Inmathematics, anaffine Lie algebra is an infinite-dimensionalLie algebra that is constructed in a canonical fashion out of a finite-dimensionalsimple Lie algebra. Given an affine Lie algebra, one can also form the associated affineKac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie algebras are interesting because theirrepresentation theory, like representation theory of finite-dimensionalsemisimple Lie algebras, is much better understood than that of general Kac–Moody algebras. As observed byVictor Kac, thecharacter formula for representations of affine Lie algebras implies certain combinatorial identities, theMacdonald identities.

Affine Lie algebras play an important role instring theory andtwo-dimensional conformal field theory due to the way they are constructed: starting from a simple Lie algebra${\displaystyle \mathfrak{g}}$, one considers theloop algebra,${\displaystyle L\mathfrak{g}}$, formed by the${\displaystyle \mathfrak{g}}$-valued functions on a circle (interpreted as the closed string) with pointwise commutator. The affine Lie algebra${\displaystyle \hat{\mathfrak{g}}}$ is obtained by adding one extra dimension to the loop algebra and modifying the commutator in a non-trivial way, which physicists call aquantum anomaly (in this case, the anomaly of theWZW model) and mathematicians acentral extension. More generally, if σ is anautomorphism of the simple Lie algebra${\displaystyle \mathfrak{g}}$ associated to an automorphism of itsDynkin diagram, thetwisted loop algebra${\displaystyle L_{\sigma }\mathfrak{g}}$ consists of${\displaystyle \mathfrak{g}}$-valued functionsf on the real line which satisfythe twisted periodicity conditionf(x + 2π) =σ f(x). Their central extensions are precisely thetwisted affine Lie algebras. The point of view of string theory helps to understand many deep properties of affine Lie algebras, such as the fact that thecharacters of their representations transform amongst themselves under themodular group.

If${\displaystyle \mathfrak{g}}$ is a finite-dimensional simple Lie algebra, the corresponding affine Lie algebra${\displaystyle \hat{\mathfrak{g}}}$ is constructed as acentral extension of the loop algebra${\displaystyle \mathfrak{g}\otimes \mathbb{C}[t,t^{-1}]}$, with one-dimensional center${\displaystyle \mathbb{C}c.}$^[1] As a vector space,

${\displaystyle \hat{\mathfrak{g}}=\mathfrak{g}\otimes \mathbb{C}[t,t^{-1}]\oplus \mathbb{C}c,}$

where${\displaystyle \mathbb{C}[t,t^{-1}]}$ is the complex vector space ofLaurent polynomials in the indeterminatet. The Lie bracket is defined by the formula

${\displaystyle [a\otimes t^n+\alpha c,b\otimes t^m+\beta c]=[a,b]\otimes t^{n+m}+⟨a|b⟩n{\delta }_{m+n,0}c}$

for all${\displaystyle a,b\in \mathfrak{g},\alpha ,\beta \in \mathbb{C}}$ and${\displaystyle n,m\in \mathbb{Z}}$, where${\displaystyle [a,b]}$ is the Lie bracket in the Lie algebra${\displaystyle \mathfrak{g}}$ and${\displaystyle ⟨\cdot |\cdot ⟩}$ is theCartan-Killing form on${\displaystyle \mathfrak{g}.}$

The affine Lie algebra corresponding to a finite-dimensional semisimple Lie algebra is the direct sum of the affine Lie algebras corresponding to its simple summands. There is a distinguished derivation of the affine Lie algebra defined by

${\displaystyle \delta (a\otimes t^m+\alpha c)=t\frac{d}{dt}(a\otimes t^m).}$

The correspondingaffine Kac–Moody algebra is defined as asemidirect product by adding an extra generatord that satisfies [d,A] =δ(A).

TheDynkin diagram of each affine Lie algebra consists of that of the corresponding simple Lie algebra plus an additional node, which corresponds to the addition of an imaginary root. Of course, such a node cannot be attached to the Dynkin diagram in just any location, but for each simple Lie algebra there exists a number of possible attachments equal to the cardinality of the group ofouter automorphisms of the Lie algebra. In particular, this group always contains the identity element, and the corresponding affine Lie algebra is called anuntwisted affine Lie algebra. When the simple algebra admits automorphisms that are not inner automorphisms, one may obtain other Dynkin diagrams and these correspond totwisted affine Lie algebras.

Dynkin diagrams for affine Lie algebras

The set of extended (untwisted) affine Dynkin diagrams, with added nodes in green

"Twisted" affine forms are named with (2) or (3) superscripts. (k is the number of nodes in the graph)

The attachment of an extra node to the Dynkin diagram of the corresponding simple Lie algebra corresponds to the following construction. An affine Lie algebra can always be constructed as acentral extension of the loop algebra of the corresponding simple Lie algebra. If one wishes to begin instead with a semisimple Lie algebra, then one needs to centrally extend by a number of elements equal to the number of simple components of the semisimple algebra. In physics, one often considers instead the direct sum of a semisimple algebra and an abelian algebra${\displaystyle {\mathbb{C}}^n}$. In this case one also needs to addn further central elements for then abelian generators.

