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We give the definition of a graded Lie algebra with examples. The free Lie algebra on a set $X$ is defined and
the enveloping algebra of quotients of free Lie algebras are studied. The Koszul dual is looked upon as the enveloping algebra of a Lie algebra in the graded commutative case with examples.
The Poincare-Birkhoff-Witt theorem is stated and as a consequence, a formula for the Hilbert series of the enveloping algebra is obtained. In the graded commutative case a coproduct $\Delta$
on $\ext_A(k,k)$ is possible to define and thereby the homotopy Lie algebra is defined as the set of $x$ such that
$\Delta(x)=$\break $x\otimes1+1\otimes x$.
Other definitions of the homotopy Lie algebra are given
by means of minimal algebra resolutions and models. The Lie subalgebra $\eta$
generated by the one-dimensional elements in the homotopy Lie algebra $\lieg$ of $A$ is studied. Here, the enveloping algebra of $\eta$ is the Koszul dual of $A$.
Examples of homotopy Lie algebras are given for
complete intersections, Golod rings and rings with the cube of the augmentation ideal equal to zero. The dimensions of a Lie algebra given the Hilbert series for its enveloping algebra can be computed by means of a logarithmic formula (see \cite{lof4}, section 2). We apply this to edge ideals. Some examples of periodic Lie subalgebras $\eta$ are given, yielding examples of irrational Poincare series. The ``holonomy" Lie algebra (see \cite{lof3}) of a hyperplane arrangement is defined, and some examples are given, such as the graphical arrangement $K_4$.
References
[1] L.L. Avramov, Local algebra and rational homotopy, Homotopie
Algebrique et Algebre Locale (Luminy 1982), Asterisque 113-114, Soc.
Math. France, Paris, (1984), 15-43.
[2] C. Lofwall, Central elements and deformations of local rings, JPAA, vol 91,
Issues 1-3, (1994), 183-192.
[3] C. Lofwall, The holonomy Lie algebra of a matroid, arXiv: 2012.12044.
[4] C. Lofwall, Cyclic homology of algebras of global dimension at most two,
arXiv: 1711.03644v2.
[5] G. Sjoodin, The Ext-algebra of a Golod ring, JPAA, vol 38, (1985), 337-351.
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