Affine Lie algebra

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

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

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

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

for all ${\displaystyle a,b\in {\mathfrak {g}},\alpha ,\beta \in \mathbb {\mathbb {C} } }$ and ${\displaystyle n,m\in \mathbb {Z} }$, where ${\displaystyle [a,b]}$ is the Lie bracket in the Lie algebra ${\displaystyle {\mathfrak {g}}}$ and ${\displaystyle \langle \cdot |\cdot \rangle }$ is the Cartan-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

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

The corresponding affine Kac–Moody algebra is defined as a semidirect product by adding an extra generator d that satisfies [d, A] = δ(A).

The Dynkin 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 of outer automorphisms of the Lie algebra. In particular, this group always contains the identity element, and the corresponding affine Lie algebra is called an untwisted affine Lie algebra. When the simple algebra admits automorphisms that are not inner automorphisms, one may obtain other Dynkin diagrams and these correspond to twisted 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 a central 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 ${\displaystyle \mathbb {\mathbb {C} } ^{n}}$. In this case one also needs to add n further central elements for the n abelian generators.

The second integral cohomology of the loop group of the corresponding simple compact Lie 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 first Chern class of the fibration. Therefore, the central extensions of an affine Lie group are classified by a single parameter k which is called the level in the physics literature, where it first appeared. Unitary highest weight representations of the affine compact groups only exist when k 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 ${\displaystyle {\mathfrak {g}}}$ with Cartan subalgebra ${\displaystyle {\mathfrak {h}}}$ and a particular root system ${\displaystyle \Delta }$. Introducing the notation ${\displaystyle X_{n}=X\otimes t^{n},}$, one can attempt to extend a Cartan–Weyl basis ${\displaystyle \{H^{i}\}\cup \{E^{\alpha }|\alpha \in \Delta \}}$ 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 ${\displaystyle E_{n}^{\alpha }.}$

The Killing 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

$${\displaystyle {\hat {\Delta }}=\{\alpha +n\delta |n\in \mathbb {Z} ,\alpha \in \Delta \}\cup \{n\delta |n\in \mathbb {Z} ,n\neq 0\}.}$$

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 extended Cartan matrix and extended Dynkin diagrams.

The representations are constructed in more detail as follows.[1]

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 ${\displaystyle {\hat {\mathfrak {g}}}}$.

The vacuum representation of rank ${\displaystyle k}$, denoted ${\displaystyle V_{k}({\mathfrak {g}})}$ where ${\displaystyle k\in \mathbb {C} }$ is the complex representation with basis

$${\displaystyle \{v_{n_{1}\cdots n_{m}}^{\rho _{1}\cdots \rho _{m}}:n_{1}\geq \cdots \geq n_{m}\geq 1,\rho _{1}\leq \cdots \leq \rho _{m}\}\cup \{\Omega \}}$$

and define the action of ${\displaystyle {\hat {\mathfrak {g}}}}$ on ${\displaystyle V=V_{k}({\mathfrak {g}})}$ by (with ${\displaystyle n>0}$)

$${\displaystyle c=k{\text{id}}_{V},\,J_{n}^{\rho }\Omega =0,}$$

$${\displaystyle J_{-n}^{\rho }\Omega =v_{n}^{\rho }\,J_{-n}^{\rho }v_{n_{1}\cdots n_{m}}^{\rho _{1}\cdots \rho _{m}}=v_{nn_{1}\cdots n_{m}}^{\rho \rho _{1}\cdots \rho _{m}}.}$$

The vacuum representation in fact can be equipped with vertex algebra structure, in which case it is called the affine 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.