07.09.23 (Quarter-life crisis over and exactness of Koszul complex via representation theory of Lie-Algebras)
Just came back from the second part of my vacation this year. I think I got some of my hope back (Just like in the phd simulator online text game, just duckduckgo it quickly). Enough psychological stuff: Let's go back to some good old math funfact. There is this thing called quadratic algebras. Basically you start with what is called a quadratic datum = a tuple (V,R) where V is a vector-space and R is a subspace of V\otimes V (One can work much more generally with quivers, but let's ignore this here). Out of such a quadratic datum one can define two nice objects: The algebra A=A(V,R) generated by V with relations R and the coalgebra C=C(V,R) cogenerated by V with corelations R. The first object is the quotient of the tensor algebra, by the ideal generated by R and the second object is some nice graded sub-coalgebra of the tensor coalgebra (looks just like the tensor algebra, but has the deconcatenation coproduct). Just that you have some intuition: A and C lie on opposite sides of the world: If A is big, C is small and if C is big, A is small. Let's look at an example.
There is the example of such quadratic datum. Namely take V any vector space and R the space of symmetric size 2 tensors. The algebra you get here is the exterior algebra and the coalgebra is the symmetric coalgebra, i.e. all the S_n invariants in V^{\otimes n} build into one nice coalgebra. One can also start with R the space of antisymmetric tensors, such that the algebra you get is the symmetric algebra and the coalgebra one get's is the exterior coalgebra. Anyways out of this one can construct the Koszul complex by tensoring the algebra A with the coalgebra C and using the grading in C as homological degree. This yields a complex of left A-modules, which sometimes is exact. If it is exact, we call the algebra A Koszul and C it's Koszul dual coalgebra. However this is not always the case, even from this description it is nontrivial that this works for the exterior algebra and symmetric coalgebra. However I want you to notice: any vector space V is a representation of the Lie algebra gl(V). By construction the Koszul complex in our example is some complex of representations of gl(V) (or sl(V) for that matter), so one can use representation theory of semisimple Lie-algebras including highest weight vectors to control everything going on in the complex.
Conclusion: If you have some functorial construction F: Vect -> some category on vector spaces, it usually turns a vector space V, into some other object F(V) which becomes a representation of gl(V). For instance the category on the right might be some chain complex which is canonically a complex of gl(V) modules. Never forget that you can use your Lie-algebra tools to throw at such constructions.
16.08.23 (Quarter-life crisis and new project on Gerstenhaber cohomology)
Just came back from the first part of my vacation this year. My first PhD year is coming to a close and I haven't published yet, which is pretty depressing. Recently I heard about what people call a quarter-life crisis - I think I have it. However I have also some fresh energy from the vacation. Me, my advisor and Niels Kowalzig work on a Gerstenhaber cohomology project. Basically trying to figure out the connection between Gerstenhaber algebras and Hopf algebras. The project is pretty structure theoretic, which is not quite my thing. However Hopf algebras and Lie algebras are nice, so in the end it might end up being good. Let's see what the future brings and look optimistically towards a time, where I published my first paper.