Lie algebra extension

In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension Template:Math is an enlargement of a given Lie algebra Template:Math by another Lie algebra Template:Math. Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types are the split extension and the central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations. Such a Lie algebra will contain central charges.

Starting with a polynomial loop algebra over finite-dimensional simple Lie algebra and performing two extensions, a central extension and an extension by a derivation, one obtains a Lie algebra which is isomorphic with an untwisted affine Kac–Moody algebra. Using the centrally extended loop algebra one may construct a current algebra in two spacetime dimensions. The Virasoro algebra is the universal central extension of the Witt algebra.

Central extensions are needed in physics, because the symmetry group of a quantized system usually is a central extension of the classical symmetry group, and in the same way the corresponding symmetry Lie algebra of the quantum system is, in general, a central extension of the classical symmetry algebra. Kac–Moody algebras have been conjectured to be symmetry groups of a unified superstring theory. The centrally extended Lie algebras play a dominant role in quantum field theory, particularly in conformal field theory, string theory and in M-theory.

A large portion towards the end is devoted to background material for applications of Lie algebra extensions, both in mathematics and in physics, in areas where they are actually useful. A parenthetical link, (background material), is provided where it might be beneficial.

History

Due to the Lie correspondence, the theory, and consequently the history of Lie algebra extensions, is tightly linked to the theory and history of group extensions. A systematic study of group extensions was performed by the Austrian mathematician Otto Schreier in 1923 in his PhD thesis and later published. The problem posed for his thesis by Otto Hölder was "given two groups Template:Mvar and Template:Mvar, find all groups Template:Mvar having a normal subgroup Template:Mvar isomorphic to Template:Mvar such that the factor group Template:Math is isomorphic to Template:Mvar".

Lie algebra extensions are most interesting and useful for infinite-dimensional Lie algebras. In 1967, Victor Kac and Robert Moody independently generalized the notion of classical Lie algebras, resulting in a new theory of infinite-dimensional Lie algebras, now called Kac–Moody algebras. They generalize the finite-dimensional simple Lie algebras and can often concretely be constructed as extensions.

Notation and proofs

Notational abuse to be found below includes Template:Math for the exponential map Template:Math given an argument, writing Template:Mvar for the element Template:Math in a direct product Template:Math (Template:Math is the identity in Template:Mvar), and analogously for Lie algebra direct sums (where also Template:Math and Template:Math are used interchangeably). Likewise for semidirect products and semidirect sums. Canonical injections (both for groups and Lie algebras) are used for implicit identifications. Furthermore, if Template:Math, Template:Math, ..., are groups, then the default names for elements of Template:Math, Template:Math, ..., are Template:Math, Template:Math, ..., and their Lie algebras are Template:Math, Template:Math, ... . The default names for elements of Template:Math, Template:Math, ..., are Template:Math, Template:Math, ... (just like for the groups!), partly to save scarce alphabetical resources but mostly to have a uniform notation.

Lie algebras that are ingredients in an extension will, without comment, be taken to be over the same field.

The summation convention applies, including sometimes when the indices involved are both upstairs or both downstairs.

Caveat: Not all proofs and proof outlines below have universal validity. The main reason is that the Lie algebras are often infinite-dimensional, and then there may or may not be a Lie group corresponding to the Lie algebra. Moreover, even if such a group exists, it may not have the "usual" properties, e.g. the exponential map might not exist, and if it does, it might not have all the "usual" properties. In such cases, it is questionable whether the group should be endowed with the "Lie" qualifier. The literature is not uniform. For the explicit examples, the relevant structures are supposedly in place.

Definition

Lie algebra extensions are formalized in terms of short exact sequences. A short exact sequence is an exact sequence of length three, Template:NumBlk2 such that Template:Mvar is a monomorphism, Template:Mvar is an epimorphism, and Template:Math. From these properties of exact sequences, it follows that (the image of) $𝔥$ is an ideal in $𝔢$. Moreover,

$𝔤\cong 𝔢/\mathrm{Im}i=𝔢/\mathrm{Ker}s,$

but it is not necessarily the case that $𝔤$ is isomorphic to a subalgebra of $𝔢$. This construction mirrors the analogous constructions in the closely related concept of group extensions.

If the situation in (Template:EquationNote) prevails, non-trivially and for Lie algebras over the same field, then one says that $𝔢$ is an extension of $𝔤$ by $𝔥$.

Properties

The defining property may be reformulated. The Lie algebra $𝔢$ is an extension of $𝔤$ by $𝔥$ if Template:NumBlk2 is exact. Here the zeros on the ends represent the zero Lie algebra (containing only the zero vector Template:Math) and the maps are the obvious ones; $\iota$ maps Template:Math to Template:Math and $\sigma$ maps all elements of $𝔤$ to Template:Math. With this definition, it follows automatically that Template:Math is a monomorphism and Template:Math is an epimorphism.

An extension of $𝔤$ by $𝔥$ is not necessarily unique. Let $𝔢,{𝔢}^{\prime }$ denote two extensions and let the primes below have the obvious interpretation. Then, if there exists a Lie algebra isomorphism $f:𝔢\to {𝔢}^{\prime }$ such that

$f\circ i=i^{\prime },s^{\prime }\circ f=s,$

then the extensions $𝔢$ and ${𝔢}^{\prime }$ are said to be equivalent extensions. Equivalence of extensions is an equivalence relation.

Extension types

Trivial

A Lie algebra extension

$𝔥\stackrel{i}{\hookrightarrow }𝔱\stackrel{s}{\twoheadrightarrow }𝔤,$

is trivial if there is a subspace Template:Math such that Template:Math and Template:Math is an ideal in Template:Math.

Split

A Lie algebra extension

$𝔥\stackrel{i}{\hookrightarrow }𝔰\stackrel{s}{\twoheadrightarrow }𝔤,$

is split if there is a subspace Template:Math such that Template:Math as a vector space and Template:Math is a subalgebra in Template:Math.

An ideal is a subalgebra, but a subalgebra is not necessarily an ideal. A trivial extension is thus a split extension.

Central

Central extensions of a Lie algebra Template:Math by an abelian Lie algebra Template:Math can be obtained with the help of a so-called (nontrivial) 2-cocycle (background) on Template:Math. Non-trivial 2-cocycles occur in the context of projective representations (background) of Lie groups. This is alluded to further down.

A Lie algebra extension

$𝔥\stackrel{i}{\hookrightarrow }𝔢\stackrel{s}{\twoheadrightarrow }𝔤,$

is a central extension if Template:Math is contained in the center Template:Math of Template:Math.

Properties

• Since the center commutes with everything, Template:Math in this case is abelian.
• Given a central extension Template:Math of Template:Math, one may construct a 2-cocycle on Template:Math. Suppose Template:Math is a central extension of Template:Math by Template:Math. Let Template:Math be a linear map from Template:Math to Template:Math with the property that Template:Math, i.e. Template:Math is a section of Template:Math. Use this section to define Template:Math by

$ϵ(G_1,G_2)=l([G_1,G_2])-[l(G_1),l(G_2)],G_1,G_2\in 𝔤.$

The map Template:Math satisfies

$ϵ(G_1,[G_2,G_3])+ϵ(G_2,[G_3,G_1])+ϵ(G_3,[G_1,G_2])=0\in 𝔢.$

To see this, use the definition of Template:Math on the left hand side, then use the linearity of Template:Math. Use Jacobi identity on Template:Math to get rid of half of the six terms. Use the definition of Template:Math again on terms Template:Math sitting inside three Lie brackets, bilinearity of Lie brackets, and the Jacobi identity on Template:Math, and then finally use on the three remaining terms that Template:Math and that Template:Math so that Template:Math brackets to zero with everything. It then follows that Template:Math satisfies the corresponding relation, and if Template:Math in addition is one-dimensional, then Template:Math is a 2-cocycle on Template:Math (via a trivial correspondence of Template:Math with the underlying field).

A central extension

$0\stackrel{\iota }{\hookrightarrow }𝔥\stackrel{i}{\hookrightarrow }𝔢\stackrel{s}{\twoheadrightarrow }𝔤\stackrel{\sigma }{\twoheadrightarrow }0$

is universal if for every other central extension

$0\stackrel{\iota }{\hookrightarrow }{𝔥}^{\prime }\stackrel{i^{\prime }}{\hookrightarrow }{𝔢}^{\prime }\stackrel{s^{\prime }}{\twoheadrightarrow }𝔤\stackrel{\sigma }{\twoheadrightarrow }0$

there exist unique homomorphisms $\mathrm{\Phi }:𝔢\to {𝔢}^{\prime }$ and $\mathrm{\Psi }:𝔥\to {𝔥}^{\prime }$ such that the diagram

commutes, i.e. Template:Math and Template:Math. By universality, it is easy to conclude that such universal central extensions are unique up to isomorphism.

Construction

By direct sum

Let $𝔤$, $𝔥$ be Lie algebras over the same field $F$. Define

$𝔢=𝔥\times 𝔤,$

and define addition pointwise on $𝔢$. Scalar multiplication is defined by

$\alpha (H,G)=(\alpha H,\alpha G),\alpha \in F,H\in 𝔥,G\in 𝔤.$

With these definitions, $𝔥\times 𝔤\equiv 𝔥\oplus 𝔤$ is a vector space over $F$. With the Lie bracket: Template:NumBlk2 $𝔢$ is a Lie algebra. Define further

$i:𝔥\hookrightarrow 𝔢;H\mapsto (H,0),s:𝔢\twoheadrightarrow 𝔤;(H,G)\mapsto G.$

It is clear that (Template:EquationNote) holds as an exact sequence. This extension of $𝔤$ by $𝔥$ is called a trivial extension. It is, of course, nothing else than the Lie algebra direct sum. By symmetry of definitions, $𝔢$ is an extension of $𝔥$ by $𝔤$ as well, but $𝔥\oplus 𝔤\ne 𝔤\oplus 𝔥$. It is clear from (Template:EquationNote) that the subalgebra $0\oplus 𝔤$ is an ideal (Lie algebra). This property of the direct sum of Lie algebras is promoted to the definition of a trivial extension.

