# Kac-Moody algebra

Let $A$ be an $n\times n$ generalized Cartan matrix. If $n-r$ is the rank of $A$, then let $\U0001d525$ be a $n+r$ dimensional complex vector space. Choose $n$ linearly independent^{} elements ${\alpha}_{1},\mathrm{\dots},{\alpha}_{n}\in {\U0001d525}^{*}$ (called *roots*), and ${\stackrel{\u02c7}{\alpha}}_{1},\mathrm{\dots},{\stackrel{\u02c7}{\alpha}}_{n}\in \U0001d525$ (called *coroots*) such that $\u27e8{\alpha}_{i},\stackrel{\u02c7}{{\alpha}_{j}}\u27e9={a}_{ij}$, where $\u27e8\cdot ,\cdot \u27e9$ is the natural pairing of ${\U0001d525}^{*}$ and $\U0001d525$. This choice is unique up to automorphisms^{} of $\U0001d525$.

Then the Kac-Moody algebra associated to $\U0001d524(A)$ is the Lie algebra^{} generated by elements ${X}_{1},\mathrm{\dots},{X}_{n},{Y}_{1},\mathrm{\dots},{Y}_{n}$ and the elements of $\U0001d525$, with the relations

$[{X}_{i},{Y}_{i}]$ | $=\stackrel{\u02c7}{{\alpha}_{i}}$ | $[{X}_{i},{Y}_{j}]$ | $=0$ | ||

$={\alpha}_{i}(h){X}_{i}$ | $[{Y}_{i},h]$ | $=-{\alpha}_{i}(h){Y}_{i}$ | |||

$\underset{1-{a}_{ij}\text{times}}{\underset{\u23df}{[{X}_{i},[{X}_{i},\mathrm{\cdots},[{X}_{i}}},{X}_{j}]\mathrm{\cdots}]]$ | $=0$ | $\underset{1-{a}_{ij}\text{times}}{\underset{\u23df}{[{Y}_{i},[{Y}_{i},\mathrm{\cdots},[{Y}_{i}}},{Y}_{j}]\mathrm{\cdots}]]$ | $=0$ |

for any $h\in \U0001d525$.

If the matrix $A$ is positive-definite, we obtain a finite dimensional semi-simple Lie algebra, and $A$ is the Cartan matrix^{} associated to a Dynkin diagram^{}. Otherwise, the algebra^{} we obtain is infinite dimensional and has an $r$-dimensional center.

Title | Kac-Moody algebra |
---|---|

Canonical name | KacMoodyAlgebra |

Date of creation | 2013-03-22 13:31:52 |

Last modified on | 2013-03-22 13:31:52 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 7 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 17B67 |