In this article, we formalize some basic facts of Z-module. In the first section, we discuss the rank of submodule of Z-module and its properties. Especially, we formally prove that the rank of any Z-module is equal to or more than that of its submodules, and vice versa, there exists a submodule with any given rank that satisfies the above condition. In the next section, we mention basic facts ...

In this paper we study Schur-Weyl duality between the symplectic group and Brauer’s centralizer algebra over an arbitrary infinite field K. We show that the natural homomorphism from the Brauer’s centralizer algebra Bn(−2m) to the endomorphism algebra of tensor space (K) as a module over the symplectic similitude group GSp2m(K) (or equivalently, as a module over the symplectic group Sp2m(K)) is...

In this paper we prove the Schur-Weyl duality between the symplectic group and the Brauer algebra over an arbitrary infinite field K. We show that the natural homomorphism from the Brauer algebra Bn(−2m) to the endomorphism algebra of the tensor space (K2m)⊗n as a module over the symplectic similitude group GSp2m(K) (or equivalently, as a module over the symplectic group Sp2m(K)) is always surj...

Example 1.1. A field homomorphism K → F is a character by restricting it to the nonzero elements of K (that is, using G = K×) and ignoring the additive aspect of a field homomorphism. In particular, when L/K is a field extension any element of Aut(L/K) is a field homomorphism L→ L and therefore is a character of L× with values in L×. Example 1.2. For any α ∈ F×, the map Z→ F× by k 7→ αk is a ch...

We study topological rigidity of algebraic dynamical systems. In the first part of this paper we give an algebraic condition for rigidity that unifies previous rigidity results and we settle an old question of Walters. In the second part we consider the rigidity of hyperbolic systems. An algebraic dynamical system (X, S) has a phase space X that is a compact abelian group and a transformation s...

1. Let G be an algebraic linear group over a field F. If G acts by linear automorphisms on some vector space M over F we say that M is a rational G-module if it is the sum of finite-dimensional G-stable subspaces V such that the representation of G on each F is a rational representation of G. The rational G-module M is said to be rationally injective if, whenever U is a rational G-module and 0 ...

We study 2-representations, i.e. actions of 2-groups on 2-vector spaces. Our main focus is character theory for 2-representations. To this end we employ the technique of extended Burnside rings. Our main theorem is that the Ganter-Kapranov 2-character is a particular mark homomorphism of the Burnside ring. As an application we give a new proof of Osorno formula for the Ganter-Kapranov 2-charact...

Let g be a complex semisimple Lie algebra and let r ⊂ g be any reductive Lie subalgebra such that B|r is nonsingular where B is the Killing form of g. Let Z(r) and Z(g) be, respectively, the centers of the enveloping algebras of r and g. Using a Harish-Chandra isomorphism one has a homomorphism η : Z(g) → Z(r) which, by a well-known result of H. Cartan, yields the the relative Lie algebra cohom...

Up to oriented homotopy equivalence, a PD–pair (X, ∂X) with aspherical boundary components is uniquely determined by the Π1–system {κi : Π1(∂Xi, ∗) → Π1(X, ∗)}i∈J , the orientation character ωX ∈ H(X;Z/2Z) and the image of the fundamental class [X, ∂X] ∈ H3(X, ∂X;Z) under the classifying map [3]. We call the triple ({κi}i∈J , ωX , [X, ∂X]) the fundamental triple of the PD–pair (X, ∂X). Using Pe...

We first study some properties‎ ‎of $A$-module homomorphisms $theta:Xrightarrow Y$‎, ‎where $X$‎ ‎and $Y$ are Fréchet $A$-modules and $A$ is a unital‎ ‎Fréchet algebra‎. ‎Then we show that if there exists a‎ ‎continued bisection of the identity for $A$‎, ‎then $theta$ is‎ ‎automatically continuous under certain condition on $X$‎. ‎In‎ ‎particular‎, ‎every homomorphism from $A$ into certain‎ ‎Fr...