By semidirect sum

Inspired by the construction of a semidirect product (background) of groups using a homomorphism Template:Math, one can make the corresponding construct for Lie algebras.

If Template:Math is a Lie algebra homomorphism, then define a Lie bracket on $𝔢=𝔥\oplus 𝔤$ by Template:NumBlk2 With this Lie bracket, the Lie algebra so obtained is denoted Template:Math and is called the semidirect sum of Template:Math and Template:Math.

By inspection of (Template:EquationNote) one sees that Template:Math is a subalgebra of Template:Math and Template:Math is an ideal in Template:Math. Define Template:Math by Template:Math and Template:Math by Template:Math. It is clear that Template:Math. Thus Template:Math is a Lie algebra extension of Template:Math by Template:Math.

As with the trivial extension, this property generalizes to the definition of a split extension.

Example
Let Template:Mvar be the Lorentz group Template:Math and let Template:Math denote the translation group in 4 dimensions, isomorphic to Template:Math, and consider the multiplication rule of the Poincaré group Template:Math

$(a_2,{\mathrm{\Lambda }}_2)(a_1,{\mathrm{\Lambda }}_1)=(a_2+{\mathrm{\Lambda }}_2{a}_1,{\mathrm{\Lambda }}_2{\mathrm{\Lambda }}_1),a_1,a_2\in \mathrm{T}\subset \mathrm{P},{\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2\in \mathrm{O}(3,1)\subset \mathrm{P},$

(where Template:Math and Template:Math are identified with their images in Template:Math). From it follows immediately that, in the Poincaré group, Template:Math. Thus every Lorentz transformation Template:Math corresponds to an automorphism Template:Math of Template:Math with inverse Template:Math and Template:Math is clearly a homomorphism. Now define

$\bar{\mathrm{P}}=\mathrm{T}{\otimes }_S\mathrm{O}(3,1),$

endowed with multiplication given by (Template:EquationNote). Unwinding the definitions one finds that the multiplication is the same as the multiplication one started with and it follows that Template:Math. From (Template:EquationNote) follows that Template:Math and then from (Template:EquationNote) it follows that Template:Math.

By derivation

Let Template:Math be a derivation (background) of Template:Math and denote by Template:Math the one-dimensional Lie algebra spanned by Template:Math. Define the Lie bracket on Template:Math by

$[G_1+H_1,G_2+H_2]=[\lambda \delta +H_1,\mu \delta +H_2]=[H_1,H_2]+\lambda \delta (H_2)-\mu \delta (H_1).$

It is obvious from the definition of the bracket that Template:Math is and ideal in Template:Math in and that Template:Math is a subalgebra of Template:Math. Furthermore, Template:Math is complementary to Template:Math in Template:Math. Let Template:Math be given by Template:Math and Template:Math by Template:Math. It is clear that Template:Math. Thus Template:Math is a split extension of Template:Math by Template:Math. Such an extension is called extension by a derivation.

If Template:Math is defined by Template:Math, then Template:Mvar is a Lie algebra homomorphism into Template:Math. Hence this construction is a special case of a semidirect sum, for when starting from Template:Mvar and using the construction in the preceding section, the same Lie brackets result.

By 2-cocycle

If Template:Math is a 2-cocycle (background) on a Lie algebra Template:Math and Template:Math is any one-dimensional vector space, let Template:Math (vector space direct sum) and define a Lie bracket on Template:Math by

$[\mu H+G_1,\nu H+G_2]=[G_1,G_2]+\epsilon (G_1,G_2)H,\mu ,\nu \in F.$

Here Template:Mvar is an arbitrary but fixed element of Template:Math. Antisymmetry follows from antisymmetry of the Lie bracket on Template:Math and antisymmetry of the 2-cocycle. The Jacobi identity follows from the corresponding properties of Template:Math and of Template:Math. Thus Template:Math is a Lie algebra. Put Template:Math and it follows that Template:Math. Also, it follows with Template:Math and Template:Math that Template:Math. Hence Template:Math is a central extension of Template:Math by Template:Math. It is called extension by a 2-cocycle.

Theorems

Below follows some results regarding central extensions and 2-cocycles.

Theorem
Let Template:Math and Template:Math be cohomologous 2-cocycles on a Lie algebra Template:Math and let Template:Math and Template:Math be respectively the central extensions constructed with these 2-cocycles. Then the central extensions Template:Math and Template:Math are equivalent extensions.
Proof
By definition, Template:Math. Define

$\psi :G+\mu c\in {𝔢}_1\mapsto G+\mu c+f(G)c\in {𝔢}_2.$

It follows from the definitions that Template:Mvar is a Lie algebra isomorphism and (Template:EquationNote) holds.

Corollary
A cohomology class Template:Math defines a central extension of Template:Math which is unique up to isomorphism.

The trivial 2-cocycle gives the trivial extension, and since a 2-coboundary is cohomologous with the trivial 2-cocycle, one has
Corollary
A central extension defined by a coboundary is equivalent with a trivial central extension.

Theorem
A finite-dimensional simple Lie algebra has only trivial central extensions.
Proof
Since every central extension comes from a 2-cocycle Template:Math, it suffices to show that every 2-cocycle is a coboundary. Suppose Template:Math is a 2-cocycle on Template:Math. The task is to use this 2-cocycle to manufacture a 1-cochain Template:Mvar such that Template:Math.

The first step is to, for each Template:Math, use Template:Mvar to define a linear map Template:Math by ${\rho }_{G_1}(G_2)\equiv \phi (G_1,G_2)$. These linear maps are elements of Template:Math. Let Template:Math be the vector space isomorphism associated to the nondegenerate Killing form Template:Math, and define a linear map Template:Math by $d(G_1)\equiv \nu ({\rho }_{G_1})$. This turns out to be a derivation (for a proof, see below). Since, for semisimple Lie algebras, all derivations are inner, one has Template:Math for some Template:Math. Then

$\phi (G_1,G_2)\equiv {\rho }_{G_1}(G_2)=K(\nu ({\rho }_{G_1}),G_2)\equiv K(d(G_1),G_2)=K({\mathrm{a}\mathrm{d}}_{G_d}(G_1),G_2)=K([G_d,G_1],G_2)=K(G_d,[G_1,G_2]).$

Let Template:Mvar be the 1-cochain defined by

$f(G)=K(G_d,G).$

Then

$\delta f(G_1,G_2)=f([G_1,G_2])=K(G_d,[G_1,G_2])=\phi (G_1,G_2),$

showing that Template:Math is a coboundary. Template:Hidden begin To verify that Template:Mvar actually is a derivation, first note that it is linear since Template:Mvar is, then compute

$\begin{array}{cc}K(d([G_1,G_2]),G_3))& =\phi ([G_1,G_2]),G_3))=\phi (G_1,[G_2,G_3])+\phi (G_2,[G_3,G_1])\\ & =K(d(G_1),[G_2,G_3])+K(d(G_1),(G_3,G_1))=K([d(G_1),G_2],G_3)+K([G_1,d(G_2)],G_3))\\ & =K([d(G_1),G_2]+[G_1,d(G_2)],G_3).\end{array}$

By appeal to the non-degeneracy of Template:Mvar, the left arguments of Template:Mvar are equal on the far left and far right. Template:Hidden end The observation that one can define a derivation Template:Math, given a symmetric non-degenerate associative form Template:Mvar and a 2-cocycle Template:Mvar, by

$K(\nu ({\rho }_{G_1}),G_2)\equiv K(d(G_1),G_2),$

or using the symmetry of Template:Math and the antisymmetry of Template:Mvar,

$K(d(G_1),G_2)=-K(G_1,d(G_2)),$

leads to a corollary.

Corollary
Let Template:Mvar be a non-degenerate symmetric associative bilinear form and let Template:Mvar be a derivation satisfying

$L(d(G_1),G_2)=-L(G_1,d(G_2)),$

then Template:Mvar defined by

$\phi (G_1,G_2)=L(d(G_1),G_2)$

is a 2-cocycle.

Proof The condition on Template:Mvar ensures the antisymmetry of Template:Mvar. The Jacobi identity for 2-cocycles follows starting with

$\phi ([G_1,G_2],G_3)=L(d[G_1,G_2],G_3)=L([d(G_1),G_2],G_3)+L([G_1,d(G_2)],G_3),$

using symmetry of the form, the antisymmetry of the bracket, and once again the definition of Template:Mvar in terms of Template:Mvar.

If Template:Math is the Lie algebra of a Lie group Template:Math and Template:Math is a central extension of Template:Math, one may ask whether there is a Lie group Template:Math with Lie algebra Template:Math. The answer is, by Lie's third theorem affirmative. But is there a central extension Template:Math of Template:Math with Lie algebra Template:Math? The answer to this question requires some machinery, and can be found in Template:Harvtxt.

Applications

The "negative" result of the preceding theorem indicates that one must, at least for semisimple Lie algebras, go to infinite-dimensional Lie algebras to find useful applications of central extensions. There are indeed such. Here will be presented affine Kac–Moody algebras and Virasoro algebras. These are extensions of polynomial loop-algebras and the Witt algebra respectively.

