Kac–Moody algebra

"Almost simultaneously in 1967, Victor Kac in the USSR and Robert Moody in Canada developed what was to become Kac–Moody algebra. Kac and Moody noticed that if Wilhelm Killing's conditions were relaxed, it was still possible to associate to the Cartan matrix a Lie algebra which, necessarily, would be infinite dimensional." – A. J. Coleman^[6]

In his 1967 thesis,

Introduction

Given an n×n

generalized Cartan matrix

${\displaystyle C={\begin{pmatrix}C_{ij}\end{pmatrix}}}$, one can construct a Lie algebra ${\displaystyle {\mathfrak {g}}'(C)}$ defined by

generators

${\displaystyle e_{i}}$, ${\displaystyle h_{i}}$, and ${\displaystyle f_{i}\left(i\in \{1,\ldots ,n\}\right)}$ and relations given by:

• ${\displaystyle \left[h_{i},h_{j}\right]=0\ }$ for all ${\displaystyle i,j\in \{1,\ldots ,n\}}$;
• ${\displaystyle \left[h_{i},e_{j}\right]=c_{ij}e_{j}}$;
• ${\displaystyle \left[h_{i},f_{j}\right]=-c_{ij}f_{j}}$;
• ${\displaystyle \left[e_{i},f_{j}\right]=\delta _{ij}h_{i}}$, where ${\displaystyle \delta _{ij}}$ is the Kronecker delta;
• If ${\displaystyle i\neq j}$ (so ${\displaystyle c_{ij}\leq 0}$) then ${\displaystyle {\textrm {ad}}(e_{i})^{1-c_{ij}}(e_{j})=0}$ and ${\displaystyle \operatorname {ad} (f_{i})^{1-c_{ij}}(f_{j})=0}$, where ${\displaystyle \operatorname {ad} :{\mathfrak {g}}\to \operatorname {End} ({\mathfrak {g}}),\operatorname {ad} (x)(y)=[x,y],}$ is the
  adjoint representation
  of ${\displaystyle {\mathfrak {g}}}$.

Under a "symmetrizability" assumption, ${\displaystyle {\mathfrak {g}}'(C)}$ identifies with the derived subalgebra ${\displaystyle {\mathfrak {g}}'(C)=[{\mathfrak {g}}(C),{\mathfrak {g}}(C)]}$ of the affine Kac-Moody algebra ${\displaystyle {\mathfrak {g}}(C)}$ defined below.[12]

Definition

Assume we are given an ${\displaystyle n\times n}$

r. For every such ${\displaystyle C}$, there exists a unique up to isomorphism realization of ${\displaystyle C}$, i.e. a triple ${\displaystyle ({\mathfrak {h}},\{\alpha _{i}\}_{i=1}^{n},\{\alpha _{i}^{\vee }\}_{i=1}^{n},}$) where ${\displaystyle {\mathfrak {h}}}$ is a complex vector space, ${\displaystyle \{\alpha _{i}^{\vee }\}_{i=1}^{n}}$ is a subset of elements of ${\displaystyle {\mathfrak {h}}}$, and ${\displaystyle \{\alpha _{i}\}_{i=1}^{n}}$ is a subset of the

${\displaystyle {\mathfrak {h}}^{*}}$

1. The vector space ${\displaystyle {\mathfrak {h}}}$ has dimension 2n − r
2. The sets ${\displaystyle \{\alpha _{i}\}_{i=1}^{n}}$ and ${\displaystyle \{\alpha _{i}^{\vee }\}_{i=1}^{n}}$ are linearly independent and
3. For every ${\displaystyle 1\leq i,j\leq n,\alpha _{i}\left(\alpha _{j}^{\vee }\right)=C_{ji}}$.

The ${\displaystyle \alpha _{i}}$ are analogue to the

${\displaystyle \alpha _{i}^{\vee }}$

Then we define the Kac-Moody algebra associated to ${\displaystyle C}$ as the Lie algebra ${\displaystyle {\mathfrak {g}}:={\mathfrak {g}}(C)}$ defined by

