Related papers: The weak commutativity construction for Lie algebr…

We continue the analysis of the weak commutativity construction for Lie algebras. This is the Lie algebra $\chi(\mathfrak{g})$ generated by two isomorphic copies $\mathfrak{g}$ and $\mathfrak{g}^{\psi}$ of a fixed Lie algebra, subject to…

The weak commutativity group $\chi(G)$ is generated by two isomorphic groups $G$ and $G^{\varphi }$ subject to the relations $[g,g^{\varphi}]=1$ for all $g \in G$. We obtain new expressions for the terms of the derived series and the lower…

The operator, $\chi $, of weak commutativity between isomorphic groups $G$ and $G^{\varphi }$ was introduced by Sidki as \begin{equation*} \chi (G)=\left\langle G \cup G^{\varphi }\mid \lbrack g,g^{\varphi }]=1\,\forall \,g\in…

Let $G$ be connected reductive algebraic group defined over an algebraically closed field of characteristic $p > 0$ and suppose that $p$ is a good prime for the root system of $G$, the derived subgroup of $G$ is simply connected and the Lie…

We define and study a pro-$p$ version of Sidki's weak commutativity construction. This is the pro-$p$ group $\mathfrak{X}_p(G)$ generated by two copies $G$ and $G^{\psi}$ of a pro-$p$ group, subject to the defining relators $[g,g^{\psi}]$…

The weak commutativity group $\chi(G)$ is generated by two isomorphic groups $G$ and $G^{\varphi }$ subject to the relations $[g,g^{\varphi}]=1$ for all $g \in G$. The group $\chi(G)$ is an extension of $D(G) = [G,G^{\varphi}]$ by $G \times…

The coadjoint representation of a connected algebraic group $Q$ with Lie algebra $\mathfrak q$ is a thrilling and fascinating object. Symmetric invariants of $\mathfrak q$ (= $\mathfrak q$-invariants in the symmetric algebra $S(\mathfrak…

As an associative algebra, the Heisenberg-Weyl algebra $\mathcal{H}$ is generated by two elements $A$, $B$ subject to the relation $AB-BA=1$. As a Lie algebra, however, where the usual commutator serves as Lie bracket, the elements $A$ and…

We study rigidity questions for pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$ admitting a post-Lie algebra structure. We show that if $\mathfrak{g}$ is semisimple and $\mathfrak{n}$ is arbitrary, then we have rigidity in the sense…

We prove that for any known Lie algebra $\frak{g}$ having none invariants for the coadjoint representation, the absence of invariants is equivalent to the existence of a left invariant exact symplectic structure on the corresponding Lie…

For a connected Lie group G it was shown by Lee, Ludwig, Samei and Spronk that its Fourier algebra A(G) is weakly amenable only if G is abelian. We extend this result to general connected locally compact groups, extending an approach…

We study Lie bialgebra structures on \emph{flat metric Lie algebras}, that is, Lie algebras $(\mathfrak{g},\langle\cdot,\cdot\rangle)$ whose associated left-invariant Riemannian metric on the simply connected Lie group $G$ has zero…

We construct the algebra of fractions of a Weak Bialgebra relative to a suitable denominator set of group-like elements that is `almost central', a condition we introduce in the present article which is sufficient in order to guarantee…

From the theory of finite dimensional Lie algebras it is known that every finite dimensional Lie algebra is decomposed into a semidirect sum of semisimple subalgebra and solvable radical. Moreover, due to work of Mal'cev the study of…

Let $G$ be a connected reductive linear algebraic group over a field $k$. Using ideas from geometric invariant theory, we study the notion of $G$-complete reducibility over $k$ for a Lie subalgebra $\mathfrak h$ of the Lie algebra…

Let ${\mathcal B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type with braiding matrix $\mathfrak{q}$, let $\mathcal{L}_{\mathfrak{q}}$ be the corresponding Lusztig algebra as in arXiv:1501.04518 and let…

Let G be a compact connected Lie group. We prove that the Fourier algebra A(G) is weakly amenable if and only if G is abelian.

We study invariant pseudo-K\"ahler structures on a solvmanifold $G$ such that the Lie algebra $\mathfrak{g}$ is almost abelian, that is $\mathfrak{g}=\mathfrak{h}\rtimes\mathbb{R}$, with $\mathfrak{h}$ abelian; comparing with the…

Let $\mathfrak g$ be a complex reductive Lie algebra and $V$ the underling vector space of a finite-dimensional representation of $\mathfrak g$. Then one can consider a new Lie algebra $\mathfrak q=\mathfrak g{\ltimes} V$, which is a…

It is shown that any Lie affgebra, that is an algebraic system consisting of an affine space together with a bi-affine bracket satisfying affine versions of the antisymmetry and Jacobi identity, is isomorphic to a Lie algebra together with…