arXiv subjects

Anthony Bak

Publications and source records attributed to Anthony Bak.

4 recordsLinked to original sources

arXiv2017

The E-normal structure of odd dimensional unitary groups

In this paper we define odd dimensional unitary groups $U_{2n+1}(R,Δ)$. These groups contain as special cases the odd dimensional general linear groups $GL_{2n+1}(R)$ where $R$ is any ring, the odd dimensional orthogonal and symplectic groups $O_{2n+1}(R)$ and $Sp_{2n+1}(R)$ where $R$ is any commutative ring and further the first author's even dimensional unitary groups $U_{2n}(R,Λ)$ where $(R,Λ)$ is any form ring. We classify the E-normal subgroups of the groups $U_{2n+1}(R,Δ)$ (i.e. the subgroups which are normalized by the elementary subgroup $EU_{2n+1}(R,Δ)$), under the condition that $R$ is either a semilocal or quasifinite ring with involution and $n\geq 3$. Further we investigate the action of $U_{2n+1}(R,Δ)$ by conjugation on the set of all E-normal subgroups.

math.KT↗

arXiv2016

Subring subgroups in symplectic groups in characteristic 2

In 2012 the second author obtained a description of the lattice of subgroupsof a Chevalley group $G(Φ,A)$, containing the elementary subgroup $E(Φ,K)$ over a subring $K\subseteq A$ provided $Φ=B_n,$ $C_n$ or $F_4$, $n\ge2$, and $2$ is invertible in $K$. It turns out that this lattice splits into a disjoint union of "sandwiches", parametrized by intermediate subrings between $K$ and $A$. In the current article a similar result is proved for $Φ=B_n$ or $C_n$, $n\ge3$, and $2=0$ in $K$. In this settings one has to introduce more sandwiches, namely, the sandwiches are parametrized by form rings $(R,Λ)$ such that $K\subseteqΛ\subseteq R\subseteq A$. In particular, this result, generalizes a part of Ya.\,N.\,Nuzhin's theorem of 2013 concerning root systems $Φ=B_n,$ $C_n$, $n\ge3$, where the same description of the subgroup lattice is obtained under the condition that $A$ is an algebraic extension of~$K$.

math.RA↗

arXiv2013

Global actions, K-theory and unimodular rows

Global actions were introduced by Bak in order to have a homotopy theory in a purely algebraic setting. In this paper we apply his techniques in a particular case: the (single domain) unimodular row global action. More precisely, we compute the the path connected component and fundamental group for the unimodular row global action. An explicit computation of the fundamental group of the (connected component of) unimodular row global action is closely related to stability questions in K-theory. This will be shown by constructing an exact sequence with the fundamental group functor as the middle term and having surjective stability for the functor K_2 on the left and injective stability for the functor K_1 on the right.

math.KT↗

arXiv2010

Exploring a new peak in the heterotic landscape

We study the existence of realistic heterotic vacua on a new Abelian surface fibered Calabi-Yau threefold X with Z_8 x Z_8 fundamental group. Our main result is a no-go theorem, which says that (under mild assumptions) there is no stable holomorphic vector bundle on X satisfying the constraints required by global consistency of the heterotic vacuum and phenomenology. To prove the theorem we explore in some detail the Fourier-Mukai transform of vector bundles on Abelian surface fibrations.

hep-th↗