Polynomial loop algebra

Let Template:Math be a polynomial loop algebra (background),

$𝔤=ℂ[\lambda ,{\lambda }^{-1}]\otimes {𝔤}_0,$

where Template:Math is a complex finite-dimensional simple Lie algebra. The goal is to find a central extension of this algebra. Two of the theorems apply. On the one hand, if there is a 2-cocycle on Template:Math, then a central extension may be defined. On the other hand, if this 2-cocycle is acting on the Template:Math part (only), then the resulting extension is trivial. Moreover, derivations acting on Template:Math (only) cannot be used for definition of a 2-cocycle either because these derivations are all inner and the same problem results. One therefore looks for derivations on Template:Math. One such set of derivations is

$d_k\equiv {\lambda }^{k+1}\frac{d}{d\lambda },k\in \mathbb{Z}.$

In order to manufacture a non-degenerate bilinear associative antisymmetric form Template:Math on Template:Math, attention is focused first on restrictions on the arguments, with Template:Math fixed. It is a theorem that every form satisfying the requirements is a multiple of the Killing form Template:Mvar on Template:Math. This requires

$L({\lambda }^m\otimes G_1,{\lambda }^n\otimes G_2)={\gamma }_{lm}K(G_1,G_2).$

Symmetry of Template:Mvar implies

${\gamma }_{mn}={\gamma }_{nm},$

and associativity yields

${\gamma }_{m+k,n}={\gamma }_{m,k+n}.$

With Template:Math one sees that Template:Math. This last condition implies the former. Using this fact, define Template:Math. The defining equation then becomes

$L({\lambda }^m\otimes G_1,{\lambda }^n\otimes G_2)=f(m+n)K(G_1,G_2).$

For every Template:Math the definition

$f(n)={\delta }_{ni}\iff {\gamma }_{mn}={\delta }_{m+n,i}$

does define a symmetric associative bilinear form

$L_i({\lambda }^m\otimes G_1,{\lambda }^n\otimes G_2)={\delta }_{m+n,i}K(G_1,G_2).$

These span a vector space of forms which have the right properties.

Returning to the derivations at hand and the condition

$L_i(d_k({\lambda }^l\otimes G_1),{\lambda }^m\otimes G_2)=-L_i({\lambda }^l\otimes G_1,d_k({\lambda }^m\otimes G_2)),$

one sees, using the definitions, that

$l{\delta }_{k+l+m,i}=-m{\delta }_{k+l+m,i},$

or, with Template:Math,

$n{\delta }_{k+n,i}=0.$

This (and the antisymmetry condition) holds if Template:Math, in particular it holds when Template:Math.

Thus choose Template:Math and Template:Math. With these choices, the premises in the corollary are satisfied. The 2-cocycle Template:Mvar defined by

$\phi (P(\lambda )\otimes G_1),Q(\lambda )\otimes G_2))=L(\lambda \frac{dP}{d\lambda }\otimes G_1,Q(\lambda )\otimes G_2)$

is finally employed to define a central extension of Template:Math,

$𝔢=𝔤\oplus ℂC,$

with Lie bracket

$[P(\lambda )\otimes G_1+\mu C,Q(\lambda )\otimes G_2+\nu C]=P(\lambda )Q(\lambda )\otimes [G_1,G_2]+\phi (P(\lambda )\otimes G_1,Q(\lambda )\otimes G_2)C.$

For basis elements, suitably normalized and with antisymmetric structure constants, one has

$\begin{array}{cc}[{\lambda }^l\otimes G_i+\mu C,{\lambda }^m\otimes G_j+\nu C]& ={\lambda }^{l+m}\otimes [G_i,G_j]+\phi ({\lambda }^l\otimes G_i,{\lambda }^m\otimes G_j)C\\ & ={\lambda }^{l+m}\otimes {C_{ij}}^k{G}_k+L(\lambda \frac{d{\lambda }^l}{d\lambda }\otimes G_i,{\lambda }^m\otimes G_j)C\\ & ={\lambda }^{l+m}\otimes {C_{ij}}^k{G}_k+lL({\lambda }^l\otimes G_i,{\lambda }^m\otimes G_j)C\\ & ={\lambda }^{l+m}\otimes {C_{ij}}^k{G}_k+l{\delta }_{l+m,0}K(G_i,G_j)C\\ & ={\lambda }^{l+m}\otimes {C_{ij}}^k{G}_k+l{\delta }_{l+m,0}{C_{ik}}^m{C_{jm}}^{k}C={\lambda }^{l+m}\otimes {C_{ij}}^k{G}_k+l{\delta }_{l+m,0}{\delta }^{ij}C.\end{array}$

This is a universal central extension of the polynomial loop algebra.

A note on terminology

In physics terminology, the algebra of above might pass for a Kac–Moody algebra, whilst it will probably not in mathematics terminology. An additional dimension, an extension by a derivation is required for this. Nonetheless, if, in a physical application, the eigenvalues of Template:Math or its representative are interpreted as (ordinary) quantum numbers, the additional superscript on the generators is referred to as the level. It is an additional quantum number. An additional operator whose eigenvalues are precisely the levels is introduced further below.

Current algebra

Murray Gell-Mann, 1969 Nobel Laureate in physics, initiated the field of current algebra in the 1960s. It exploits known local symmetries even without knowledge of the underlying dynamics to extract predictions, e.g. the Adler–Weisberger sum rule.

Template:Main As an application of a central extension of polynomial loop algebra, a current algebra of a quantum field theory is considered (background). Suppose one has a current algebra, with the interesting commutator being Template:NumBlk2 with a Schwinger term. To construct this algebra mathematically, let Template:Math be the centrally extended polynomial loop algebra of the previous section with

$[{\lambda }^l\otimes G_i+\mu C,{\lambda }^m\otimes G_j+\nu C]={\lambda }^{l+m}\otimes {C_{ij}}^k{G}_k+l{\delta }_{l+m,0}{\delta }_{ij}C$

as one of the commutation relations, or, with a switch of notation (Template:Math) with a factor of Template:Math under the physics convention,

$[T_a^m,T_b^n]=i{C_{ab}}^c{T}_c^{m+n}+m{\delta }_{m+n,0}{\delta }_{ab}C.$

Define using elements of Template:Math,

$J_a(x)=\frac{\hslash }{L}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\frac{2\pi inx}{L}}{T}_a^{-n},x\in ℝ.$

One notes that

$J_a(x+L)=J_a(x)$

so that it is defined on a circle. Now compute the commutator,

$\begin{array}{cc}[][J_a(x),J_b(y)]& ={\left(\frac{\hslash }{L}\right)}^2[\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\frac{2\pi inx}{L}}{T}_a^{-n},\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\frac{2\pi imy}{L}}{T}_b^{-m}]\\ & ={\left(\frac{\hslash }{L}\right)}^2\sum _{m,n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\frac{2\pi inx}{L}}{e}^{\frac{2\pi imy}{L}}[T_a^{-n},T_b^{-m}].\end{array}$

For simplicity, switch coordinates so that Template:Math and use the commutation relations,

$\begin{array}{cc}[][J_a(z),J_b(0)]& ={\left(\frac{\hslash }{L}\right)}^2\sum _{m,n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\frac{2\pi inz}{L}}[i{C_{ab}}^c{T}_c^{-m-n}+m{\delta }_{m+n,0}{\delta }_{ab}C]\\ & ={\left(\frac{\hslash }{L}\right)}^2\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\frac{2\pi i(-m)z}{L}}\sum _{l=-\mathrm{\infty }}^{\mathrm{\infty }}i{e}^{\frac{2\pi i(l)z}{L}}{C_{ab}}^c{T}_c^{-l}+{\left(\frac{\hslash }{L}\right)}^2\sum _{m,n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\frac{2\pi inz}{L}}m{\delta }_{m+n,0}{\delta }_{ab}C\\ & =\left(\frac{\hslash }{L}\right)\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\frac{2\pi imz}{L}}i{C_{ab}}^c{J}_c(z)-{\left(\frac{\hslash }{L}\right)}^2\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\frac{2\pi inz}{L}}n{\delta }_{ab}C\end{array}$

Now employ the Poisson summation formula,

$\frac{1}{L}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\frac{-2\pi inz}{L}}=\frac{1}{L}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\delta (z+nL)=\delta (z)$

for Template:Mvar in the interval Template:Math and differentiate it to yield

$-\frac{2\pi i}{L^2}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}n{e}^{\frac{-2\pi inz}{L}}={\delta }^{\prime }(z),$

and finally

$[J_a(x-y),J_b(0)]=i\hslash {C_{ab}}^c{J}_c(x-y)\delta (x-y)+\frac{i{\hslash }^2}{2\pi }{\delta }_{ab}C{\delta }^{\prime }(x-y),$

or

$[J_a(x),J_b(y)]=i\hslash {C_{ab}}^c{J}_c(x)\delta (x-y)+\frac{i{\hslash }^2}{2\pi }{\delta }_{ab}C{\delta }^{\prime }(x-y),$

since the delta functions arguments only ensure that the arguments of the left and right arguments of the commutator are equal (formally Template:Math).

By comparison with Template:EquationNote, this is a current algebra in two spacetime dimensions, including a Schwinger term, with the space dimension curled up into a circle. In the classical setting of quantum field theory, this is perhaps of little use, but with the advent of string theory where fields live on world sheets of strings, and spatial dimensions are curled up, there may be relevant applications.

Kac–Moody algebra

Robert Moody (left), Fellow of the Royal Society of Canada, is a Canadian mathematician at University of Alberta. He is co-discoverer of the Kac–Moody algebra together with Victor Kac, Fellow of the American Mathematical Society, a Russian mathematician working at MIT.

