Research ideas on Homological Algebra

Question

I am planning on doing some personal studies and some poster research for some future conferences in Homological Algebra, does there exist a current list of outstanding problems in the field of Homological Algebra?

Answer 1

There is indeed a list of problems in homological algebra (also for the cohomology of groups and algebras). For a list by Brown see here.

One of the open questions is the Toral Rank Conjecture (TRC) for nilpotent Lie algebras. For a bit of context, recall the TRC by S. Halperin in 1987 from topology:

Toral Rank Conjecture (TRC). Let $T_r\rightarrow E\rightarrow B$ be a fibre bundle, where $T_r=S^1\times \cdots \times S^1$ is the $r$-dimensional torus and $B$ is a compact and simply connected space. Then the dimension of the cohomology of the total space $E$ satisfies $$ \dim H^*(E)\ge 2^r. $$ The version for complex nilpotent Lie algebras is as follows:

Conjecture (TRC for nilpotent Lie algebras): The sum of the Betti numbers of a nilpotent Lie algebra $L$ satisfies $$ \dim H^*(L)\ge 2^{\dim Z}, $$ where $Z$ denotes the center of $L$.