Loop algebra

Not to be confused with quasigroup with an identity element, also called an algebraic loop.

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

Definition

For a Lie algebra ${\displaystyle {\mathfrak {g}}}$ over a field ${\displaystyle K}$, if ${\displaystyle K[t,t^{-1}]}$ is the space of Laurent polynomials, then

$${\displaystyle L{\mathfrak {g}}:={\mathfrak {g}}\otimes K[t,t^{-1}],}$$

with the inherited bracket

$${\displaystyle [X\otimes t^{m},Y\otimes t^{n}]=[X,Y]\otimes t^{m+n}.}$$

Geometric definition

If ${\displaystyle {\mathfrak {g}}}$ is a Lie algebra, the tensor product of ${\displaystyle {\mathfrak {g}}}$ with C^∞(S¹), the algebra of (complex) smooth functions over the circle manifold S¹ (equivalently, smooth complex-valued periodic functions of a given period),

$${\displaystyle {\mathfrak {g}}\otimes C^{\infty }(S^{1}),}$$

is an infinite-dimensional Lie algebra with the Lie bracket given by

$${\displaystyle [g_{1}\otimes f_{1},g_{2}\otimes f_{2}]=[g_{1},g_{2}]\otimes f_{1}f_{2}.}$$

Here g₁ and g₂ are elements of ${\displaystyle {\mathfrak {g}}}$ and f₁ and f₂ are elements of C^∞(S¹).

This isn't precisely what would correspond to the direct product of infinitely many copies of ${\displaystyle {\mathfrak {g}}}$, 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 ${\displaystyle {\mathfrak {g}}}$; a smooth parametrized loop in ${\displaystyle {\mathfrak {g}}}$, in other words. This is why it is called the loop algebra.

Gradation

Defining ${\displaystyle {\mathfrak {g}}_{i}}$ to be the linear subspace ${\displaystyle {\mathfrak {g}}_{i}={\mathfrak {g}}\otimes t^{i}<L{\mathfrak {g}},}$ the bracket restricts to a product

$${\displaystyle [\cdot \,,\,\cdot ]:{\mathfrak {g}}_{i}\times {\mathfrak {g}}_{j}\rightarrow {\mathfrak {g}}_{i+j},}$$

hence giving the loop algebra a ${\displaystyle \mathbb {Z} }$-graded Lie algebra structure.

In particular, the bracket restricts to the 'zero-mode' subalgebra ${\displaystyle {\mathfrak {g}}_{0}\cong {\mathfrak {g}}}$.

Derivation

See also: Derivation (differential algebra)

There is a natural derivation on the loop algebra, conventionally denoted ${\displaystyle d}$ acting as

$${\displaystyle d:L{\mathfrak {g}}\rightarrow L{\mathfrak {g}}}$$

$${\displaystyle d(X\otimes t^{n})=nX\otimes t^{n}}$$

and so can be thought of formally as ${\displaystyle d=t{\frac {d}{dt}}}$.

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 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.

Affine Lie algebras as central extension of loop algebras

See also: Lie algebra extension § Polynomial loop-algebra, and Affine Lie algebra

If ${\displaystyle {\mathfrak {g}}}$ is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra ${\displaystyle L{\mathfrak {g}}}$ gives rise to an affine Lie algebra. Furthermore this central extension is unique. ^[1]

The central extension is given by adjoining a central element ${\displaystyle {\hat {k}}}$, that is, for all ${\displaystyle X\otimes t^{n}\in L{\mathfrak {g}}}$,

$${\displaystyle [{\hat {k}},X\otimes t^{n}]=0,}$$

and modifying the bracket on the loop algebra to

$${\displaystyle [X\otimes t^{m},Y\otimes t^{n}]=[X,Y]\otimes t^{m+n}+mB(X,Y)\delta _{m+n,0}{\hat {k}},}$$

where ${\displaystyle B(\cdot ,\cdot )}$ is the Killing form.

The central extension is, as a vector space, ${\displaystyle L{\mathfrak {g}}\oplus \mathbb {C} {\hat {k}}}$ (in its usual definition, as more generally, ${\displaystyle \mathbb {C} }$ can be taken to be an arbitrary field).

Cocycle

See also: Lie algebra extension § Central

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

$${\displaystyle \varphi :L{\mathfrak {g}}\times L{\mathfrak {g}}\rightarrow \mathbb {C} }$$

satisfying

$${\displaystyle \varphi (X\otimes t^{m},Y\otimes t^{n})=mB(X,Y)\delta _{m+n,0}.}$$

Then the extra term added to the bracket is ${\displaystyle \varphi (X\otimes t^{m},Y\otimes t^{n}){\hat {k}}.}$

Affine Lie algebra

In physics, the central extension ${\displaystyle L{\mathfrak {g}}\oplus \mathbb {C} {\hat {k}}}$ is sometimes referred to as the affine Lie algebra. In mathematics, this is insufficient, and the full affine Lie algebra is the vector space ^[2]

$${\displaystyle {\hat {\mathfrak {g}}}=L{\mathfrak {g}}\oplus \mathbb {C} {\hat {k}}\oplus \mathbb {C} d}$$

where ${\displaystyle d}$ is the derivation defined above.

On this space, the Killing form can be extended to a non-degenerate form, and so allows a root system analysis of the affine Lie algebra.

References

1. Kac, V.G. (1990). Infinite-dimensional Lie algebras (3rd ed.). Cambridge University Press. Exercise 7.8. ISBN   978-0-521-37215-2.
2. P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, 1997, ISBN   0-387-94785-X

• Fuchs, Jurgen (1992), Affine Lie Algebras and Quantum Groups, Cambridge University Press, ISBN   0-521-48412-X