Do you want to publish a course? Click here

Group gradings on finitary simple Lie algebras

131   0   0.0 ( 0 )
 Added by Mikhail Kotchetov
 Publication date 2011
  fields
and research's language is English




Ask ChatGPT about the research

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.



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.
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 $n$.
For any abelian group $G$, we classify up to isomorphism all $G$-gradings on the classical central simple Lie algebras, except those of type $D_4$, over the field of real numbers (or any real closed field).
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.
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 $mathrm{char} mathbb{F}$.
comments
Fetching comments Fetching comments
Sign in to be able to follow your search criteria
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا