Loop algebra

Loop algebra

In mathematics, loop algebras are certain types of Lie algebra, of particular interest in theoretical physics.

If mathfrak{g} is a Lie algebra, the tensor product of mathfrak{g} with C^infty(S^1),

:mathfrak{g}otimes C^infty(S^1),

the algebra of (complex) smooth functions over the circle manifold S1 is an infinite-dimensional Lie algebra with the Lie bracket given by

: [g_1otimes f_1,g_2 otimes f_2] = [g_1,g_2] otimes f_1 f_2.

Here g1 and g2 are elements of mathfrak{g} and f1 and f2 are elements of C^infty(S^1).

This isn't precisely what would correspond to the direct product of infinitely many copies of mathfrak{g}, one for each point in S1, because of the smoothness restriction. Instead, it can be thought of in terms of smooth map from S1 to mathfrak{g}; a smooth parameterized loop in mathfrak{g}, in other words. This is why it is called the loop algebra.

We can take the Fourier transform on this loop algebra by defining

:gotimes t^n

as

:gotimes e^{-insigma}

where

:0 &le; &sigma; <2&pi;

is a coordinatization of S1.

If mathfrak{g} is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra gives rise to an affine Kac-Moody algebra.

Similarly, a set of all smooth maps from S1 to a Lie group G 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.


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Loop — A loop is generally something that closes back on itself such as a circle. The closing can appear in time or in space.cience and technology*Loop (algebra), a quasigroup with an identity element *Loop (graph theory), an edge that begins and ends… …   Wikipedia

  • Loop group — In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise. Specifically, let LG denote the space of continuous maps S^1 o G equipped with the compact open topology. An element of LG is called… …   Wikipedia

  • Loop (topology) — In mathematics, a loop in a topological space X is a path f from the unit interval I = [0,1] to X such that f (0) = f (1). In other words, it is a path whose initial point is equal to the terminal point.A loop may also be seen as a continuous map …   Wikipedia

  • Loop quantum gravity — Not to be confused with the path integral formulation of LQG, see spin foam. This article is about LQG in its Canonical formulation.. Beyond the Standard Model …   Wikipedia

  • Affine Lie algebra — In mathematics, an affine Lie algebra is an infinite dimensional Lie algebra that is constructed in a canonical fashion out of a finite dimensional simple Lie algebra. It is a Kac–Moody algebra whose generalized Cartan matrix is positive semi… …   Wikipedia

  • List of abstract algebra topics — Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this …   Wikipedia

  • Lorentz invariance in loop quantum gravity — Loop quantum gravity (LQG) is a quantization of a classical Lagrangian field theory. It is equivalent to the usual Einstein Cartan theory in that it leads to the same equations of motion describing general relativity with torsion. As such, it can …   Wikipedia

  • Moufang loop — In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang. Contents 1 Definition 2 Examples 3 Properties …   Wikipedia

  • Moufang-Loop — In der Mathematik ist eine Quasigruppe eine nichtleere Menge Q mit einer binären Verknüpfung , in der für alle a und b in Q die Gleichungen und jeweils genau eine Lösung haben, d.h., die Lösung existiert und ist eindeutig. Eine Quasigruppe ist… …   Deutsch Wikipedia

  • Allgemeine Algebra — Der Begriff algebraische Struktur, missverständlich auch „universelle Algebra“, „allgemeine Algebra“ oder „Algebra“ genannt, bezeichnet ein mathematisches Objekt. Das Synonym allgemeine Algebra bezeichnet gleichzeitig auch den Teilbereich der… …   Deutsch Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”