  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 $K_4$.