${\displaystyle e_{i}}$

${\displaystyle f_{i}\left(i\in \{1,\ldots ,n\}\right)}$

${\displaystyle {\mathfrak {h}}}$

• ${\displaystyle \left[h,h'\right]=0\ }$ for ${\displaystyle h,h'\in {\mathfrak {h}}}$;
• ${\displaystyle \left[h,e_{i}\right]=\alpha _{i}(h)e_{i}}$, for ${\displaystyle h\in {\mathfrak {h}}}$;
• ${\displaystyle \left[h,f_{i}\right]=-\alpha _{i}(h)f_{i}}$, for ${\displaystyle h\in {\mathfrak {h}}}$;
• ${\displaystyle \left[e_{i},f_{j}\right]=\delta _{ij}\alpha _{i}^{\vee }}$, where ${\displaystyle \delta _{ij}}$ is the Kronecker delta;
• If ${\displaystyle i\neq j}$ (so ${\displaystyle c_{ij}\leq 0}$) then ${\displaystyle {\textrm {ad}}(e_{i})^{1-c_{ij}}(e_{j})=0}$ and ${\displaystyle \operatorname {ad} (f_{i})^{1-c_{ij}}(f_{j})=0}$, where ${\displaystyle \operatorname {ad} :{\mathfrak {g}}\to \operatorname {End} ({\mathfrak {g}}),\operatorname {ad} (x)(y)=[x,y],}$ is the
  adjoint representation
  of ${\displaystyle {\mathfrak {g}}}$.

A real (possibly infinite-dimensional) Lie algebra is also considered a Kac–Moody algebra if its complexification is a Kac–Moody algebra.

Root-space decomposition of a Kac–Moody algebra

${\displaystyle {\mathfrak {h}}}$ is the analogue of a Cartan subalgebra for the Kac–Moody algebra ${\displaystyle {\mathfrak {g}}}$.

If ${\displaystyle x\neq 0}$ is an element of ${\displaystyle {\mathfrak {g}}}$ such that

${\displaystyle \forall h\in {\mathfrak {h}},[h,x]=\lambda (h)x}$

for some ${\displaystyle \lambda \in {\mathfrak {h}}^{*}\backslash \{0\}}$, then ${\displaystyle x}$ is called a root vector and ${\displaystyle \lambda }$ is a root of ${\displaystyle {\mathfrak {g}}}$. (The zero functional is not considered a root by convention.) The set of all roots of ${\displaystyle {\mathfrak {g}}}$ is often denoted by ${\displaystyle \Delta }$ and sometimes by ${\displaystyle R}$. For a given root ${\displaystyle \lambda }$, one denotes by ${\displaystyle {\mathfrak {g}}_{\lambda }}$ the root space of ${\displaystyle \lambda }$; that is,

${\displaystyle {\mathfrak {g}}_{\lambda }=\{x\in {\mathfrak {g}}:\forall h\in {\mathfrak {h}},[h,x]=\lambda (h)x\}}$.

It follows from the defining relations of ${\displaystyle {\mathfrak {g}}}$ that ${\displaystyle e_{i}\in {\mathfrak {g}}_{\alpha _{i}}}$ and ${\displaystyle f_{i}\in {\mathfrak {g}}_{-\alpha _{i}}}$. Also, if ${\displaystyle x_{1}\in {\mathfrak {g}}_{\lambda _{1}}}$ and ${\displaystyle x_{2}\in {\mathfrak {g}}_{\lambda _{2}}}$, then ${\displaystyle \left[x_{1},x_{2}\right]\in {\mathfrak {g}}_{\lambda _{1}+\lambda _{2}}}$ by the Jacobi identity.

A fundamental result of the theory is that any Kac–Moody algebra can be decomposed into the direct sum of ${\displaystyle {\mathfrak {h}}}$ and its root spaces, that is

${\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus \bigoplus _{\lambda \in \Delta }{\mathfrak {g}}_{\lambda }}$,

and that every root ${\displaystyle \lambda }$ can be written as ${\displaystyle \lambda =\sum _{i=1}^{n}z_{i}\alpha _{i}}$ with all the ${\displaystyle z_{i}}$ being

• A
  positive definite matrix S gives rise to a finite-dimensional simple Lie algebra
  .
• A
  positive semidefinite matrix S gives rise to an infinite-dimensional Kac–Moody algebra of affine type, or an affine Lie algebra
  .
• An
  indefinite matrix
  S gives rise to a Kac–Moody algebra of indefinite type.
• Since the diagonal entries of C and S are positive, S cannot be
  negative definite
  or negative semidefinite.

Among the Kac–Moody algebras of indefinite type, most work has focused on those hyperbolic type, for which the matrix S is indefinite, but for each proper subset of I, the corresponding submatrix is positive definite or positive semidefinite. Hyperbolic Kac–Moody algebras have rank at most 10, and they have been completely classified.[15] There are infinitely many of rank 2, and 238 of ranks between 3 and 10.

See also

• Weyl–Kac character formula
• Generalized Kac–Moody algebra
• Integrable module
• Monstrous moonshine

Citations