In a 1988 paper "The Lie algebras $\mathfrak{gl}(\lambda)$ and cohomologies of Lie algebras of differential operators", Feigin defined the analytic continuation of $\mathfrak{sl}(n)$ algebras (over $\mathbb{C}$) to real $n$. He started by considering the universal enveloping algebra of $\mathfrak{sl}_2$ which he factored by an ideal generated by $\Delta-const$ where $\Delta$ is the Casimir. The result is an infinite-dimensional Lie algebra $\mathfrak{gl}(\lambda)$, and factoring it further by 1 gets $\mathfrak{sl}(\lambda)$. For integer $\lambda$ this algebra has an ideal factoring by which one gets $\mathfrak{sl}(n)$. Feigin's paper is terse; a more detailed discussion can be found in the paper by Post and Van den Hijligenberg.

I was wondering if a similar construction is possible for other classical Lie algebras. For example, what is the analytic continuation of $\mathfrak{so}(n)$?



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