The second integral cohomology of the loop group of the corresponding simple compactLie group is isomorphic to the integers. Central extensions of the affine Lie group by a single generator are topologically circle bundles over this free loop group, which are classified by a two-class known as the firstChern class of thefibration. Therefore, the central extensions of an affine Lie group are classified by a single parameterk which is called thelevel in the physics literature, where it first appeared. Unitary highest weight representations of the affine compact groups only exist whenk is a natural number. More generally, if one considers a semi-simple algebra, there is a central charge for each simple component.

As in the finite case, determining the Cartan–Weyl basis is an important step in determining the structure of affine Lie algebras.

Fix a finite-dimensional, simple, complex Lie algebra${\displaystyle \mathfrak{g}}$ withCartan subalgebra${\displaystyle \mathfrak{h}}$ and a particular root system${\displaystyle \mathrm{\Delta }}$. Introducing the notation${\displaystyle X_n=X\otimes t^n,}$, one can attempt to extend a Cartan–Weyl basis${\displaystyle \{H^i\}\cup \{E^{\alpha }|\alpha \in \mathrm{\Delta }\}}$ for${\displaystyle \mathfrak{g}}$ to one for the affine Lie algebra, given by${\displaystyle \{H_n^i\}\cup \{c\}\cup \{E_n^{\alpha }\}}$, with${\displaystyle \{H_0^i\}\cup \{c\}}$ forming an abelian subalgebra.

The eigenvalues of${\displaystyle ad(H_0^i)}$ and${\displaystyle ad(c)}$ on${\displaystyle E_n^{\alpha }}$ are${\displaystyle {\alpha }^i}$ and${\displaystyle 0}$ respectively and independently of${\displaystyle n}$. Therefore the root${\displaystyle \alpha }$ is infinitely degenerate with respect to this abelian subalgebra. Appending the derivation described above to the abelian subalgebra turns the abelian subalgebra into a Cartan subalgebra for the affine Lie algebra, with eigenvalues${\displaystyle ({\alpha }^1,\cdots ,{\alpha }^{dim\mathfrak{h}},0,n)}$ for${\displaystyle E_n^{\alpha }.}$

TheKilling form can almost be completely determined using its invariance property. Using the notation${\displaystyle B}$ for the Killing form on${\displaystyle \mathfrak{g}}$ and${\displaystyle \hat{B}}$ for the Killing form on the affine Kac–Moody algebra,$${\displaystyle \hat{B}(X_n,Y_m)=B(X,Y){\delta }_{n+m,0},}$$$${\displaystyle \hat{B}(X_n,c)=0,\hat{B}(X_n,d)=0}$$$${\displaystyle \hat{B}(c,c)=0,\hat{B}(c,d)=1,\hat{B}(d,d)=0,}$$where only the last equation is not fixed by invariance and instead chosen by convention. Notably, the restriction of${\displaystyle \hat{B}}$ to the${\displaystyle c,d}$ subspace gives a bilinear form with signature${\displaystyle (+,-)}$.

Write the affine root associated with${\displaystyle E_n^{\alpha }}$ as${\displaystyle \hat{\alpha }=(\alpha ;0;n)}$. Defining${\displaystyle \delta =(0,0,1)}$, this can be rewritten$${\displaystyle \hat{\alpha }=\alpha +n\delta .}$$

The full set of roots is$${\displaystyle \hat{\mathrm{\Delta }}=\{\alpha +n\delta |n\in \mathbb{Z},\alpha \in \mathrm{\Delta }\}\cup \{n\delta |n\in \mathbb{Z},n\ne 0\}.}$$Then${\displaystyle \delta }$ is unusual as it has zero length:${\displaystyle (\delta ,\delta )=0}$ where${\displaystyle (\cdot ,\cdot )}$ is the bilinear form on the roots induced by the Killing form.

In order to obtain a basis of simple roots for the affine algebra, an extra simple root must be appended, and is given by$${\displaystyle {\alpha }_0=-\theta +\delta }$$where${\displaystyle \theta }$ is the highest root of${\displaystyle \mathfrak{g}}$, using the usual notion of height of a root. This allows definition of the extendedCartan matrix and extendedDynkin diagrams.

