分次代数的Hilbert级数、Poincaré级数和同伦李代数--一个讨论会
Hilbert series, Poincaré series and homotopy Lie algebras of graded algebras -- a seminar
摘要：
首先证明了一个关于分次代数$A$的Hilbert级数与二元分次代数$A$的Poincaré级数的关系式.这给出了Fröberg公式的情况下，双分次$Tor^A(k,k)$是集中在对角线上，我们采取的定义$A$是“Koszul”。我们看看在交换的情况下，从Koszul复杂的“平凡的Golod”的情况下获得的决议。用不同的方法介绍了$Ext_A(k,k)$的代数结构。由一维元素生成的子代数定义为$A$的“Koszul”对偶。我们定义了“广义Koszul复形”，并在$A$的增广理想的立方为零的情况下构造了一个最小分辨率.以上结果至少有45年的历史，大部分可以在我的论文中找到。第二部分定义了分次李代数。引入了自由李代数和包络代数。在分次交换情形下，Koszul对偶被看作是李代数的包络代数，并举例说明.提出了Poincaré-Birkhoff-Witt定理，给出了包络代数的Hilbert级数的一个公式.同伦李代数有不同的定义，特别是由一维元素生成的李子代数。对于完全交、Golod环和增广理想的立方等于零的环给出了同伦李代数的例子。给出了一个由Hilbert级数表示的李代数的维数的对数公式，并给出了它的包络代数的维数。给出了周期李子代数的例子，得到无理庞加莱级数。定义了超平面排列的“holonomy”李代数，并以图形排列$K_4$为例进行了讨论.
英文摘要：
We begin with proving a formula relating the Hilbert series of a graded algebra $A$ and the Poincaré series for $A$ in two variables. This gives the Fröberg formula in the case where the bigraded $Tor^A(k,k)$ is concentrated on the diagonal, which we take as definition of $A$ being "Koszul". We look at a resolution in the commutative case obtained from the Koszul complex in the "trivially Golod" case. The algebra structure of $Ext_A(k,k)$ is introduced in different ways. Its subalgebra generated by the one-dimensional elements is by definition the "Koszul" dual of $A$. We define the"generalized Koszul complex" and construct a minimal resolution in the case where the cube of the augmentation ideal of $A$ is zero. The above results are at least 45 years old and most of it can be found in my thesis. In the second part graded Lie algebras are defined. Free Lie algebras and enveloping algebras are introduced. The Koszul dual is looked upon as the enveloping algebra of a Lie algebra in the graded commutative case with examples. The Poincaré-Birkhoff-Witt theorem is stated giving a formula for the Hilbert series of the enveloping algebra. The homotopy Lie algebra is defined in different ways, in particular the Lie subalgebra generated by the one-dimensional elements. Examples of homotopy Lie algebras are given for complete intersections, Golod rings and rings with the cube of the augmentation ideal equal to zero. A logarithmic formula for the dimensions of a Lie algebra given the Hilbert series for its enveloping algebra is given. Examples of periodic Lie subalgebras are given, yielding irrational Poincaré series. The "holonomy" Lie algebra of a hyperplane arrangement is defined examplified by the graphical arrangement $K_4$.
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