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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
![{\ displaystyle \ rho (n) = \ left ({\ begin {array} {cc} 1 & n \\ 0 & 1 \ end {array}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/528ca5fe0c351c115430e2006a381e7d59f12f04)
- given representation is reducible , but not semi-simple.
![{\ displaystyle \ rho \ colon \ mathbb {Z} \ to SL (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b5cce8a65ede3e9b14714e647a21ca65d0fa53)
- 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41f509b5c0167b3cfd7f697786ef4d7d5518f333)
![\ xi](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db)
![{\ displaystyle \ xi ^ {4} - \ xi ^ {2} + 1 = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3433f3b5d821f1e565213f41d5d52b694c8af511)
![{\ 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7ed0d1ce95b846e702777285b82e5b7fcd19357)
- given representation is reducible, but not semi-simple.
![{\ displaystyle \ rho \ colon \ Gamma \ to SL (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/234e6b8bb3d73ff1e931bb787bdd06a4ab3eceb0)
See also
literature
- Serge Lang, algebra
- Nathan Jacobson, Basic Algebra II