The derivation Template:Math used in the construction of the 2-cocycle Template:Mvar in the previous section can be extended to a derivation Template:Mvar on the centrally extended polynomial loop algebra, here denoted by Template:Math in order to realize a Kac–Moody algebra (background). Simply set

$D(P(\lambda )\otimes G+\mu C)=\lambda \frac{dP(\lambda )}{d\lambda }\otimes G.$

Next, define as a vector space

$𝔢=ℂd+𝔤.$

The Lie bracket on Template:Math is, according to the standard construction with a derivation, given on a basis by

$\begin{array}{cc}[{\lambda }^m\otimes G_1+\mu C+\nu D,{\lambda }^n\otimes G_2+{\mu }^{\prime }C+{\nu }^{\prime }D]& ={\lambda }^{m+n}\otimes [G_1,G_2]+m{\delta }_{m+n,0}K(G_1,G_2)C+\nu D({\lambda }^n\otimes G_1)-{\nu }^{\prime }D({\lambda }^m\otimes G_2)\\ & ={\lambda }^{m+n}\otimes [G_1,G_2]+m{\delta }_{m+n,0}K(G_1,G_2)C+\nu n{\lambda }^n\otimes G_1-{\nu }^{\prime }m{\lambda }^m\otimes G_2.\end{array}$

For convenience, define

$G_i^m\leftrightarrow {\lambda }^m\otimes G_i.$

In addition, assume the basis on the underlying finite-dimensional simple Lie algebra has been chosen so that the structure coefficients are antisymmetric in all indices and that the basis is appropriately normalized. Then one immediately through the definitions verifies the following commutation relations.

$\begin{array}{cc}[G_i^m,G_j^n]& ={C_{ij}}^k{G}_k^{m+n}+m{\delta }_{ij}{\delta }^{m+n,0}C,\\ [C,G_i^m]& =0,1\le i,j,N,m,n\in \mathbb{Z}\\ [D,G_i^m]& =m{G}_i^m\\ [D,C]& =0.\end{array}$

These are precisely the short-hand description of an untwisted affine Kac–Moody algebra. To recapitulate, begin with a finite-dimensional simple Lie algebra. Define a space of formal Laurent polynomials with coefficients in the finite-dimensional simple Lie algebra. With the support of a symmetric non-degenerate alternating bilinear form and a derivation, a 2-cocycle is defined, subsequently used in the standard prescription for a central extension by a 2-cocycle. Extend the derivation to this new space, use the standard prescription for a split extension by a derivation and an untwisted affine Kac–Moody algebra obtains.

Virasoro algebra

Template:Main The purpose is to construct the Virasoro algebra (named after Miguel Angel Virasoro) as a central extension by a 2-cocycle Template:Mvar of the Witt algebra Template:Math (background). The Jacobi identity for 2-cocycles yields Template:NumBlk2 Letting Template:Math and using antisymmetry of Template:Mvar one obtains

$(m+p){\eta }_{mp}=(m-p){\eta }_{m+p,0}.$

In the extension, the commutation relations for the element Template:Math are

$[d_0+\mu C,d_m+\nu C{]}_{\phi }=-m{d}_m+{\eta }_{0m}C=-m(d_m-\frac{{\eta }_{0m}}{m}C).$

It is desirable to get rid of the central charge on the right hand side. To do this define

$f:W\to ℂ;d_m\to \frac{\phi (d_0,d_m)}{m}=\frac{{\eta }_{0m}}{m}.$

Then, using Template:Math as a 1-cochain,

$\eta {\text{'}}_{0n}={\phi }^{\prime }(d_0,d_n)=\phi (d_0,d_n)+\delta f([d_0,d_n])=\phi (d_0,d_n)-n\frac{{\eta }^{0n}}{n}=0,$

so with this 2-cocycle, equivalent to the previous one, one has

$[d_0+\mu C,d_m+\nu C{]}_{{\phi }^{\prime }}=-m{d}_m.$

With this new 2-cocycle (skip the prime) the condition becomes

$(n+p){\eta }_{mp}=(n-p){\eta }_{m+p,0}=0,$

and thus

${\eta }_{mp}=a(m){\delta }_{m.-p},a(-m)=-a(m),$

where the last condition is due to the antisymmetry of the Lie bracket. With this, and with Template:Math (cutting out a "plane" in Template:Math), (Template:EquationNote) yields

$(2m+p)a(p)+(m-p)a(m+p)+(m+2p)a(m)=0,$

that with Template:Math (cutting out a "line" in Template:Math) becomes

$(m-1)a(m+1)-(m+2)a(m)+(2m+1)a(1)=0.$

This is a difference equation generally solved by

$a(m)=\alpha m+\beta m^3.$

The commutator in the extension on elements of Template:Math is then

$[d_l,d_m]=(l-m)d_{l+m}+(\alpha m+\beta m^3){\delta }_{l,-m}C.$

With Template:Mvar it is possible to change basis (or modify the 2-cocycle by a 2-coboundary) so that

$[d{\text{'}}_l,d{\text{'}}_m]=(l-m)d_{l+m},$

with the central charge absent altogether, and the extension is hence trivial. (This was not (generally) the case with the previous modification, where only Template:Math obtained the original relations.) With Template:Mvar the following change of basis,

$d{\text{'}}_l=d_l+{\delta }_{0l}\frac{\alpha +\gamma }{2}C,$

the commutation relations take the form

$[d{\text{'}}_l,d{\text{'}}_m]=(l-m)d{\text{'}}_{l+m}+(\gamma m+\beta m^3){\delta }_{l,-m}C,$

showing that the part linear in Template:Mvar is trivial. It also shows that Template:Math is one-dimensional (corresponding to the choice of Template:Mvar). The conventional choice is to take Template:Math and still retaining freedom by absorbing an arbitrary factor in the arbitrary object Template:Math. The Virasoro algebra Template:Math is then

$𝒱=𝒲+ℂC,$

with commutation relations

$[d_l+\mu C,d_m+\nu C]=(l-m)d_{l+m}+\frac{(m-m^3)}{12}{\delta }_{l,-m}C.$

Bosonic open strings

Template:Main The relativistic classical open string (background) is subject to quantization. This roughly amounts to taking the position and the momentum of the string and promoting them to operators on the space of states of open strings. Since strings are extended objects, this results in a continuum of operators depending on the parameter Template:Mvar. The following commutation relations are postulated in the Heisenberg picture.

$\begin{array}{cc}[X^I(\tau ,\sigma ),{𝒫}^{\tau J}(\tau ,\sigma )]& =i{\eta }^{IJ}\delta (\sigma -{\sigma }^{\prime }),\\ [x_0^{-}(\tau ),p^{+}(\tau )]& =-i.\end{array}$

All other commutators vanish.

Because of the continuum of operators, and because of the delta functions, it is desirable to express these relations instead in terms of the quantized versions of the Virasoro modes, the Virasoro operators. These are calculated to satisfy

$[{\alpha }_m^I,{\alpha }_n^J]=m{\eta }^{IJ}{\delta }_{m+n,0}$

They are interpreted as creation and annihilation operators acting on Hilbert space, increasing or decreasing the quantum of their respective modes. If the index is negative, the operator is a creation operator, otherwise it is an annihilation operator. (If it is zero, it is proportional to the total momentum operator.) Since the light cone plus and minus modes were expressed in terms of the transverse Virasoro modes, one must consider the commutation relations between the Virasoro operators. These were classically defined (then modes) as

$L_n=\frac{1}{2}\sum _{p\in \mathbb{Z}}{\alpha }_{n-p}^I{\alpha }_p^I.$

Since, in the quantized theory, the alphas are operators, the ordering of the factors matter. In view of the commutation relation between the mode operators, it will only matter for the operator Template:Math (for which Template:Math). Template:Math is chosen normal ordered,

$L_0=\frac{1}{2}{\alpha }_0^I{\alpha }_0^I+\sum _{p=1}^{\mathrm{\infty }}{\alpha }_{-p}^I{\alpha }_p^I,={\alpha }^{\prime }{p}^I{p}^I+\sum _{p=1}^{\mathrm{\infty }}p{\alpha }_p^{I†}{\alpha }_p^I+c$

where Template:Mvar is a possible ordering constant. One obtains after a somewhat lengthy calculation the relations

$[L_m,L_n]=(m-n)L_{m+n},m+n\ne 0.$

If one would allow for Template:Math above, then one has precisely the commutation relations of the Witt algebra. Instead one has

$[L_m,L_n]=(m-n)L_{m+n}+\frac{D-2}{12}(m^3-m){\delta }_{m+n,0},\mathrm{\forall }m,n\in \mathbb{Z}.$

upon identification of the generic central term as Template:Math times the identity operator, this is the Virasoro algebra, the universal central extension of the Witt algebra.

The operator Template:Math enters the theory as the Hamiltonian, modulo an additive constant. Moreover, the Virasoro operators enter into the definition of the Lorentz generators of the theory. It is perhaps the most important algebra in string theory. The consistency of the Lorentz generators, by the way, fixes the spacetime dimensionality to 26. While this theory presented here (for relative simplicity of exposition) is unphysical, or at the very least incomplete (it has, for instance, no fermions) the Virasoro algebra arises in the same way in the more viable superstring theory and M-theory.

Group extension

Template:Main A projective representation Template:Math of a Lie group Template:Mvar (background) can be used to define a so-called group extension Template:Math.

In quantum mechanics, Wigner's theorem asserts that if Template:Math is a symmetry group, then it will be represented projectively on Hilbert space by unitary or antiunitary operators. This is often dealt with by passing to the universal covering group of Template:Math and take it as the symmetry group. This works nicely for the rotation group Template:Math and the Lorentz group Template:Math, but it does not work when the symmetry group is the Galilean group. In this case one has to pass to its central extension, the Bargmann group, which is the symmetry group of the Schrödinger equation. Likewise, if Template:Math, the group of translations in position and momentum space, one has to pass to its central extension, the Heisenberg group.

