We consider finite-dimensional irreducible transitive graded Lie algebras $L = \sum_{i=-q}^rL_i$ over algebraically closed fields of characteristic three. We assume that the null component $L_0$ is classical and reductive. The adjoint representation of $L$ on itself induces a representation of the commutator subalgebra $L_0'$ of the null component on the minus-one component $L_{-1}.$ We show that if the depth $q$ of $L$ is greater than one, then this representation must be restricted.
Let $S$ be a partial groupoid, that is, a set with a partial binary operation. An $S$-graded ring $R$ is said to be graded von Neumann regular if $x\in xRx$ for every homogeneous element $x\in R.$ Under the assumption that $S$ is cancellative, we characterize $S$-graded rings which are graded von Neumann regular. If a ring is $S$-graded von Neumann regular, and if $S$ is cancellative, then $S$ is such that for every $s\in S,$ there exist $s^{-1}\in S$ and idempotent elements $e,$ $f\in S$ for which $es=sf=s,$ $fs^{-1}=s^{-1}e=s^{-1},$ $ss^{-1}=e$ and $s^{-1}s=f,$ which is a property enjoyed by…
We develop the theory of a category ${\mathscr C}_A$ which is a generalisation to non-restricted ${\mathfrak g}$-modules of a category famously studied by Andersen, Jantzen and Soergel for restricted ${\mathfrak g}$-modules, where ${\mathfrak g}$ is the Lie algebra of a reductive group $G$ over an algebraically closed field ${\mathbb K}$ of characteristic $p>0$. Its objects are certain graded bimodules. On the left, they are graded modules over an algebra $U_χ$ associated to ${\mathfrak g}$ and to $χ\in{\mathfrak g}^{*}$ in standard Levi form. On the right, they are modules over a commutative …
In this paper, we describe the automorphisms of solvable Leibniz algebras with null-filiform nilradical. Moreover we describe the automorphisms of solvable Leibniz algebras with naturally graded non-Lie filiform nilradicals, whose the dimension of complementary space is maximal.
We study the finite-dimensional irreducible representations of the nullity 2 centreless core $\mathfrak{g}_{2n,ρ}(\mathbb{C}_q)$ by investigating the structure of the $\mathrm{BC}_n$-graded Lie algebra $\mathfrak{g}_{2n,ρ}(R)$, where $R$ is a unital involutory associative algebra over a field $k$ of characteristic zero.
Let $R$ be the homogeneous coordinate ring of the Grassmannian $\mathbb{G}=Gr(2,n)$ defined over an algebraically closed field $k$ of characteristic $p \geq \max\{n-2,3\}$. In this paper we give a description of the decomposition of $R$, considered as graded $R^{p^r}$-module, for $r \geq 2$. This is a companion paper to our earlier paper, where the case $r=1$ was treated, and taken together, our results imply that $R$ has finite F-representation type (FFRT). Though it is expected that all rings of invariants for reductive groups have FFRT, ours is the first non-trivial example of such a ring f…
