# representation theory of $\U0001d530{\U0001d529}_{2}\u2102$

The special linear Lie algebra of $2\times 2$ matricies, denoted by $\U0001d530{\U0001d529}_{2}\u2102$, is defined to be the span (over $\u2102$) of the matricies

$$E=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right),H=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{array}\right),F=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 0\hfill \end{array}\right)$$ |

with Lie bracket given by the commutator^{} of matricies: $[X,Y]:=X\cdot Y-Y\cdot X$. The matricies $E,F,H$ satisfy the commutation relations: $[E,F]=H,[H,E]=2E,[H,F]=-2F$.

The representation theory of $\U0001d530{\U0001d529}_{2}\u2102$ is a very important tool for understanding the structure theory and representation theory of other Lie algebras^{} (semi-simple^{} finite dimensional Lie algebras, as well as infinite dimensional Kac-Moody Lie algebras).

The finite dimensional, irreducible, representations of $\U0001d530{\U0001d529}_{2}\u2102$ are in bijection with the non-negative integers ${\mathbb{Z}}_{\ge 0}$ as follows. Let $k\in {\mathbb{Z}}_{\ge 0}$, $V$ be a $\u2102$-vector space^{} spanned by vectors ${v}_{0},\mathrm{\dots},{v}_{k}$. The following action of $E,H,F$ on $V$ define the unique (up to isomorphism^{}) irreducible representation of $\U0001d530{\U0001d529}_{2}\u2102$ of dimension^{} $k+1$ (or of highest weight $k$):

$$ |

The main points are that the one dimensional spaces $\u2102\cdot {v}_{i}$ are eigenspaces^{} for $H$ with eigenvalue^{} $k-2i$, the operator corresponding to $E$ kills ${v}_{0}$ and otherwise sends $\u2102\cdot {v}_{i}\to \u2102\cdot {v}_{i-1}$, while $F$ kills ${v}_{k}$ and otherwise sends $\u2102\cdot {v}_{i}\to \u2102\cdot {v}_{i+1}$. The operator corresponding to $E$ is often called a *raising operator* since it raises the eigenvalue for $H$, and that of $F$ is called a *lowering operator* since it lowers the eigenvalue for $H$.

$\U0001d530{\U0001d529}_{2}\u2102$ is a simple Lie algebra, thus by Weyl’s Theorem all finite dimensional representations for $\U0001d530{\U0001d529}_{2}\u2102$ are completely reducible. So any finite dimensional representation of $\U0001d530{\U0001d529}_{2}\u2102$ splits into a direct sum^{} of irreducible representations for various non-negative integers as described above.

Title | representation theory of $\U0001d530{\U0001d529}_{2}\u2102$ |
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Canonical name | RepresentationTheoryOfmathfraksl2mathbbC |

Date of creation | 2013-03-22 15:30:29 |

Last modified on | 2013-03-22 15:30:29 |

Owner | benjaminfjones (879) |

Last modified by | benjaminfjones (879) |

Numerical id | 7 |

Author | benjaminfjones (879) |

Entry type | Definition |

Classification | msc 22E60 |

Classification | msc 22E47 |

Defines | sl_2 |

Defines | special linear Lie algebra of 2x2 matricies |