Affine current algebra

{title2=$L\mathfrak g=\mathfrak g\otimes_k k[t,t^{-1}]$}
{wiki}

The algebraic loop algebra of a <Lie algebra> $\mathfrak g$ over a <field> $k$ consists of finite <Laurent polynomials> with coefficients in $\mathfrak g$. Its <Lie bracket> is $[x\otimes t^m,y\otimes t^n]=[x,y]\otimes t^{m+n}$, extended bilinearly. Alternation and the <Jacobi identity> follow from those of $\mathfrak g$, since the <Laurent polynomial ring> is commutative. Appropriate smooth or formal completions give other loop-algebra variants; the <affine current algebra> adds a central extension to these Fourier modes. For the affine extension, see https://math.mit.edu/~kac/not-easily-available/kac-and-wakimoto-1990.pdf[Kac and Wakimoto, Section 0.2].


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