Let Template:Mvar be the 2-cocycle on Template:Math induced by Template:Math. Define

$G_{\mathrm{e}\mathrm{x}}={ℂ}^{*}\times G=\{(\lambda ,g)|\lambda \in ℂ,g\in G\}$

as a set and let the multiplication be defined by

$({\lambda }_1,g_1)({\lambda }_2,g_2)=({\lambda }_1{\lambda }_2\omega (g_1,g_2),g_1{g}_2).$

Associativity holds since Template:Mvar is a 2-cocycle on Template:Math. One has for the unit element

$(1,e)(\lambda ,g)=(\lambda \omega (e,g),g)=(\lambda ,g)=(\lambda ,g)(1,e),$

and for the inverse

$(\lambda ,g{)}^{-1}=(\frac{1}{\lambda \omega (g,g^{-1})},g^{-1}).$

The set Template:Math is an abelian subgroup of Template:Math. This means that Template:Math is not semisimple. The center of Template:Mvar, Template:Math includes this subgroup. The center may be larger.

At the level of Lie algebras it can be shown that the Lie algebra Template:Math of Template:Math is given by

${𝔤}_{\mathrm{e}\mathrm{x}}=ℂC\oplus 𝔤,$

as a vector space and endowed with the Lie bracket

$[\mu C+G_1,\nu C+G_2]=[G_1,G_2]+\eta (G_1,G_2)C.$

Here Template:Mvar is a 2-cocycle on Template:Math. This 2-cocycle can be obtained from Template:Mvar albeit in a highly nontrivial way.

Now by using the projective representation Template:Math one may define a map Template:Math by

${\mathrm{\Pi }}_{\mathrm{e}\mathrm{x}}((\lambda ,g))=\lambda \mathrm{\Pi }(g).$

It has the properties

${\mathrm{\Pi }}_{\mathrm{e}\mathrm{x}}(({\lambda }_1,g_1)){\mathrm{\Pi }}_{\mathrm{e}\mathrm{x}}(({\lambda }_2,g_2))={\lambda }_1{\lambda }_2\mathrm{\Pi }(g_1)\mathrm{\Pi }(g_2)={\lambda }_1{\lambda }_2\omega (g_1,g_2)\mathrm{\Pi }(g_1{g}_2)={\mathrm{\Pi }}_{\mathrm{e}\mathrm{x}}({\lambda }_1{\lambda }_2\omega (g_1,g_2),g_1{g}_2)={\mathrm{\Pi }}_{\mathrm{e}\mathrm{x}}(({\lambda }_1,g_1)({\lambda }_2,g_2)),$

so Template:Math is a bona fide representation of Template:Math.

In the context of Wigner's theorem, the situation may be depicted as such (replace Template:Math by Template:Math); let Template:Math denote the unit sphere in Hilbert space Template:Mvar, and let Template:Math be its inner product. Let Template:Math denote ray space and Template:Math the ray product. Let moreover a wiggly arrow denote a group action. Then the diagram

commutes, i.e.

${\pi }_2\circ {\mathrm{\Pi }}_{\mathrm{e}\mathrm{x}}((\lambda ,g))(\psi )=\mathrm{\Pi }\circ \pi (g)({\pi }_1(\psi )),\psi \in S\mathscr{H}.$

Moreover, in the same way that Template:Math is a symmetry of Template:Math preserving Template:Math, Template:Math is a symmetry of Template:Math preserving Template:Math. The fibers of Template:Math are all circles. These circles are left invariant under the action of Template:Math. The action of Template:Math on these fibers is transitive with no fixed point. The conclusion is that Template:Math is a principal fiber bundle over Template:Math with structure group Template:Math.

Background material

In order to adequately discuss extensions, structure that goes beyond the defining properties of a Lie algebra is needed. Rudimentary facts about these are collected here for quick reference.

Derivations

A derivation Template:Mvar on a Lie algebra Template:Math is a map

$\delta :𝔤\to 𝔤$

such that the Leibniz rule

$\delta [G_1,G_2]=[\delta G_1,G_2]+[G_1,\delta G_2]$

holds. The set of derivations on a Lie algebra Template:Math is denoted Template:Math. It is itself a Lie algebra under the Lie bracket

$[{\delta }_1,{\delta }_2]={\delta }_1\circ {\delta }_2-{\delta }_2\circ {\delta }_1.$

It is the Lie algebra of the group Template:Math of automorphisms of Template:Math. One has to show

$\delta [G_1,G_1]=[\delta G_1,G_2]+[G_1,\delta G_2]\iff e^{t\delta }[G_1,G_2]=[e^{t\delta }{G}_1,e^{t\delta }{G}_2],\mathrm{\forall }t\in ℝ.$

If the rhs holds, differentiate and set Template:Math implying that the lhs holds. If the lhs holds Template:Math, write the rhs as

$[G_1,G_2]\stackrel{?}{=}{e}^{-t\delta }[e^{t\delta }{G}_1,e^{t\delta }{G}_2],$

and differentiate the rhs of this expression. It is, using Template:Math, identically zero. Hence the rhs of this expression is independent of Template:Mvar and equals its value for Template:Math, which is the lhs of this expression.

If Template:Math, then Template:Math, acting by Template:Math, is a derivation. The set Template:Math is the set of inner derivations on Template:Math. For finite-dimensional simple Lie algebras all derivations are inner derivations.

Semidirect product (groups)

Template:Main Consider two Lie groups Template:Mvar and Template:Mvar and Template:Math, the automorphism group of Template:Mvar. The latter is the group of isomorphisms of Template:Mvar. If there is a Lie group homomorphism Template:Math, then for each Template:Math there is a Template:Math with the property Template:Math. Denote with Template:Mvar the set Template:Math and define multiplication by Template:NumBlk2 Then Template:Math is a group with identity Template:Math and the inverse is given by Template:Math. Using the expression for the inverse and equation (Template:EquationNote) it is seen that Template:Mvar is normal in Template:Mvar. Denote the group with this semidirect product as Template:Math.

Conversely, if Template:Math is a given semidirect product expression of the group Template:Math, then by definition Template:Math is normal in Template:Math and Template:Math for each Template:Math where Template:Math and the map Template:Math is a homomorphism.

Now make use of the Lie correspondence. The maps Template:Math each induce, at the level of Lie algebras, a map Template:Math. This map is computed by Template:NumBlk2 For instance, if Template:Math and Template:Math are both subgroups of a larger group Template:Mvar and Template:Math, then Template:NumBlk2 and one recognizes Template:Math as the adjoint action Template:Math of Template:Mvar on Template:Math restricted to Template:Mvar. Now Template:Math [ Template:Math if Template:Math is finite-dimensional] is a homomorphism, and appealing once more to the Lie correspondence, there is a unique Lie algebra homomorphism Template:Math. This map is (formally) given by Template:NumBlk2 for example, if Template:Math, then (formally) Template:NumBlk2 where a relationship between Template:Math and the adjoint action Template:Math rigorously proved in here is used.

Lie algebra
The Lie algebra is, as a vector space, Template:Math. This is clear since Template:Math generates Template:Math and Template:Math. The Lie bracket is given by

$[H_1+G_1,H_2+G_2{]}_{𝔢}=[H_1,H_2{]}_{𝔥}+{\psi }_{G_1}(H_2)-{\psi }_{G_2}(H_1)+[G_1,G_2{]}_{𝔤}.$

Template:Hidden begin To compute the Lie bracket, begin with a surface in Template:Math parametrized by Template:Math and Template:Math. Elements of Template:Math in Template:Math are decorated with a bar, and likewise for Template:Math.

$\begin{array}{cc}{e}^{e^{t\bar{G}}s\bar{H}{e}^{-t\bar{G}}}& =e^{t\bar{G}}{e}^{s\bar{H}}{e}^{-t\bar{G}}=(1,e^{tG})(e^{sH},1)(1,e^{-tG})\\ & =({\varphi }_{e^{tG}}(e^{sH}),e^{tG})(1,e^{-tG})=({\varphi }_{e^{tG}}(e^{sH}){\varphi }_{e^{tG}}(1),1)\\ & =({\varphi }_{e^{tG}}(e^{sH}),1)\end{array}$

One has

$\frac{d}{ds}{e^{A{d}_{e^{t\bar{G}}}s\bar{H}}|}_{s=0}=A{d}_{e^{t\bar{G}}}\bar{H}$

and

$\frac{d}{ds}{({\varphi }_{e^{tG}}(e^{sH}),1)|}_{s=0}=({\mathrm{\Psi }}_{e^{tG}}(H),0)$

by Template:EquationNote and thus

$A{d}_{e^{t\bar{G}}}\bar{H}=({\mathrm{\Psi }}_{e^{tG}}(H),0).$

Now differentiate this relationship with respect to Template:Mvar and evaluate at Template:Math:

$\frac{d}{dt}{e^{t\bar{G}}\bar{H}{e}^{-t\bar{G}}|}_{t=0}=[\bar{G},\bar{H}]$

and

$\frac{d}{dt}{({\mathrm{\Psi }}_{e^{tG}}(H),0)|}_{t=0}=({\psi }_G(H),0)$

by Template:EquationNote and thus

$[H_1+G_1,H_2+G_2{]}_{𝔢}=[H_1,H_2{]}_{𝔥}+[G_1,H_2]+[H_1,G_2]+[G_1,G_2{]}_{𝔤}=[H_1,H_2{]}_{𝔥}+{\psi }_{G_1}(H_2)-{\psi }_{G_2}(H_1)+[G_1,G_2{]}_{𝔤}.$

Cohomology

Template:Main For the present purposes, consideration of a limited portion of the theory Lie algebra cohomology suffices. The definitions are not the most general possible, or even the most common ones, but the objects they refer to are authentic instances of more the general definitions.

