黑克代数,又名黑克环,是对称群环(group ring for the symmetric group)
的
形变,在代数数论及表示论都会出现。
设
![{\displaystyle \epsilon \in \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5214c752c471014da8c65cbea859f931ef23042e)
![{\displaystyle l\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08b7892aeda0492fd284389aaa5775b55dff75c1)
黑克环
产生自:
![{\displaystyle \sigma _{1},\sigma _{2},......,\sigma _{l-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75e47a00b6cd061858cbfa84a76ef2c68204f48e)
而
要符合:
![{\displaystyle \sigma _{i}\sigma _{i}^{-1}=\sigma _{i}^{-1}\sigma _{i}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/876bc1f770d765566a489f1460e7349586ce0b97)
- 当|i-j|>1,就有
![{\displaystyle \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/708c02826f49f8a68897517be43947d257972333)
- 当j=i+1,就有
![{\displaystyle \sigma _{i}\sigma _{j}\sigma _{i}=\sigma _{j}\sigma _{i}\sigma _{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/622a2013463a745c18010a999b3700a771fc1dc0)
![{\displaystyle (\sigma _{i}+1)(\sigma _{i}-\epsilon )=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a8e66e082a788f29cd2df9a95e74ddbe4894768)
当l=1时,就约定
。
留意:最后一项条件中当
时,
,此所谓形变。
- Vyjayanthi Chari / Andrew Pressley (1994), "A Guide to Quantum Groups", Cambridge, ISBN 0-521-55884-0 , p.332