Semi-simple representation

Semi-simple representations are a term from the mathematical field of representation theory .

definition

A representation is called semi-simple if it is the direct sum of irreducible representations .

Examples

• Any representation of a finite group is semi-simple.
• Any representation of a compact Lie group is semi-simple.
• Any representation of a semi-simple Lie group is semi-simple.
• By

${\displaystyle \rho (n)=\left({\begin{array}{cc}1&n\\0&1\end{array}}\right)}$

given representation is reducible , but not semi-simple. ${\displaystyle \rho \colon \mathbb {Z} \to SL(2,\mathbb {C} )}$

• Let be the knot group of the trefoil loop and a primitive 12th root of unity with . By ${\displaystyle \Gamma =\langle A,B\mid ABA=BAB\rangle }$ ${\displaystyle \xi }$ ${\displaystyle \xi ^{4}-\xi ^{2}+1=0}$

${\displaystyle \rho (A)=\left({\begin{array}{cc}\xi &0\\0&\xi ^{-1}\end{array}}\right),\rho (B)=\left({\begin{array}{cc}\xi &1\\0&\xi ^{-1}\end{array}}\right)}$

given representation is reducible, but not semi-simple. ${\displaystyle \rho \colon \Gamma \to SL(2,\mathbb {C} )}$

See also

• Semi-simple module

literature

• Serge Lang, algebra
• Nathan Jacobson, Basic Algebra II