Factor Language Blog

I proved Conjecture 5.9 on page 14 of Barry Jessup’s paper in the case of two-step nilpotent Lie algebras. I had a proof for the case where the dimension of the center was equal to 2 a few weeks ago, and discovered a nice generalization this morning after my supervisor suggested some simplified notation to clarify the argument. The proof is nothing special; there are some tricky bookkeeping details and it runs to about 5 pages since it is a double induction, however it does not use any advanced tools from algebraic topology. I’m still proud I managed to prove it, because I’ve been working on this since April. Generalizing it to the case of an arbitrary nilpotent Lie algebra (or finding a counter-example) should be fun, and might require the use of spectral sequences. The current proof relies on what is essentially a spectral sequence collapsing at the first term. This is the second original result of my thesis. My previous result shows that the central representation is faithful for a certain class of two-step algebras. This latest result shows that it is non-trivial for all two-step algebras (however we know of two-steps where the central representation is not faithful).