2-cocycles
The objects of primary interest are the 2-cocycles on Template:Math, defined as bilinear alternating functions,

$\varphi :𝔤\times 𝔤\to F,$

that are alternating,

$\varphi (G_1,G_2)=-\varphi (G_2,G_1),$

and having a property resembling the Jacobi identity called the Jacobi identity for 2-cycles,

$\varphi (G_1,[G_2,G_3])+\varphi (G_2,[G_3,G_1])+\varphi (G_3,[G_1,G_2])=0.$

The set of all 2-cocycles on Template:Math is denoted Template:Math.

2-cocycles from 1-cochains
Some 2-cocycles can be obtained from 1-cochains. A 1-cochain on Template:Math is simply a linear map,

$f:𝔤\to F$

The set of all such maps is denoted Template:Math and, of course (in at least the finite-dimensional case) Template:Math. Using a 1-cochain Template:Mvar, a 2-cocycle Template:Math may be defined by

$\delta f(G_1,G_2)=f([G_1,G_2]).$

The alternating property is immediate and the Jacobi identity for 2-cocycles is (as usual) shown by writing it out and using the definition and properties of the ingredients (here the Jacobi identity on Template:Math and the linearity of Template:Mvar). The linear map Template:Math is called the coboundary operator (here restricted to Template:Math).

The second cohomology group
Denote the image of Template:Math of Template:Mvar by Template:Math. The quotient

$H^2(𝔤,𝔽)=Z^2(𝔤,𝔽)/B^2(𝔤,𝔽)$

is called the second cohomology group of Template:Math. Elements of Template:Math are equivalence classes of 2-cocycles and two 2-cocycles Template:Math and Template:Math are called equivalent cocycles if they differ by a 2-coboundary, i.e. if Template:Math for some Template:Math. Equivalent 2-cocycles are called cohomologous. The equivalence class of Template:Math is denoted Template:Math.

These notions generalize in several directions. For this, see the main articles.

Structure constants

Template:Main Let Template:Math be a Hamel basis for Template:Math. Then each Template:Math has a unique expression as

$G=\sum _{\alpha \in A}{c}_{\alpha }{G}_{\alpha },c_{\alpha }\in F,G_{\alpha }\in B$

for some indexing set Template:Mvar of suitable size. In this expansion, only finitely many Template:Math are nonzero. In the sequel it is (for simplicity) assumed that the basis is countable, and Latin letters are used for the indices and the indexing set can be taken to be Template:Math. One immediately has

$[G_i,G_j]={C_{ij}}^k{G}_k$

for the basis elements, where the summation symbol has been rationalized away, the summation convention applies. The placement of the indices in the structure constants (up or down) is immaterial. The following theorem is useful:

Theorem:There is a basis such that the structure constants are antisymmetric in all indices if and only if the Lie algebra is a direct sum of simple compact Lie algebras and Template:Math Lie algebras. This is the case if and only if there is a real positive definite metric Template:Mvar on Template:Math satisfying the invariance condition

$g_{\alpha \beta }{C^{\beta }}_{\gamma \delta }=-g_{\gamma \beta }{C^{\beta }}_{\alpha \delta }.$

in any basis. This last condition is necessary on physical grounds for non-Abelian gauge theories in quantum field theory. Thus one can produce an infinite list of possible gauge theories using the Cartan catalog of simple Lie algebras on their compact form (i.e., Template:Math, etc. One such gauge theory is the Template:Math gauge theory of the standard model with Lie algebra Template:Math.

Killing form

Template:Main The Killing form is a symmetric bilinear form on Template:Math defined by

$K(G_1,G_2)=\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}({\mathrm{a}\mathrm{d}}_{G_1}{\mathrm{a}\mathrm{d}}_{G_2}).$

Here Template:Math is viewed as a matrix operating on the vector space Template:Math. The key fact needed is that if Template:Math is semisimple, then, by Cartan's criterion, Template:Mvar is non-degenerate. In such a case Template:Mvar may be used to identify Template:Math and Template:Math. If Template:Math, then there is a Template:Math such that

$⟨\lambda ,G⟩=K(G_{\lambda },G)\mathrm{\forall }G\in 𝔤.$

This resembles the Riesz representation theorem and the proof is virtually the same. The Killing form has the property

$K([G_1,G_2],G_3)=K(G_1,[G_2,G_3]),$

which is referred to as associativity. By defining Template:Math and expanding the inner brackets in terms of structure constants, one finds that the Killing form satisfies the invariance condition of above.

Loop algebra

Template:Main A loop group is taken as a group of smooth maps from the unit circle Template:Math into a Lie group Template:Math with the group structure defined by the group structure on Template:Math. The Lie algebra of a loop group is then a vector space of mappings from Template:Math into the Lie algebra Template:Math of Template:Math. Any subalgebra of such a Lie algebra is referred to as a loop algebra. Attention here is focused on polynomial loop algebras of the form

$\{h:S^1\to 𝔤|h(\lambda )=\sum {\lambda }^n{G}_n,n\in \mathbb{Z},\lambda =e^{i\theta }\in S^1,G_n\in 𝔤\}.$

Template:Hidden begin To see this, consider elements Template:Math near the identity in Template:Math for Template:Math in the loop group, expressed in a basis Template:Math for Template:Math

$H(\lambda )=e^{h^k(\lambda )G_k}=e_G+h^k(\lambda )G_k+\dots ,$

where the Template:Math are real and small and the implicit sum is over the dimension Template:Math of Template:Math. Now write

$h^k(\lambda )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\theta }_{-n}^k{\lambda }^n$

to obtain

$e^{h^k(\lambda )G_k}=1_G+\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\theta }_{-n}^k{\lambda }^n{G}_k+\dots .$

Thus the functions

$h:S^1\to 𝔤;h(\lambda )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{k=1}^K{\theta }_{-n}^k{\lambda }^n{G}_k\equiv \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\lambda }^n{G}_n$

constitute the Lie algebra. Template:Hidden end

A little thought confirms that these are loops in Template:Math as Template:Mvar goes from Template:Math to Template:Math. The operations are the ones defined pointwise by the operations in Template:Math. This algebra is isomorphic with the algebra

$C[\lambda ,{\lambda }^{-1}]\otimes 𝔤,$

where Template:Math is the algebra of Laurent polynomials,

$\sum {\lambda }^k{G}_k\leftrightarrow \sum {\lambda }^k\otimes G_k.$

The Lie bracket is

$[P(\lambda )\otimes G_1,Q(\lambda )\otimes G_2]=P(\lambda )Q(\lambda )\otimes [G_1,G_2].$

In this latter view the elements can be considered as polynomials with (constant!) coefficients in Template:Math. In terms of a basis and structure constants,

$[{\lambda }^m\otimes G_i,{\lambda }^n\otimes G_j]={C_{ij}}^k{\lambda }^{m+n}\otimes G_k.$

It is also common to have a different notation,

${\lambda }^m\otimes G_i\cong {\lambda }^m{G}_i\leftrightarrow T_i^m(\lambda )\equiv T_i^m,$

where the omission of Template:Mvar should be kept in mind to avoid confusion; the elements really are functions Template:Math. The Lie bracket is then

$[T_i^m,T_j^n]={C_{ij}}^k{T}_k^{m+n},$

which is recognizable as one of the commutation relations in an untwisted affine Kac–Moody algebra, to be introduced later, without the central term. With Template:Math, a subalgebra isomorphic to Template:Math is obtained. It generates (as seen by tracing backwards in the definitions) the set of constant maps from Template:Math into Template:Mvar, which is obviously isomorphic with Template:Math when Template:Math is onto (which is the case when Template:Mvar is compact. If Template:Math is compact, then a basis Template:Math for Template:Math may be chosen such that the Template:Math are skew-Hermitian. As a consequence,

$T_i^{n†}=({\lambda }^n{G}_i{)}^{†}=-{\lambda }^{-n}{G}_i=-T_i^{-n}.$

Such a representation is called unitary because the representatives

$H(\lambda )=e^{{\theta }_n^k{T}_k^{-n}}\in G$

are unitary. Here, the minus on the lower index of Template:Mvar is conventional, the summation convention applies, and the Template:Mvar is (by the definition) buried in the Template:Maths in the right hand side.

Current algebra (physics)

Current algebras arise in quantum field theories as a consequence of global gauge symmetry. Conserved currents occur in classical field theories whenever the Lagrangian respects a continuous symmetry. This is the content of Noether's theorem. Most (perhaps all) modern quantum field theories can be formulated in terms of classical Lagrangians (prior to quantization), so Noether's theorem applies in the quantum case as well. Upon quantization, the conserved currents are promoted to position dependent operators on Hilbert space. These operators are subject to commutation relations, generally forming an infinite-dimensional Lie algebra. A model illustrating this is presented below.

To enhance the flavor of physics, factors of Template:Math will appear here and there as opposed to in the mathematical conventions.

