リー環の拡大
B を g のハメル基底とする.このとき各 G ∈ g は適切な大きさのある添え字集合 A に対して
G = ∑ α ∈ A c α G α , c α ∈ F , G α ∈ B {\displaystyle G=\sum _{\alpha \in A}c_{\alpha }G_{\alpha },\quad c_{\alpha }\in F,G_{\alpha }\in B}
と一意的に書ける.この表示において,有限個の cα だけが 0 でない.以下では(簡単のため)基底は可算であり,添え字にはラテン文字が使われ,添え字集合は N = 1, 2, ... にとれると仮定する.ただちに基底元に対して
[ G i , G j ] = C i j k G k {\displaystyle [G_{i},G_{j}]={C_{ij}}^{k}G_{k}}
が分かる,ただしアインシュタインの和の規約を用いている.構造定数の添え字の配置(上か下か)は重要ではない.次の定理は有用である:
多項式ループ代数の中心拡大の応用として,量子的場の理論のカレント代数(英語版)が考えられる(背景).Suppose one has a current algebra, with the interesting commutator being
with a Schwinger term. To construct this algebra mathematically, let g be the centrally extended polynomial loop algebra of the previous section with
[ λ l ⊗ G i + μ C , λ m ⊗ G j + ν C ] = λ l + m ⊗ C i j k G k + l δ l + m , 0 δ i j C {\displaystyle [\lambda ^{l}\otimes G_{i}+\mu C,\lambda ^{m}\otimes G_{j}+\nu C]=\lambda ^{l+m}\otimes {C_{ij}}^{k}G_{k}+l\delta _{l+m,0}\delta _{ij}C}
as one of the commutation relations, or, with a switch of notation (l→m, m→n, i→a, j→b, λ⊗Ga→Ta) with a factor of i under the physics convention,
[ T a m , T b n ] = i C a b c T c m + n + m δ m + n , 0 δ a b C .
A dot over a quantity denotes differentiation with respect to τ and a prime differentiation with respect to σ. A dot between quantities denotes the relativistic inner product.
These rather formidable equations simplify considerably with a clever choice of parametrization called the light cone gauge. In this gauge, the equations of motion become
X ̈ μ − X μ ′′ = 0 , {\displaystyle {\ddot {X}}^{\mu }-{X^{\mu }}''=0,}
the ordinary wave equation. The price to be paid is that the light cone gauge imposes constraints,
X ̇ μ ⋅ X μ ′ = 0 , ( X ̇ ) 2 + ( X ′ ) 2 = 0 , {\displaystyle {\dot {X}}^{\mu }\cdot {X^{\mu }}'=0,\quad ({\dot {X}})^{2}+(X')^{2}=0,}
so that one cannot simply take arbitrary solutions of the wave equation to represent the strings.
とおき,交換関係を
[ G i m , G j n ] = C i j k G k m + n + m δ i j δ m + n , 0 C , [ C , G i m ] = 0 , 1 ≤ i , j , N , m , n ∈ Z [ D , G i m ] = m G i m [ D , C ] = 0 {\displaystyle {\begin{aligned}{}[G_{i}^{m},G_{j}^{n}]&={C_{ij}}^{k}G_{k}^{m+n}+m\delta _{ij}\delta ^{m+n,0}C,\\{}[C,G_{i}^{m}]&=0,\quad 1\leq i,j,N,\quad m,n\in \mathbb {Z} \\{}[D,G_{i}^{m}]&=mG_{i}^{m}\\{}[D,C]&=0\end{aligned}}}
と定める.C = D = 0 ならば,Gi で張られる部分代数は明らかに上の多項式ループ代数と同一である.
L 0 = 1 2 α 0 I α 0 I + ∑ p = 1 ∞ α − p I α p I , = α ′ p I p I + ∑ p = 1 ∞ p α p I † α p I + c {\displaystyle L_{0}={\frac {1}{2}}\alpha _{0}^{I}\alpha _{0}^{I}+\sum _{p=1}^{\infty }\alpha _{-p}^{I}\alpha _{p}^{I},=\alpha 'p^{I}p^{I}+\sum _{p=1}^{\infty }p\alpha _{p}^{I\dagger }\alpha _{p}^{I}+c}
where c is a possible ordering constant. One obtains after a somewhat lengthy calculation the relations
[ L m , L n ] = ( m − n ) L m + n , m + n ≠ 0. {\displaystyle [L_{m},L_{n}]=(m-n)L_{m+n},\quad m+n\neq 0.}
If one would allow for m + n = 0 above, then one has precisely the commutation relations of the Witt algebra. Instead one has
[ L m , L n ] = ( m − n ) L m + n + D − 2 12 ( m 3 − m ) δ m + n , 0 , ∀ m , n ∈ Z . {\displaystyle [L_{m},L_{n}]=(m-n)L_{m+n}+{\frac {D-2}{12}}(m^{3}-m)\delta _{m+n,0},\quad \forall m,n\in \mathbb {Z} .}