论文标题
Schreier的类型公式和两个级别的代数和组的生长量表
Schreier's type formulae and two scales for growth of Lie algebras and groups
论文作者
论文摘要
让$ g $是一组自由等级$ n $和$ h \ subset g $其有限指数子组。然后，$ h $也是一个免费组，排名$ m $ $ h $由Schreier的公式$ M-1 =（N-1）\ CDOT | G：H |。$确定。 自由谎言代数的任何子代数也是免费的。但是，尚不存在Schreier公式的自由谎言代数的直接类似物，因为有限的编成编成的任何子代数都有无限数量的发电机。 但是，在正式的力量系列方面存在适当的Schreier自由谎言代数的公式。还存在一个版本，就指数生成功能而言。这是一项有关如何应用这些公式研究的调查。首先，这些公式允许指定明确的公式，以生成各种类型的函数，以示代代数。其次，这些用于生成功能的明确公式用于为这两种类型的生长得出渐近的公式。这些结果可以看作是自由谎言代数和组的Witt公式的类似物。如果是谎言代数，我们获得了两种量表，以供各自的生长类型。我们还不久就提到了其他类型的线性代数增长情况。
Let $G$ be a free group of rank $n$ and $H\subset G$ its subgroup of finite index. Then $H$ is also a free group and the rank $m$ of $H$ is determined by Schreier's formula $m-1=(n-1)\cdot|G:H|.$ Any subalgebra of a free Lie algebra is also free. But a straightforward analogue of Schreier's formula for free Lie algebras does not exist, because any subalgebra of finite codimension has an infinite number of generators. But the appropriate Schreier's formula for free Lie algebras exists in terms of formal power series. There exists also a version in terms of exponential generating functions. This is a survey on how these formulas are applied to study 1) growth of finitely generated Lie algebras and groups and 2) the codimension growth of varieties of Lie algebras. First, these formulae allow to specify explicit formulas for generating functions of respective types for free solvable (or more generally, polynilpotent) Lie algebras. Second, these explicit formulas for generating functions are used to derive asymptotic for these two types of the growth. These results can be viewed as analogues of the Witt formula for free Lie algebras and groups. In case of Lie algebras, we obtain two scales for respective types of growth. We also shortly mention the situation on growth for other types of linear algebras.
