Editing Loop algebra (section)

==Definition==
For a Lie algebra <math>\mathfrak{g}</math> over a field <math>K</math>, if <math>K[t,t^{-1}]</math> is the space of [[Laurent polynomials]], then
<math display=block>L\mathfrak{g} := \mathfrak{g}\otimes K[t,t^{-1}],</math>
with the inherited bracket
<math display=block>[X\otimes t^m, Y\otimes t^n] = [X,Y]\otimes t^{m+n}.</math>

=== Geometric definition ===

If <math>\mathfrak{g}</math> is a Lie algebra, the [[tensor product]] of <math>\mathfrak{g}</math> with {{math|''C''<sup>∞</sup>(''S''<sup>1</sup>)}}, the [[associative algebra|algebra]] of (complex) [[smooth function]]s over the [[n-sphere|circle]] [[manifold]] {{math|''S''<sup>1</sup>}} (equivalently, smooth complex-valued [[Periodic function|periodic functions]] of a given period),

<math display=block>\mathfrak{g}\otimes C^\infty(S^1),</math>

is an infinite-dimensional Lie algebra with the [[Lie bracket of vector fields|Lie bracket]] given by

<math display=block>[g_1\otimes f_1,g_2 \otimes f_2]=[g_1,g_2]\otimes f_1 f_2.</math>

Here {{math|''g''<sub>1</sub>}} and {{math|''g''<sub>2</sub>}} are elements of <math>\mathfrak{g}</math> and {{math|''f''<sub>1</sub>}} and {{math|''f''<sub>2</sub>}} are elements of {{math|''C''<sup>∞</sup>(''S''<sup>1</sup>)}}.

This isn't precisely what would correspond to the [[direct product]] of infinitely many copies of <math>\mathfrak{g}</math>, one for each point in {{math|''S''<sup>1</sup>}}, because of the smoothness restriction. Instead, it can be thought of in terms of [[smooth map]] from {{math|''S''<sup>1</sup>}} to <math>\mathfrak{g}</math>; a smooth parametrized loop in <math>\mathfrak{g}</math>, in other words. This is why it is called the '''loop algebra'''.