Finite separable field extensions with prescribed extensions of valuations. II

Let A¹,...,A^k be pairwise independent valuation rings of K. Prescribing extensions Δ _i ^j . of the value group Γ^j and extensions \(\mathfrak{L}_i^j\) of the residue field \(H^j\) of A^j (i=1,...,r^j) such that \(\sum\limits_{i = 1}^{r^j } {(\Delta _i^j :\Gamma ^j )} \cdot [\mathfrak{L}_i^j :H^j ] = n\) , we provide sufficient conditions for the existence of a separable field extension L of K of degree n with exactly r^j pairwise independent valuation rings B _i ^j lying over A^j, which have Δ _i ^j as value groups and \(\mathfrak{L}_i^j\) as residue fields.     

Complete reducibility and Steinberg endomorphisms

Let G be a connected reductive algebraic group defined over an algebraically closed field of positive characteristic. We study a generalization of the notion of G-complete reducibility in the context of Steinberg endomorphisms of G. Our main theorem extends a special case of a rationality result in this setting.     

Isomorphism problems for Hopf–Galois structures on separable field extensions

Let $L/K$ be a finite separable extension of fields whose Galois closure $E/K$ has Galois group G. Greither and Pareigis use Galois descent to show that a Hopf algebra giving a Hopf–Galois structure on $L/K$ has the form $E{[N]}^G$ for some group N of order $[L:K]$. We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as K-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and K-algebras that appear in the classification of Hopf–Galois structures on a cyclic extension of degree $p^n$, for p an odd prime number.     

Complete reducibility of gyrogroup representations

Abstract

In this article, we show that any finite gyrogroup can be represented on a space of complex-valued functions. In particular, we prove that any linear representation of a finite gyrogroup on a finite-dimensional complex inner product space is unitary and hence is completely reducible using strong connections between linear actions of groups and gyrogroups. Also, we provide an example of a unitary representation of an arbitrary finite gyrogroup, which resembles the group-theoretic left regular representation.     

Smooth extensions of functions on separable Banach spaces

Let X be a Banach space with a separable dual X*. Let ${Y\subset X}Let X be a Banach space with a separable dual X*. Let Y ì X{Y\subset X} be a closed subspace, and f:Y?\mathbbR{f:Y\rightarrow\mathbb{R}} a C ¹-smooth function. Then we show there is a C ¹ extension of f to X.     

Complete extensions of topological spaces

We consider complete extensions of locally connected and peripherally bicompact spaces. We formulate a necessary and sufficient condition for a one-point bicompactification of a locally bicompact space to be a unique bicompactification with zero-dimensional growth.     

Homomorphisms,separable extensions,and Morita maps for weak module algebras

By using a trace one element, we give a sufficient and necessary condition for a weak module algebra A to be a projective left A#H-module, where A#H denotes the weak smash product. We also give some differentiated conditions for the weak smash product to be a separable extension on the weak module algebra A and get the weak structure theorem in the category of weak (H,A)-Hopf modules.     

Splitting of field extensions

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