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I need to work with the polynomial current Lie algebra of the general linear Lie algebra, that is $\mathfrak{gl}_N[x]$, which I consider as a complex Lie algebra. In fact I would like to work with its universal enveloping algebra. But the Lie algebra in question is infinite-dimensional. How can I define it by specifying the structure constants?

I need to work with the polynomial current Lie algebra of the general linear Lie algebra, that is $\mathfrak{gl}_N[x]$, which I consider as a complex Lie algebra. In fact I would like to work with its universal enveloping algebra. But the Lie algebra in question is infinite-dimensional. How can I define it by specifying the structure constants?