Observations about the Lie algebra g2⊂so(7), associative 3-planes,and so(4) subalgebras

We make several observations relating the Lie algebra ${\mathfrak{g}}_2?\mathfrak{so}\left(7\right)$, associative 3-planes, and $\mathfrak{so}\left(4\right)$ subalgebras. Some are likely well-known but not easy to find in the literature, while other results are new. We show that an element $X\in {\mathfrak{g}}_2$ cannot have rank 2, and if it has rank 4 then its kernel is an associative subspace. We prove a canonical form theorem for elements of ${\mathfrak{g}}_2$. Given an associative 3-plane $P$ in ${\mathbb{R}}^7$, we construct a Lie subalgebra $\Theta \left(P\right)$ of $\mathfrak{so}\left(7\right)={\Lambda }^2\left({\mathbb{R}}^7\right)$ that is isomorphic to $\mathfrak{so}\left(4\right)$. This $\mathfrak{so}\left(4\right)$ subalgebra differs from other known constructions of $\mathfrak{so}\left(4\right)$ subalgebras of $\mathfrak{so}\left(7\right)$ determined by an associative 3-plane. These are results of an NSERC undergraduate research project. The paper is written so as to be accessible to a wide audience.     

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.     

Virtual resolutions of points in P1×P1

We explore explicit virtual resolutions, as introduced by Berkesch, Erman, and Smith, for ideals of finite sets of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$. Specifically, we describe a virtual resolution for a sufficiently general set of points X in ${\mathbb{P}}^1\times {\mathbb{P}}^1$ that only depends on $|X|$. We also improve an existence result of Berkesch, Erman, and Smith in the special case of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$; more precisely, we give an effective bound for their construction that gives a virtual resolution of length two for any set of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$.     

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.     

Positive solutions for a class of singular quasilinear Schrödinger equations with critical Sobolev exponent

We prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growth

$?\mathrm{\Delta }u?\lambda c(x)u?\kappa \alpha (\mathrm{\Delta }(|u{|}^{2\alpha }))|u{|}^{2\alpha ?2}u=|u{|}^{q?2}u+|u{|}^{2^{?}?2}u,u\in D^{1,2}({\mathbb{R}}^N),$

via variational methods, where $\lambda \ge 0$, $c:{\mathbb{R}}^N\to {\mathbb{R}}^{+}$, $\kappa >0$, $0<\alpha <1/2$, $2<q<2^{?}$. It is interesting that we do not need to add a weight function to control $|u{|}^{q?2}u$.