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Commutative Algebra Webinar وبینار جبر جابجایی




TITLE  
On the Homotopy Lie Algebra of a Graded Commutative Algebra


LECTURER  
Clas Lofwall  
Stockholm University, Sweden  
 


TIME  
Thursday, February 18, 2021,   11:00 - 13:00


VENUE   (Online)



SUMMARY

 

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.

Access link:
https://meet.google.com/ndk-abim-gxy

 




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