Let $R$ be a continuous von Neumann regular ring with no nonzero abelian idempotents. We prove that every element of $R$ is a single additive commutator. In particular, every operator affiliated with a type $\mathrm{II}_1$ von Neumann algebra is a commutator of operators affiliated with the same algebra. Our commutator theorem, together with a theorem of Jaikin-Zapirain and L\'opez-\'Alvarez, implies that, if $R$ is a nonzero unital algebra over a field $\mathbb{K}$ of characteristic zero, the quotient division ring of the first Weyl algebra over $\mathbb{K}$ embeds unitally into $R$. We also give a direct proof of this consequence.
If $G$ is a connected reductive algebraic group over a field $k$ of characteristic $\ell$, we show that both the Lie-theoretic and group-theoretic commuting schemes associated to $G$ are normal and Cohen--Macaulay when $\ell = 0$, or $\ell$ is larger than the Coxeter numbers of all simple factors of $G$, and that $\ell...
Let $R$ be a nonzero commutative ring with identity, $n\geq3$, and $\sigma\in GL_n(R)$. We prove that there exist nonidentity transvections $\tau_1,\tau_2\in GL_n(R)$ such that $\big[[\sigma,\tau_1],\tau_2\big]=I_n$. Hence, Kourovka Notebook Problem 10.46 has a stronger affirmative answer over commutative rings with id...
Let $A$ be a split finite-dimensional algebra over a field whose radical $J$ satisfies $J^3=0$, and let $s$ be the number of isomorphism classes of simple $A$-modules. We prove that a non-projective module $M$ with ${\rm Ext}_A^i(M,A)=0$ for all $i>0$ has a non-zero self-extension in some degree between $1$ and $3s+1$....
Semigroups in which right divisibility is a partial order, that is, $\mathscr R$-trivial semigroups, have been studied by people in algebraic combinatorics in connection with Markov chains (especially left regular bands). Margolis and the author showed that the quiver of the algebra of an $\mathscr R$-trivial monoid is...
Let $R$ be a compact discrete valuation ring and $G = R^d \rtimes \operatorname{GL}_d(R)$. We show that the central Fourier algebra $\operatorname{ZA}(G)$ is not amenable, reaffirming a conjecture of Alaghmandan and Spronk. Our methods reduce to studying the Fourier algebra of the commutative orbit hypergroup $H = R^d/...
Let $L$ be a finite-dimensional pointed Hopf algebra over an algebraically closed field. We establish an intrinsic characterization of index-two extensions: every Hopf algebra $H$ containing $L$ as a Hopf subalgebra and satisfying $H_0=L_0$ and $\dim H=2\dim L$ is a normalized quadratic extension of $L$. For a fixed su...
Rong-Chuan Xiong· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
