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Register a new userOn Hermitian and skew-Hermitian matrix algebras over octonions
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We prove simplicity, and compute $delta$-derivations and symmetric associative forms of algebras in the title.
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Given a grading $Gamma: A=oplus_{gin G}A_g$ on a nonassociative algebra $A$ by an abelian group $G$, we have two subgroups of the group of automorphisms of $A$: the automorphisms that stabilize each homogeneous component $A_g$ (as a subspace) and the automorphisms that permute the components. By the Weyl group of $Gamma$ we mean the quotient of the latter subgroup by the former. In the case of a Cartan decomposition of a semisimple complex Lie algebra, this is the automorphism group of the root system, i.e., the so-called extended Weyl group. A grading is called fine if it cannot be refined. We compute the Weyl groups of all fine gradings on matrix algebras, octonions and the Albert algebra over an algebraically closed field (of characteristic different from 2 in the case of the Albert algebra).
We characterize derivations and 2-local derivations from $M_{n}(mathcal{A})$ into $M_{n}(mathcal{M})$, $n ge 2$, where $mathcal{A}$ is a unital algebra over $mathbb{C}$ and $mathcal{M}$ is a unital $mathcal{A}$-bimodule. We show that every derivation $D: M_{n}(mathcal{A}) to M_{n}(mathcal{M})$, $n ge 2,$ is the sum of an inner derivation and a derivation induced by a derivation from $mathcal{A}$ to $mathcal{M}$. We say that $mathcal{A}$ commutes with $mathcal{M}$ if $am=ma$ for every $ainmathcal{A}$ and $minmathcal{M}$. If $mathcal{A}$ commutes with $mathcal{M}$ we prove that every inner 2-local derivation $D: M_{n}(mathcal{A}) to M_{n}(mathcal{M})$, $n ge 2$, is an inner derivation. In addition, if $mathcal{A}$ is commutative and commutes with $mathcal{M}$, then every 2-local derivation $D: M_{n}(mathcal{A}) to M_{n}(mathcal{M})$, $n ge 2$, is a derivation.
This is an expository paper. Its purpose is to explain the linear algebra that underlies Donaldson-Thomas theory and the geometry of Riemannian manifolds with holonomy in $G_2$ and ${rm Spin}(7)$.
We study the free objects in the variety of semigroups and variety of monoids generated by the monoid of all $n times n$ upper triangular matrices over a commutative semiring. We obtain explicit representations of these, as multiplicative subsemigroups of quiver algebras over polynomial semirings. In the $2 times 2$ case this also yields a representation as a subsemigroup of a semidirect product of commutative monoids. In particular, from the case where $n=2$ and the semiring is the tropical semifield, we obtain a representation of the free objects in the monoid and semigroup varieties generated by the bicyclic monoid (or equivalently, by the free monogenic inverse monoid), inside a semidirect product of a commutative monoid acting on a semilattice. We apply these representations to answer several questions, including that of when the given varieties are locally finite.
Known classification results allow us to find the number of (equivalence classes of) fine gradings on matrix algebras and on classical simple Lie algebras over an algebraically closed field $mathbb{F}$ (assuming $mathrm{char} mathbb{F} e 2$ in the Lie case). The computation is easy for matrix algebras and especially for simple Lie algebras of type $B_r$ (the answer is just $r+1$), but involves counting orbits of certain finite groups in the case of Series $A$, $C$ and $D$. For $Xin{A,C,D}$, we determine the exact number of fine gradings, $N_X(r)$, on the simple Lie algebras of type $X_r$ with $rle 100$ as well as the asymptotic behaviour of the average, $hat N_X(r)$, for large $r$. In particular, we prove that there exist positive constants $b$ and $c$ such that $exp(br^{2/3})lehat N_X(r)leexp(cr^{2/3})$. The analogous average for matrix algebras $M_n(mathbb{F})$ is proved to be $aln n+O(1)$ where $a$ is an explicit constant depending on $mathrm{char} mathbb{F}$.
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