Therepresentation theory for affine Lie algebras is usually developed usingVerma modules. Just as in the case of semi-simple Lie algebras, these arehighest weight modules. There are no finite-dimensional representations; this follows from the fact that thenull vectors of a finite-dimensional Verma module are necessarily zero; whereas those for the affine Lie algebras are not. Roughly speaking, this follows because theKilling form is Lorentzian in the${\displaystyle c,\delta }$ directions, thus${\displaystyle (z,\overline{z})}$ are sometimes called "lightcone coordinates" on the string. The "radially ordered"current operator products can be understood to be time-likenormal ordered by taking${\displaystyle z=\exp (\tau +i\sigma )}$ with${\displaystyle \tau }$ the time-like direction along the stringworld sheet and${\displaystyle \sigma }$ the spatial direction.

The representations are constructed in more detail as follows.^[2]

Fix a Lie algebra${\displaystyle \mathfrak{g}}$ and basis${\displaystyle \{J^{\rho }\}}$. Then${\displaystyle \{J_n^{\rho }\}=\{J^{\rho }\otimes t^n\}}$ is a basis for the corresponding loop algebra, and${\displaystyle \{J_n^{\rho }\}\cup \{c\}}$ is a basis for the affine Lie algebra${\displaystyle \hat{\mathfrak{g}}}$.

Thevacuum representation of rank${\displaystyle k}$, denoted by${\displaystyle V_k(\mathfrak{g})}$ where${\displaystyle k\in \mathbb{C}}$, is the complex representation with basis$${\displaystyle \{v_{n_1\cdots n_m}^{{\rho }_1\cdots {\rho }_m}:n_1\ge \cdots \ge n_m\ge 1,{\rho }_1\le \cdots \le {\rho }_m\}\cup \{\mathrm{\Omega }\},}$$and where the action of${\displaystyle \hat{\mathfrak{g}}}$ on${\displaystyle V=V_k(\mathfrak{g})}$ is given by:$${\displaystyle c=k{\text{id}}_V,J_n^{\rho }\mathrm{\Omega }=0,\mathrm{f}\mathrm{o}\mathrm{r}n\ge 0,J_{-n}^{\rho }\mathrm{\Omega }=v_n^{\rho },\mathrm{f}\mathrm{o}\mathrm{r}n>0,}$$$${\displaystyle \mathrm{a}\mathrm{n}\mathrm{d}{J}_{-n}^{\rho }{v}_{n_1\cdots n_m}^{{\rho }_1\cdots {\rho }_m}=v_{n{n}_1\cdots n_m}^{\rho {\rho }_1\cdots {\rho }_m}.}$$

The vacuum representation in fact can be equipped with vertex algebra structure, in which case it is calledtheaffine vertex algebra of rank${\displaystyle k}$. The affine Lie algebra naturally extends to the Kac–Moody algebra, with the differential${\displaystyle d}$ represented by the translation operator${\displaystyle T}$ in the vertex algebra.

TheWeyl group of an affine Lie algebra can be written as asemi-direct product of the Weyl group of the zero-mode algebra (the Lie algebra used to define theloop algebra) and thecoroot lattice.

TheWeyl character formula of thealgebraic characters of the affine Lie algebras generalizes to theWeyl-Kac character formula. A number of interesting constructions follow from these. One may construct generalizations of theJacobi theta function. These theta functions transform under themodular group. The usual denominator identities of semi-simple Lie algebras generalize as well; because the characters can be written as "deformations" orq-analogs of the highest weights, this led to many new combinatoric identities, include many previously unknown identities for theDedekind eta function. These generalizations can be viewed as a practical example of theLanglands program.

Due to theSugawara construction, the universal enveloping algebra of any affine Lie algebra has theVirasoro algebra as a subalgebra. This allows affine Lie algebras to serve as symmetry algebras ofconformal field theories such asWZW models or coset models. As a consequence, affine Lie algebras also appear in the worldsheet description ofstring theory.

TheHeisenberg algebra^[3] defined by generators${\displaystyle a_n,n\in \mathbb{Z}}$ satisfying commutation relations$${\displaystyle [a_m,a_n]=m{\delta }_{m+n,0}c}$$can be realized as the affine Lie algebra${\displaystyle \hat{\mathfrak{u}}(1)}$.

• Fuchs, Jurgen (1992),Affine Lie Algebras and Quantum Groups, Cambridge University Press,ISBN 0-521-48412-X
• Goddard, Peter; Olive, David (1988),Kac-Moody and Virasoro algebras: A Reprint Volume for Physicists, Advanced Series in Mathematical Physics, vol. 3, World Scientific,ISBN 9971-5-0419-7
• Kac, Victor (1990),Infinite dimensional Lie algebras (3 ed.), Cambridge University Press,ISBN 0-521-46693-8
• Kohno, Toshitake (1998),Conformal Field Theory and Topology, American Mathematical Society,ISBN 0-8218-2130-X
• Pressley, Andrew;Segal, Graeme (1986),Loop groups, Oxford University Press,ISBN 0-19-853535-X

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