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Are there naturally-arising Lie algebras(or superalgebras) over $\mathbf{C}(t)$? I have such an algebra, and I want to compute its cohomology–but I'm out of ideas. Are there known methods or theory for dealing with this problem?

My Lie superalgebra is generated by $u_i $ and $ v_i,$ $i\in\mathbb{N},$ $|u_i|=1,|v_i|=0$. The bracket is as follows: $[u_i,u_j]=(i−j)u_{i+j}, [u_i,v_j]=-(t+i−j)v_{i+j},[v_i,v_j]=0$.

It arose as a tool for computing stable Betty numbers of $L_1/L_n$.

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    $\begingroup$ Why don't you say more about your Lie algebra, to provide some context? $\endgroup$
    – Victor Protsak
    Commented Apr 15, 2017 at 17:31
  • $\begingroup$ Computation of cohomology is not sensitive on the ground field, so at least in principle you can reduce to the case of an algebraically closed field. Now I'm not sure what you have in mind, which boils down to Victor's request. $\endgroup$
    – YCor
    Commented Apr 15, 2017 at 22:27
  • $\begingroup$ As regards your first question, there are naturally occurring "1-parameter families" of complex Lie algebras, for instance with basis $(u,v,w)$ and brackets $[u,v]=v$, $[u,w]=tw$, $[v,w]=0$ (where $t$ is a complex parameter; these Lie algebras are pairwise non-isomorphic except changing $t$ to $1/t$). It sounds natural to define the corresponding Lie algebra over $\mathbf{C}(t)$ with the same formula ($t$ being now the indeterminate). $\endgroup$
    – YCor
    Commented Apr 15, 2017 at 23:31
  • $\begingroup$ It is graded in $\mathbf{Z}^2$, with $u_i$ of degree $(i,0)$ and $v_i$ of degree $(i,1)$. The tensorial algebra, and hence the exterior algebra and in turn the (co)homolgy then inherits of a grading: it's enough to compute it in each fixed degree $(i,j)$ with $i,j\ge 0$ (clearly $H_n(\mathfrak{g})_{(i,j)}=0$ if $n<j$). In each case ($(i,j,n)$ fixed) it boils down to an explicit linear algebra problem. $\endgroup$
    – YCor
    Commented Apr 19, 2017 at 11:38
  • $\begingroup$ Even though I made a mistake in the description of my algebra(which Ifixed already), the problem didn't become any simpler. The complexity is that i have infinitely many such explicit algebraic problems. And i want to solve all of them in less than a year. So I clearly need some more sophisticated way of solving them. For example cohomologies of degree $(i,0)$ are just the cohomologies of $L_1$ which were computed in a paper by Goncharova. $\endgroup$
    – user83406
    Commented Apr 19, 2017 at 15:41

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