In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics. Definition For a Lie algebra over a field , if is the space of Laurent polynomials, then with the inherited bracket Geometric definition If is a Lie algebra, the tensor product of with , the algebra of (complex) smooth functions over the circle manifold (equivalently, smooth complex-valued periodic functions of a given period), is an infinite-dimensional Lie algebra with the Lie bracket given by Here and are elements of and and are elements of . This isn't precisely what would correspond to the direct product of infinitely many copies of , one for each point in , because of the smoothness restriction. Instead, it can be thought of in terms of smooth map from to ; a smooth parametrized loop in , in other words. This is why it is called the loop algebra. Gradation Defining to be the linear subspace the bracket restricts to a product hence giving the loop algebra a -graded Lie algebra structure. In particular, the bracket restricts to the 'zero-mode' subalgebra . Derivation There is a natural derivation on the loop algebra, conventionally denoted acting as and so can be thought of formally as . It is required to define affine Lie algebras, which are used in physics, particularly conformal field theory. Loop group Similarly, a set of all smooth maps from to a Lie group forms an infinite-dimensional Lie group (Lie group in the sense we can define functional derivatives over it) called the loop group. The Lie algebra of a loop group is the corresponding loop algebra. Affine Lie algebras as central extension of loop algebras If is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra gives rise to an affine Lie algebra. Furthermore this central extension is unique. The central extension is given by adjoining a central element , that is, for all , and modifying the bracket on the loop algebra to where is the Killing form. The central extension is, as a vector space, (in its usual definition, as more generally, can be taken to be an arbitrary field). Cocycle Using the language of Lie algebra cohomology, the central extension can be described using a 2-cocycle on the loop algebra. This is the map satisfying Then the extra term added to the bracket is Affine Lie algebra In physics,