论文标题
代数产生的群体及其谎言代数
Algebraically generated groups and their lie algebras
论文作者
论文摘要
仿射品种X的自动形态组AUT（X）是一个组。它的谎言代数在X上的矢量场的lie代数VF（X）中嵌入。我们研究了AUT（X）的亚组与VF（X）的Lie子代理之间的关系。 我们表明，当Aut（X）的连接代数子组G_I家族产生的AUT（X）的亚组G是代数时，并且仅当Lie代数LIE lie G_i G_i产生有限的Vf（x）的有限尺寸lie lie lie lie lie lie subalgebra。 扩展了Cohen-Draisma的结果，我们证明了由局部nilpotent载体场产生的VF（X）的局部有限的代数l是代数，即l = l = lie g，对于AUT（x）的代数亚组G。 按照相同的行，我们证明，如果有限的许多连接代数组生成的AUT（x）的亚组G是可溶解的，那么它是一个可解决的代数组。 我们还表明，派生的长度是AUT（x）的单位代数亚组u在上方由DIM X界定。此结果基于以下三角剖分定理： AUT（A^n）的每个独立代数子组在A^n中具有致密轨道，都与DeJonquières子组的子组相连。 此外，我们给出了一个由两个代数元素生成的AUT（A^2）的自由亚组F的示例，以便F的Zariski闭合是两个嵌套交换封闭的单位ind-Ind-subgroups的免费产物。 对于任何仿射Indgroup g，一个人都可以关联一个规范的理想l_g \ subset lie g。它是由分区空间T_E X线性生成的所有代数子集X \ subset G，它在e中平滑。它具有重要的属性，即对于过滤的同态同态ϕ：g \ h t诱导的同构dϕ_e：l_g \ to l_h也是溢流的。此外，如果H \ subset G是有限的codimension的下一个正常的IND-SUBGROUP，则L_H在L_G中具有有限的codimension。
The automorphism group Aut(X) of an affine variety X is an ind-group. Its Lie algebra is canonically embedded into the Lie algebra VF(X) of vector fields on X. We study the relations between subgroups of Aut(X) and Lie subalgebras of VF(X). We show that a subgroup G of Aut(X) generated by a family of connected algebraic subgroups G_i of Aut(X) is algebraic if and only if the Lie algebras Lie G_i generate a finite dimensional Lie subalgebra of VF(X). Extending a result by Cohen-Draisma we prove that a locally finite Lie algebra L of VF(X) generated by locally nilpotent vector fields is algebraic, i.e. L = Lie G for an algebraic subgroup G of Aut(X). Along the same lines we prove that if a subgroup G of Aut(X) generated by finitely many connected algebraic groups is solvable, then it is a solvable algebraic group. We also show that the derived length a unipotent algebraic subgroup U of Aut(X) is bounded above by dim X. This result is based on the following triangulation theorem: Every unipotent algebraic subgroup of Aut(A^n) with a dense orbit in A^n is conjugate to a subgroup of the de Jonquières subgroup. Furthermore, we give an example of a free subgroup F of Aut(A^2) generated by two algebraic elements such that the Zariski closure of F is a free product of two nested commutative closed unipotent ind-subgroups. To any affine ind-group G one can associate a canonical ideal L_G \subset Lie G. It is linearly generated by the tangent spaces T_e X for all algebraic subsets X \subset G which are smooth in e. It has the important property that for a surjective homomorphism ϕ: G \to H the induced homomorphism dϕ_e : L_G \to L_H is surjective as well. Moreover, if H \subset G is a subnormal closed ind-subgroup of finite codimension, then L_H has finite codimension in L_G.
