Current algebra

Summary

A current algebra is a Lie algebra of maps from a space into a fixed Lie algebra $\mathfrak{g}$, with the bracket taken pointwise. Algebraically it is the tensor product $\mathfrak{g} \otimes A$ of $\mathfrak{g}$ with a commutative algebra $A$ of functions. The name comes from quantum field theory, where the commutation relations of current operators take this form. Loop algebras, where $A$ is the ring of Laurent polynomials, are the most important case.