Consider a column vector Template:Math of scalar fields Template:Math. Let the Lagrangian density be

$ℒ={\partial }_{\mu }{\varphi }^{†}{\partial }^{\mu }\varphi -m^2{\varphi }^{†}\varphi .$

This Lagrangian is invariant under the transformation

$\varphi \mapsto e^{-i\sum _{a=1}^r{\alpha }^a{F}_a}\varphi ,$

where Template:Math are generators of either Template:Math or a closed subgroup thereof, satisfying

$[F_a,F_b]=i{C_{ab}}^c{F}_c.$

Noether's theorem asserts the existence of Template:Math conserved currents,

$J_a^{\mu }=-{\pi }^{\mu }i{F}_a\varphi ,{\pi }^{k\mu }=\frac{\partial ℒ}{\partial ({\partial }_{\mu }{\varphi }_k)},$

where Template:Math is the momentum canonically conjugate to Template:Math. The reason these currents are said to be conserved is because

${\partial }_{\mu }{J}_a^{\mu }=0,$

and consequently

$Q_a(t)=\int J_a^0{d}^{3}x=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\equiv Q_a,$

the charge associated to the charge density Template:Math is constant in time. This (so far classical) theory is quantized promoting the fields and their conjugates to operators on Hilbert space and by postulating (bosonic quantization) the commutation relations

$\begin{array}{cc}[{\varphi }_k(t,x),{\pi }^l(t,x)]& =i\delta (x-y){\delta }_k^l,\\ [{\varphi }_k(t,x),{\varphi }_l(t,x)]& =[{\pi }^k(t,x),{\pi }^l(t,x)]=0.\end{array}$

The currents accordingly become operators They satisfy, using the above postulated relations, the definitions and integration over space, the commutation relations

$\begin{array}{cc}[J_a^0(t,𝐱),J_b^0(t,𝐲)]& =i\delta (𝐱-𝐲){C_{ab}}^c{J}_c^0(ct,𝐱)\\ [Q_a,Q_b]& =i{Q_{ab}}^c{Q}_c\\ [Q_a,J_b^{\mu }(t,𝐱)]& =i{C_{ab}}^c{J}_c^{\mu }(t,𝐱),\end{array}$

where the speed of light and the reduced Planck constant have been set to unity. The last commutation relation does not follow from the postulated commutation relations (these are fixed only for Template:Math, not for Template:Math), except for Template:Math For Template:Math the Lorentz transformation behavior is used to deduce the conclusion. The next commutator to consider is

$[J_a^0(t,𝐱),J_b^i(t,𝐲)]=i{C_{ab}}^c{J}_c^i(t,𝐱)\delta (𝐱-𝐲)+S_{ab}^{ij}{\partial }_j\delta (𝐱-𝐲)+....$

The presence of the delta functions and their derivatives is explained by the requirement of microcausality that implies that the commutator vanishes when Template:Math. Thus the commutator must be a distribution supported at Template:Math. The first term is fixed due to the requirement that the equation should, when integrated over Template:Math, reduce to the last equation before it. The following terms are the Schwinger terms. They integrate to zero, but it can be shown quite generally that they must be nonzero.

Template:Hidden begin Consider a conserved current Template:NumBlk2 with a generic Schwinger term

$[J^0(t,𝐱),J^i(t,𝐲)]=i\delta (𝐱-𝐲)J^i(t,𝐱)+C^i(𝐱,𝐲).$

By taking the vacuum expectation value (VEV),

$⟨0|C^i(𝐱,𝐲)|0⟩=⟨0|[J^0(t,𝐱),J^i(t,𝐲)]|0⟩,$

one finds

$\begin{array}{cc}⟨0|\frac{\partial C^i(𝐱,𝐲)}{{\partial }_{y^i}}|0⟩& =⟨0|[J^0(t,𝐱),\frac{\partial J^i(t,𝐲)}{{\partial }_{y^i}}]|0⟩\\ & =-⟨0|[J^0(t,𝐱),\frac{\partial J^0(t,𝐲)}{{\partial }_t}]|0⟩=i⟨0|[J^0(t,𝐱),[J^0(t,𝐲),H]]|0⟩\\ & =-i⟨0|J^0(t,𝐱)H{J}^0(t,𝐲)+J^0(t,𝐱)H{J}^0(t,𝐱)|0⟩,\end{array}$

where Template:EquationNote and Heisenberg's equation of motion have been used as well as Template:Math and its conjugate.

Multiply this equation by Template:Math and integrate with respect to Template:Math and Template:Math over all space, using integration by parts, and one finds

$-i\int \int d𝐱d𝐲⟨0|C^i(𝐱,𝐲)|0⟩f(𝐱)\frac{\partial f}{\partial y^i}f(𝐱)=2⟨0|FHF|⟩,F=\int J^0(𝐱)f(𝐱).$

Now insert a complete set of states, Template:Math

$⟨0|FHF|⟩=\sum _{mn}⟨0|F|m⟩⟨m|H|n⟩⟨n|F|0⟩=\sum _{mn}⟨0|F|m⟩E_n{\delta }_{mn}⟨n|F|0⟩)\sum _{n\ne 0}|⟨0|F|n⟩{|}^2{E}_n>0\Rightarrow C^i(𝐱,𝐲)\ne 0.$

Here hermiticity of Template:Mvar and the fact that not all matrix elements of Template:Mvar between the vacuum state and the states from a complete set can be zero. Template:Hidden end

Affine Kac–Moody algebra

Template:Main Let Template:Math be an Template:Math-dimensional complex simple Lie algebra with a dedicated suitable normalized basis such that the structure constants are antisymmetric in all indices with commutation relations

$[G_i,G_j]={C_{ij}}^k{G}_k,1\le i,j,N.$

An untwisted affine Kac–Moody algebra Template:Math is obtained by copying the basis for each Template:Math (regarding the copies as distinct), setting

$\overline{𝔤}=FC\oplus FD\oplus \underset{1\le i\le \mathbb{N},m\in \mathbb{Z}}{⨁}F{G}_m^i$

as a vector space and assigning the commutation relations

$\begin{array}{cc}[G_i^m,G_j^n]& ={C_{ij}}^k{G}_k^{m+n}+m{\delta }_{ij}{\delta }^{m+n,0}C,\\ [C,G_i^m]& =0,1\le i,j,N,m,n\in \mathbb{Z}\\ [D,G_i^m]& =m{G}_i^m\\ [D,C]& =0.\end{array}$

If Template:Math, then the subalgebra spanned by the Template:Math is obviously identical to the polynomial loop algebra of above.

Witt algebra

Ernst Witt (1911–1991), German mathematician. The Witt algebras, studied by him over finite fields in the 1930s, were first examined in the complex case by Cartan in 1909.

Template:Main The Witt algebra, named after Ernst Witt, is the complexification of the Lie algebra Template:Math of smooth vector fields on the circle Template:Math. In coordinates, such vector fields may be written

$X=f(\phi )\frac{d}{d\phi },$

and the Lie bracket is the Lie bracket of vector fields, on Template:Math simply given by

$[X,Y]=[f\frac{d}{d\phi },g\frac{d}{d\phi }]=(f\frac{dg}{d\phi }-g\frac{df}{d\phi })\frac{d}{d\phi }.$

The algebra is denoted Template:Math. A basis for Template:Mvar is given by the set

$\{d_n,n\in \mathbb{Z}\}=\{i{e}^{in\phi }\frac{d}{d\phi }=-z^{n+1}\frac{d}{dz}|n\in \mathbb{Z}\}.$

This basis satisfies

$[d_l,d_m]=(l-m)d_{l+m}\equiv {C_{lm}}^n{d}_n=(l-m){\delta }_{l+m}^n{d}_n,l,m,n\in \mathbb{Z}.$

This Lie algebra has a useful central extension, the Virasoro algebra. It has Template:Mathdimensional subalgebras isomorphic with Template:Math and Template:Math. For each Template:Math, the set Template:Math spans a subalgebra isomorphic to Template:Math.

Template:Hidden begin For Template:Math one has

$[d_0,d_{-1}]=d_{-1},[d_0,d_1]=-d_1,[d_1,d_{-1}]=2d_0.$

These are the commutation relations of Template:Math with

$d_0\leftrightarrow H=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),d_{-1}\leftrightarrow X=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),d_1\leftrightarrow Y=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),H,X,Y\in 𝔰𝔩(2,ℝ).$

The groups Template:Math and Template:Math are isomorphic under the map

$SU(1,1)=\left(\begin{array}{cc}1& -i\\ 1& i\end{array}\right)SL(2,ℝ){\left(\begin{array}{cc}1& -i\\ 1& i\end{array}\right)}^{-1},$

and the same map holds at the level of Lie algebras due to the properties of the exponential map. A basis for Template:Math is given, see classical group, by

$U_0=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),U_1=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),U_2=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right)$

Now compute

$\begin{array}{cc}{H}_{𝔰𝔲(1,1)}& =\left(\begin{array}{cc}1& -i\\ 1& i\end{array}\right)H{\left(\begin{array}{cc}1& -i\\ 1& i\end{array}\right)}^{-1}=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)=U_0,\\ X_{𝔰𝔲(1,1)}& =\left(\begin{array}{cc}1& -i\\ 1& i\end{array}\right)X{\left(\begin{array}{cc}1& -i\\ 1& i\end{array}\right)}^{-1}=\frac{1}{2}\left(\begin{array}{cc}i& -i\\ i& -i\end{array}\right)=\frac{1}{2}(U_1+U_2),\\ Y_{𝔰𝔲(1,1)}& =\left(\begin{array}{cc}1& -i\\ 1& i\end{array}\right)Y{\left(\begin{array}{cc}1& -i\\ 1& i\end{array}\right)}^{-1}=\frac{1}{2}\left(\begin{array}{cc}-i& -i\\ i& i\end{array}\right)=\frac{1}{2}(U_1-U_2).\end{array}$

The map preserves brackets and there are thus Lie algebra isomorphisms between the subalgebra of Template:Mvar spanned by Template:Math with real coefficients, Template:Math and Template:Math. The same holds for any subalgebra spanned by Template:Math, this follows from a simple rescaling of the elements (on either side of the isomorphisms). Template:Hidden end

Projective representation

Template:Main If Template:Mvar is a matrix Lie group, then elements Template:Mvar of its Lie algebra m can be given by

$X=\frac{d}{dt}{(g(t))|}_{t=0},$

where Template:Mvar is a differentiable path in Template:Mvar that goes through the identity element at Template:Math. Commutators of elements of the Lie algebra can be computed as

$[X_1,X_2]={\frac{d}{dt}|}_{t=0}{\frac{d}{ds}|}_{s=0}{e}^{t{X}_1}{e}^{s{X}_2}{e}^{-t{X}_1}.$

Likewise, given a group representation Template:Math, its Lie algebra Template:Math is computed by

$\begin{array}{cc}[][Y_1,Y_2]& ={\frac{d}{dt}|}_{t=0}{\frac{d}{ds}|}_{s=0}U(e^{t{X}_1})U(e^{s{X}_2})U(e^{-t{X}_1})\\ & ={\frac{d}{dt}|}_{t=0}{\frac{d}{ds}|}_{s=0}U(e^{t{X}_1}{e}^{s{X}_2}{e}^{-t{X}_1})\end{array},$

where $Y_1={\frac{d}{dt}|}_{t=0}U(e^{t{X}_1})$ and $Y_2={\frac{d}{ds}|}_{s=0}U(e^{s{X}_2})$. Then there is a Lie algebra isomorphism between Template:Math and Template:Math sending bases to bases, so that Template:Math is a faithful representation of Template:Math.

