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Register a new userOn the algebraic set of singular elements in a complex simple Lie algebra
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Let $G$ be a complex simple Lie group and let $g = hbox{rm Lie},G$. Let $S(g)$ be the $G$-module of polynomial functions on $g$ and let $hbox{rm Sing},g$ be the closed algebraic cone of singular elements in $g$. Let ${cal L}s S(g)$ be the (graded) ideal defining $hbox{rm Sing},g$ and let $2r$ be the dimension of a $G$-orbit of a regular element in $g$. Then ${cal L}^k = 0$ for any $k<r$. On the other hand, there exists a remarkable $G$-module $Ms {cal L}^r$ which already defines $hbox{rm Sing},g$. The main results of this paper are a determination of the structure of $M$.
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We show that if $B$ is a block of a finite group algebra $kG$ over an algebraically closed field $k$ of prime characteristic $p$ such that $HH^1(B)$ is a simple Lie algebra and such that $B$ has a unique isomorphism class of simple modules, then $B$ is nilpotent with an elementary abelian defect group $P$ of order at least $3$, and $HH^1(B)$ is in that case isomorphic to the Jacobson-Witt algebra $HH^1(kP)$. In particular, no other simple modular Lie algebras arise as $HH^1(B)$ of a block $B$ with a single isomorphism class of simple modules.
Let $L$ be a Lie algebra of Block type over $C$ with basis ${L_{alpha,i},|,alpha,iinZ}$ and brackets $[L_{alpha,i},L_{beta,j}]=(beta(i+1)-alpha(j+1))L_{alpha+beta,i+j}$. In this paper, we shall construct a formal distribution Lie algebra of $L$. Then we decide its conformal algebra $B$ with $C[partial]$-basis ${L_alpha(w),|,alphainZ}$ and $lambda$-brackets $[L_alpha(w)_lambda L_beta(w)]=(alphapartial+(alpha+beta)lambda)L_{alpha+beta}(w)$. Finally, we give a classification of free intermediate series $B$-modules.
We generalize earlier results about connected components of idempotents in Banach algebras, due to B. SzH{o}kefalvi Nagy, Y. Kato, S. Maeda, Z. V. Kovarik, J. Zemanek, J. Esterle. Let $A$ be a unital complex Banach algebra, and $p(lambda) = prodlimits_{i = 1}^n (lambda - lambda_i)$ a polynomial over $Bbb C$, with all roots distinct. Let $E_p(A) := {a in A mid p(a) = 0}$. Then all connected components of $E_p(A)$ are pathwise connected (locally pathwise connected) via each of the following three types of paths: 1)~similarity via a finite product of exponential functions (via an exponential function); 2)~a polynomial path (a cubic polynomial path); 3)~a polygonal path (a polygonal path consisting of $n$ segments). If $A$ is a $C^*$-algebra, $lambda_i in Bbb R$, let $S_p(A):= {ain A mid a = a^*$, $p(a) = 0}$. Then all connected components of $S_p(A)$ are pathwise connected (locally pathwise connected), via a path of the form $e^{-ic_mt}dots e^{-ic_1t} ae^{ic_1t}dots e^{ic_mt}$, where $c_i = c_i^*$, and $t in [0, 1]$ (of the form $e^{-ict} ae^{ict}$, where $c = c^*$, and $t in [0,1]$). For (self-adjoint) idempotents we have by these old papers that the distance of different connected components of them is at least~$1$. For $E_p(A)$, $S_p(A)$ we pose the problem if the distance of different connected components is at least $min bigl{|lambda_i - lambda_j| mid 1 leq i,j leq n, i eq jbigr}$. For the case of $S_p(A)$, we give a positive lower bound for these distances, that depends on $lambda_1, dots, lambda_n$. We show that several local and global lifting theorems for analytic families of idempotents, along analytic families of surjective Banach algebra homomorphisms, from our recent paper with B. Aupetit and M. Mbekhta, have analogues for elements of $E_p(A)$ and $S_p(A)$.
The commuting variety of a reductive Lie algebra ${goth g}$ is the underlying variety of a well defined subscheme of $gg g{}$. In this note, it is proved that this scheme is normal. In particular, its ideal of definition is a prime ideal.
Let $A$ be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of $A$ is a simple directed graph, then $HH^1(A)$ is a solvable Lie algebra. The second main result shows that if the Ext-quiver of $A$ has no loops and at most two parallel arrows in any direction, and if $HH^1(A)$ is a simple Lie algebra, then char(k) is not equal to $2$ and $HH^1(A)cong$ $sl_2(k)$. The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.
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