Representations of Lie algebras by non-skewselfadjoint operators in Hilbert space

We study non-skewselfadjoint representations of a finite dimensional real Lie algebra $\mathfrak{g}$. To this end we embed a non-skewselfadjoint representation of $\mathfrak{g}$ into a more complicated structure, that we call a $\mathfrak{g}$-operator vessel and that is associated to an overdetermined linear conservative input/state/output system on the corresponding simply connected Lie group $\mathfrak{G}$. We develop the frequency domain theory of the system in terms of representations of $\mathfrak{G}$, and introduce the joint characteristic function of a $\mathfrak{g}$-operator vessel which is the analogue of the classical notion of the characteristic function of a single non-selfadjoint operator. As the first non-commutative example, we apply the theory to the Lie algebra of the $ax+b$ group, the group of affine transformations of the line.     

The discriminant problem for Artin-Schreier extensions: Explicit results

Given a global function field K of characteristic p, for all effective divisors $\mathfrak{D}$ in the divisor group of K we count the number of cyclic extensions $F\supset K$ of degree p where the relative discriminant ${\mathrm{\text{Disc}}}_K(F)=(p-1)\mathfrak{D}$.     

Applications and homological properties of local rings with decomposable maximal ideals

We construct a local Cohen–Macaulay ring R with a prime ideal $\mathfrak{p}\in \mathrm{Spec}(R)$ such that R satisfies the uniform Auslander condition (UAC), but the localization $R_{\mathfrak{p}}$ does not satisfy Auslander's condition (AC). Given any positive integer n, we also construct a local Cohen–Macaulay ring R with a prime ideal $\mathfrak{p}\in \mathrm{Spec}(R)$ such that R has exactly two non-isomorphic semidualizing modules, but the localization $R_{\mathfrak{p}}$ has $2^n$ non-isomorphic semidualizing modules. Each of these examples is constructed as a fiber product of two local rings over their common residue field. Additionally, we characterize the non-trivial Cohen–Macaulay fiber products of finite Cohen–Macaulay type.     

Projective well orders and coanalytic witnesses

We further develop a forcing notion known as Coding with Perfect Trees and show that this poset preserves, in a strong sense, definable P-points, definable tight MAD families and definable selective independent families. As a result, we obtain a model in which $\mathfrak{a}=\mathfrak{u}=\mathfrak{i}={\mathrm{?}}_1<2^{{\mathrm{?}}_0}={\mathrm{?}}_2$, each of $\mathfrak{a}$, $\mathfrak{u}$, $\mathfrak{i}$ has a ${\mathrm{\Pi }}_1^1$ witness and there is a ${\mathrm{\Delta }}_3^1$ well-order of the reals. Note that both the complexity of the witnesses of the above combinatorial cardinal characteristics, as well as the complexity of the well-order are optimal. In addition, we show that the existence of a ${\mathrm{\Delta }}_3^1$ well-order of the reals is consistent with $\mathfrak{c}={\mathrm{?}}_2$ and each of the following: $\mathfrak{a}=\mathfrak{u}<\mathfrak{i}$, $\mathfrak{a}=\mathfrak{i}<\mathfrak{u}$, $\mathfrak{a}<\mathfrak{u}=\mathfrak{i}$, where the smaller cardinal characteristics have co-analytic witnesses.Our methods allow the preservation of only sufficiently definable witnesses, which significantly differs from other preservation results of this type.     

Regular characters of classical groups over complete discrete valuation rings

Let $\mathfrak{o}$ be a complete discrete valuation ring with finite residue field $\mathsf{k}$ of odd characteristic, and let G be a symplectic or special orthogonal group scheme over $\mathfrak{o}$. For any $?\in \mathbb{N}$ let $G^{?}$ denote the ?-th principal congruence subgroup of $\mathbf{G}(\mathfrak{o})$. An irreducible character of the group $\mathbf{G}(\mathfrak{o})$ is said to be regular if it is trivial on a subgroup $G^{?+1}$ for some ?, and if its restriction to $G^{?}/G^{?+1}?\mathrm{Lie}(\mathbf{G})(\mathsf{k})$ consists of characters of minimal $\mathbf{G}({\mathsf{k}}^{\mathrm{alg}})$-stabilizer dimension. In the present paper we consider the regular characters of such classical groups over $\mathfrak{o}$, and construct and enumerate all regular characters of $\mathbf{G}(\mathfrak{o})$, when the characteristic of $\mathsf{k}$ is greater than two. As a result, we compute the regular part of their representation zeta function.     

The Z2-orbifold of the universal affine vertex algebra

Let $\mathfrak{g}$ be a simple, finite-dimensional complex Lie algebra, and let $V^k(\mathfrak{g})$ denote the universal affine vertex algebra associated to $\mathfrak{g}$ at level k. The Cartan involution on $\mathfrak{g}$ lifts to an involution on $V^k(\mathfrak{g})$, and we denote by $V^k{(\mathfrak{g})}^{{\mathbb{Z}}_2}$ the orbifold, or fixed-point subalgebra, under this involution. Our main result is an explicit minimal strong finite generating set for $V^k{(\mathfrak{g})}^{{\mathbb{Z}}_2}$ for generic values of k.     

R+ˆ is surprisingly an integral domain

It is shown that if $(R,\mathfrak{m},k)$ is a complete local domain with $\mathrm{char}k=p>0$ and $R^{+}$ is its integral closure in an algebraic closure of the quotient field, then both the $\mathfrak{m}$-adic and p-adic completions of $R^{+}$ are integral domains. More generally, this theorem remains true if the completeness assumption is relaxed to allow R to be an analytically irreducible Henselian local ring. It is also shown that these rings, which are Cohen-Macaulay R-modules (even balanced in the $\mathfrak{m}$-adic case), will have dimension larger than the dimension of R unless $\dim ?R\le 1$.     

A local–global principle for surjective polynomial maps

Let R be an affine domain of characteristic zero with finite quotients. We prove that a polynomial map over R is surjective if and only if it is surjective over $\stackrel{?}{R_{\mathfrak{m}}}$, the completion of R with respect to $\mathfrak{m}$, for every maximal ideal $\mathfrak{m}?R$. In fact, the completions $\stackrel{?}{R_{\mathfrak{m}}}$ may be replaced by arbitrary subrings containing R. We use this result to yield a characterization of surjective polynomial maps, and remark that there does not exist a similar principle for injective polynomial maps.