Loop algebra

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

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

If g {\displaystyle {\mathfrak {g}}} is a Lie algebra, the tensor product of g {\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),

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

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

[ g 1 ⊗ f 1 , g 2 ⊗ f 2 ] = [ g 1 , g 2 ] ⊗ f 1 f 2 . {\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 g {\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 g {\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 g {\displaystyle {\mathfrak {g}}} ; a smooth parametrized loop in g {\displaystyle {\mathfrak {g}}} , in other words. This is why it is called the loop algebra.

Defining g i {\displaystyle {\mathfrak {g}}_{i}} to be the linear subspace g i = g ⊗ t i < L g , {\displaystyle {\mathfrak {g}}_{i}={\mathfrak {g}}\otimes t^{i}<L{\mathfrak {g}},} the bracket restricts to a product [ ⋅ , ⋅ ] : g i × g j → g i + j , {\displaystyle [\cdot \,,\,\cdot ]:{\mathfrak {g}}_{i}\times {\mathfrak {g}}_{j}\rightarrow {\mathfrak {g}}_{i+j},} hence giving the loop algebra a Z {\displaystyle \mathbb {Z} } -graded Lie algebra structure.

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

There is a natural derivation on the loop algebra, conventionally denoted d {\displaystyle d} acting as d : L g → L g {\displaystyle d:L{\mathfrak {g}}\rightarrow L{\mathfrak {g}}} d ( X ⊗ t n ) = n X ⊗ t n {\displaystyle d(X\otimes t^{n})=nX\otimes t^{n}} and so can be thought of formally as d = t d d t {\displaystyle d=t{\frac {d}{dt}}} .

It is required to define affine Lie algebras, which are used in physics, particularly conformal field theory.

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.

If g {\displaystyle {\mathfrak {g}}} is a semisimple Lie algebra, then a nontrivial central extension of its loop algebra L g {\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 k ^ {\displaystyle {\hat {k}}} , that is, for all X ⊗ t n ∈ L g {\displaystyle X\otimes t^{n}\in L{\mathfrak {g}}} , [ k ^ , X ⊗ t n ] = 0 , {\displaystyle [{\hat {k}},X\otimes t^{n}]=0,} and modifying the bracket on the loop algebra to [ X ⊗ t m , Y ⊗ t n ] = [ X , Y ] ⊗ t m + n + m B ( X , Y ) δ m + n , 0 k ^ , {\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 B ( ⋅ , ⋅ ) {\displaystyle B(\cdot ,\cdot )} is the Killing form.

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

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 φ : L g × L g → C {\displaystyle \varphi :L{\mathfrak {g}}\times L{\mathfrak {g}}\rightarrow \mathbb {C} } satisfying φ ( X ⊗ t m , Y ⊗ t n ) = m B ( X , Y ) δ m + n , 0 . {\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 φ ( X ⊗ t m , Y ⊗ t n ) k ^ . {\displaystyle \varphi (X\otimes t^{m},Y\otimes t^{n}){\hat {k}}.}

In physics, the central extension L g ⊕ C k ^ {\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] g ^ = L g ⊕ C k ^ ⊕ C d {\displaystyle {\hat {\mathfrak {g}}}=L{\mathfrak {g}}\oplus \mathbb {C} {\hat {k}}\oplus \mathbb {C} d} where d {\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.