∩ ζ(K) of ζ on H ∩K to ...

We show that, in any Mal’tsev (and a fortiori protomodular) category E, not only the fibre GrdXE of internal groupoids above the object X is a naturally Mal’tsev category, but moreover it shares with the category Ab of abelian groups the property following which the domain of any split epimorphism is isomorphic with the direct sum of its codomain with its kernel. This allows us to point at a ne...

We explicitly construct a star product for the complex Grassmann manifolds using the method of phase space reduction. Functions on C(p+q)·p ∗, the space of (p + q)× p matrices of rank p, invariant under the right action of Gl(p, C) can be regarded as functions on the Grassmann manifold Gp,q(C), but do not form a subalgebra whereas functions only invariant under the unitary subgroup U(p) ⊂ Gl(p,...

Let Modg be the modular group of surfaces of genus g. Each element [h] ∈ Modg induces in the integer homology of a surface of genus g a symplectic automorphism H([h]) and Poincaré shown that H : Modg → Sp(2g,Z) is an epimorphism. The theory of real algebraic curves justify the definition of real Riemann surface as a Riemann surface S with an anticonformal involution σ. Let (S, σ) be a real Riem...

In this paper, we investigate the structure and representations of the quantum group U(∞) = Uυ(gl∞). We will present a realization for U(∞), following Beilinson–Lusztig–MacPherson (BLM) [1], and show that the natural algebra homomorphism ζr from U(∞) to the infinite q-Schur algebra S(∞, r) is not surjective for any r > 1. We will give a BLM type realization for the image U(∞, r) := ζr(U(∞)) and...

We notice that for a Hopf algebra H , its antipode S is both an epimorphism and a monomorphism from H to H in the category of Hopf algebras over a field. Together with the existence of Hopf algebras with non-injective or non-surjective antipode, this proves the existence of non-surjective epimorphisms and non-injective monomorphisms in the category of Hopf algebras. Using Schauenburg’s free Hop...

A simplicial complex L on n vertices determines a subcomplex TL of the ntorus, with fundamental group the right-angled Artin groupGL. Given an epimorphism χ : GL → Z, let T L be the corresponding cover, with fundamental group the Artin kernel Nχ. We compute the cohomology jumping loci of the toric complex TL, as well as the homology groups of T L with coefficients in a field k, viewed as module...

A homomorphism of a hypergroup (H, ◦) is a function f : H → H satisfying f(x ◦ y) ⊆ f(x) ◦ f(y) for all x, y ∈ H. A homomorphism of a hypergroup (H, ◦) is called an epimorphism if f(H) = H. Denote by Hom(H, ◦) and Epi(H, ◦) the set of all homomorphisms and the set of all epimorphisms of a hypergroup (H, ◦), respectively. In this paper, the elements of Hom(Zn, ◦m) and Epi(Zn, ◦m) are characteriz...