Let $F_m$ be the free metabelian Lie algebra of rank $m$ over a field $K$ characteristic 0. An automorphism $\varphi$ is called central if commutes with every inner $F_m$. Such automorphisms form centralizer $\text{\rm C}(\text{\rm Inn}(F_m))$ group Inn}(F_m)$ in Aut}(F_m)$. We provide an elementary proof to show that Inn}(F_m))=\text{\rm Inn}(F_m)$.

this is a survey article on centralizers of finite‎ ‎subgroups in locally finite‎, ‎simple groups or lfs-groups as we‎ ‎will call them‎. ‎we mention some of the open problems about‎ ‎centralizers of subgroups in lfs-groups and applications of the‎ ‎known information about the centralizers of subgroups to the‎ ‎structure of the locally finite group‎. ‎we also prove the‎ ‎following‎: ‎let $g$ be...

The higher rank Racah algebra $R(n)$ introduced recently is recalled. A quotient of this by central elements, which we call the special $sR(n)$, then introduced. Using results from classical invariant theory, $sR(n)$ shown to be isomorphic centralizer $Z_{n}(\mathfrak{sl}_2)$ diagonal embedding $U(\mathfrak{sl}_2)$ in $U(\mathfrak{sl}_2)^{\otimes n}$. This leads a first and novel presentation t...

‎let $r$ be a ring with involution $*$‎. ‎an additive mapping $t:rto r$ is called a left(respectively right) centralizer if $t(xy)=t(x)y$ (respectively $t(xy)=xt(y)$) for all $x,yin r$‎. ‎the purpose of this paper is to examine the commutativity of prime rings with involution satisfying certain identities involving left centralizers.

Let g=g0¯⊕g1¯ be a basic classical Lie superalgebra over C, e∈g0¯ nilpotent element and ge the centralizer of e in g. We study various properties elements g, which have previously only been considered case algebras. In particular, we prove that is reachable if satisfies Panyushev property for g=sl(m|n), m≠n or psl(n|n) osp(m|2n). For exceptional superalgebras g=D(2,1;α), G(3), F(4), give classi...

We show there is a residual set of non-Anosov C∞ Axiom A diffeomorphisms with the no cycles property whose elements have trivial centralizer. If M is a surface and 2 ≤ r ≤ ∞, then we will show there exists an open and dense set of of Cr Axiom A diffeomorphisms with the no cycles property whose elements have trivial centralizer. Additionally, we examine commuting diffeomorphisms preserving a com...

Using the geometric Satake correspondence, Mirkovic-Vilonen cycles in affine Grasssmannian give bases for representations of a semisimple group G . We prove that these are perfect, i.e. compatible with action Chevelley generators positive half Lie algebra g. compute this terms intersection multiplicities Grassmannian. stitch together to basis C[N] regular functions on unipotent subgroup. multip...

Given an n×n matrix c over a unitary ring R, the centralizer of in full Mn(R) is called principal ring, denoted by Sn(c,R). We investigate its structure and prove: (1) If invertible with c-free point, or if R has no zero-divisors Jordan-similar all eigenvalues center then separable Frobenius extension Sn(c,R) sense Kasch. (2) integral domain matrix, cellular R-algebra Graham Lehrer. In particul...

We construct a series of inclusions of hyperfinite subfactors Pi in the centralizer R of Powers’ IIIt factor, Pi+1 ⊂ Pi ⊂ R. It is proven that: ∀i ∈ N, P ′ i ∩R = C, [Pi : Pi+1] ≥ λ = (1 + t) t Via an outer action of Z2 on R, we also construct an inclusion of hyperfinite II1 factors, M3 ⊂ M1 with the index equal to λ, such that the relative commutant M1 ∩M ′ 3 is four dimensional.