If however Template:Math is an admissible set of representatives of a projective unitary representation, i.e. a unitary representation up to a phase factor, then the Lie algebra, as computed from the group representation, is not isomorphic to Template:Math. For Template:Math, the multiplication rule reads

$U(g_1)U(g_2)=\omega (g_1,g_2)U(g_1{g}_2)=e^{i\xi (g_1,g_2)}U(g_1{g}_2).$

The function Template:Mvar,often required to be smooth, satisfies

$\begin{array}{cc}\omega (g,e)& =\omega (e,g)=1,\\ \omega (g_1,g_2{g}_3)\omega (g_2,g_3)& =\omega (g_1,g_2)\omega (g_1{g}_2,g_3)\\ \omega (g,g^{-1})& =\omega (g^{-1},g).\end{array}$

It is called a 2-cocycle on Template:Mvar.

From the above equalities, $(U(g){)}^{-1}=\frac{1}{\omega (g,g^{-1})}U(g^{-1})$, so one has

$\begin{array}{cc}[][Y_1,Y_2]& ={\frac{d}{dt}|}_{t=0}{\frac{d}{ds}|}_{s=0}U(e^{t{X}_1})U(e^{s{X}_2})(U(e^{t{X}_1}){)}^{-1}\\ & ={\frac{d}{dt}|}_{t=0}{\frac{d}{ds}|}_{s=0}\frac{1}{\omega (e^{t{X}_1},e^{-t{X}_1})}U(e^{t{X}_1})U(e^{s{X}_2})U(e^{-t{X}_1})\\ & ={\frac{d}{dt}|}_{t=0}{\frac{d}{ds}|}_{s=0}\frac{\omega (e^{t{X}_1},e^{s{X}_2})\omega (e^{t{X}_1}{e}^{s{X}_2},e^{-t{X}_1})}{\omega (e^{t{X}_1},e^{-t{X}_1})}U(e^{t{X}_1}{e}^{s{X}_2}{e}^{-t{X}_1})\\ & \equiv {\frac{d}{dt}|}_{t=0}{\frac{d}{ds}|}_{s=0}\mathrm{\Omega }(e^{t{X}_1},e^{s{X}_2})U(e^{t{X}_1}{e}^{s{X}_2}{e}^{-t{X}_1})\\ & ={\frac{d}{dt}|}_{t=0}{\frac{d}{ds}|}_{s=0}U(e^{t{X}_1}{e}^{s{X}_2}{e}^{-t{X}_1})+{\frac{d}{dt}|}_{t=0}{\frac{d}{ds}|}_{s=0}\mathrm{\Omega }(e^{t{X}_1},e^{s{X}_2})I,\end{array}$

because both Template:Math and Template:Mvar evaluate to the identity at Template:Math. For an explanation of the phase factors Template:Mvar, see Wigner's theorem. The commutation relations in Template:Math for a basis,

$[X_i,X_j]=C_{ij}^k{X}_k$

become in Template:Math

$[Y_i,Y_j]=C_{ij}^k{Y}_k+D_{ij}I,$

so in order for Template:Math to be closed under the bracket (and hence have a chance of actually being a Lie algebra) a central charge Template:Math must be included.

Relativistic classical string theory

Template:Main A classical relativistic string traces out a world sheet in spacetime, just like a point particle traces out a world line. This world sheet can locally be parametrized using two parameters Template:Mvar and Template:Mvar. Points Template:Math in spacetime can, in the range of the parametrization, be written Template:Math. One uses a capital Template:Mvar to denote points in spacetime actually being on the world sheet of the string. Thus the string parametrization is given by Template:Math. The inverse of the parametrization provides a local coordinate system on the world sheet in the sense of manifolds.

The equations of motion of a classical relativistic string derived in the Lagrangian formalism from the Nambu–Goto action are

$\frac{\partial {𝒫}_{\mu }^{\tau }}{\partial \tau }+\frac{\partial {𝒫}_{\mu }^{\sigma }}{\partial \sigma }=0,{𝒫}_{\mu }^{\tau }=-\frac{T_0}{c}\frac{(\dot{X}\cdot X^{\prime })X{\text{'}}_{\mu }-(X^{\prime }{)}^2{\dot{X}}_{\mu }}{\sqrt{(\dot{X}\cdot X^{\prime }{)}^2-(\dot{X}{)}^2(X^{\prime }{)}^2}},{𝒫}_{\mu }^{\sigma }=-\frac{T_0}{c}\frac{(\dot{X}\cdot X^{\prime })X{\text{'}}_{\mu }-(\dot{X}{)}^{2}X{\text{'}}_{\mu }}{\sqrt{(\dot{X}\cdot X^{\prime }{)}^2-(\dot{X}{)}^2(X^{\prime }{)}^2}}.$

A dot over a quantity denotes differentiation with respect to Template:Mvar and a prime differentiation with respect to Template:Mvar. 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

${\ddot{X}}^{\mu }-{X^{\mu }}^{″}=0,$

the ordinary wave equation. The price to be paid is that the light cone gauge imposes constraints,

${\dot{X}}^{\mu }\cdot {X^{\mu }}^{\prime }=0,(\dot{X}{)}^2+(X^{\prime }{)}^2=0,$

so that one cannot simply take arbitrary solutions of the wave equation to represent the strings. The strings considered here are open strings, i.e. they don't close up on themselves. This means that the Neumann boundary conditions have to be imposed on the endpoints. With this, the general solution of the wave equation (excluding constraints) is given by

$X^{\mu }(\sigma ,\tau )=x_0^{\mu }+2{\alpha }^{\prime }{p}_0^{\mu }\tau -i\sqrt{2{\alpha }^{\prime }}\sum _{n=1}(a_n^{\mu *}{e}^{in\tau }-a_n^{\mu }{e}^{-in\tau })\frac{\cos n\sigma }{\sqrt{n}},$

where Template:Math is the slope parameter of the string (related to the string tension). The quantities Template:Math and Template:Math are (roughly) string position from the initial condition and string momentum. If all the Template:Math are zero, the solution represents the motion of a classical point particle.

This is rewritten, first defining

${\alpha }_0^{\mu }=\sqrt{2{\alpha }^{\prime }}{a}_{\mu },{\alpha }_n^{\mu }=a_n^{\mu }\sqrt{n},{\alpha }_{-n}^{\mu }=a_n^{\mu *}\sqrt{n},$

and then writing

$X^{\mu }(\sigma ,\tau )=x_0^{\mu }+\sqrt{2{\alpha }^{\prime }}{\alpha }_0^{\mu }\tau +i\sqrt{2{\alpha }^{\prime }}\sum _{n\ne 0}\frac{1}{n}{\alpha }_n^{\mu }{e}^{-in\tau }\cos n\sigma .$

In order to satisfy the constraints, one passes to light cone coordinates. For Template:Math, where Template:Math is the number of space dimensions, set

$\begin{array}{cc}{X}^I(\sigma ,\tau )& =x_0^I+\sqrt{2{\alpha }^{\prime }}{\alpha }_0^I\tau +i\sqrt{2{\alpha }^{\prime }}\sum _{n\ne 0}\frac{1}{n}{\alpha }_n^I{e}^{-in\tau }\cos n\sigma ,\\ X^{+}(\sigma ,\tau )& =\sqrt{2{\alpha }^{\prime }}{\alpha }_0^{+}\tau ,\\ X^{-}(\sigma ,\tau )& =x_0^{-}+\sqrt{2{\alpha }^{\prime }}{\alpha }_0^{-}\tau +i\sqrt{2{\alpha }^{\prime }}\sum _{n\ne 0}\frac{1}{n}{\alpha }_n^{-}{e}^{-in\tau }\cos n\sigma .\end{array}$

Not all Template:Math are independent. Some are zero (hence missing in the equations above), and the "minus coefficients" satisfy

$\sqrt{2{\alpha }^{\prime }}{\alpha }_n^{-}=\frac{1}{2p^{+}}\sum _{p\in \mathbb{Z}}{\alpha }_{n-p}^I{\alpha }_p^I.$

The quantity on the left is given a name,

$\sqrt{2{\alpha }^{\prime }}{\alpha }_n^{-}\equiv \frac{1}{p^{+}}{L}_n,L_n=\frac{1}{2}\sum _{p\in \mathbb{Z}}{\alpha }_{n-p}^I{\alpha }_p^I,$

the transverse Virasoro mode.

When the theory is quantized, the alphas, and hence the Template:Math become operators.

See also

• Group cohomology
• Group contraction (Inönu–Wigner contraction)
• Group extension
• Lie algebra cohomology

References

Books

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Journals

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Web

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