Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn group gradings on PI-algebras
207
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We show that there exists a constant K such that for any PI- algebra W and any nondegenerate G-grading on W where G is any group (possibly infinite), there exists an abelian subgroup U of G with $[G : U] leq exp(W)^K$. A G-grading $W = bigoplus_{g in G}W_g$ is said to be nondegenerate if $W_{g_1}W_{g_2}... W_{g_r} eq 0$ for any $r geq 1$ and any $r$ tuple $(g_1, g_2,..., g_r)$ in $G^r$.
rate research
Read More
We classify, up to isomorphism, gradings by abelian groups on nilpotent filiform Lie algebras of nonzero rank. In case of rank 0, we describe conditions to obtain non trivial $Z_k$-gradings.
We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically closed field of characteristic different from 2.
For a given abelian group G, we classify the isomorphism classes of G-gradings on the simple restricted Lie algebras of types W(m;1) and S(m;1) (m>=2), in terms of numerical and group-theoretical invariants. Our main tool is automorphism group schemes, which we determine for the simple restricted Lie algebras of types S(m;1) and H(m;1). The ground field is assumed to be algebraically closed of characteristic p>3.
In this paper we consider gradings by a finite abelian group $G$ on the Lie algebra $mathfrak{sl}_n(F)$ over an algebraically closed field $F$ of characteristic different from 2 and not dividing $n$.
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 $mathrm{char} mathbb{F}$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
