Hilbert series, Poincar\'e series and homotopy Lie algebras of graded algebras -- a seminar

TL;DR

This seminar explores the relationships between Hilbert series, Poincaré series, and homotopy Lie algebras of graded algebras, including formulas, dualities, and examples in both algebraic and Lie algebra contexts.

Contribution

It introduces new formulas and constructions connecting Hilbert and Poincaré series with homotopy Lie algebras, extending classical results and providing new examples.

Findings

Derived a formula relating Hilbert and Poincaré series for graded algebras.

Constructed minimal resolutions in specific algebraic cases.

Provided examples of homotopy Lie algebras with irrational Poincaré series.

Abstract

We begin with proving a formula relating the Hilbert series of a graded algebra $A$ and the Poincar\'{e} series for $A$ in two variables. This gives the Fr\"oberg 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 Poincar\'{e}-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 Poincar\'{e} series. The "holonomy" Lie algebra of a hyperplane arrangement is defined examplified by the graphical arrangement $K_4$.