Generalized projective representations for sl(n+1)
Generalized projective representations for sl(n+1)
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DOI：
10.1016/j.jalgebra.2010.07.009
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发表时间：
2010-06
影响因子：
0.9
通讯作者：
Yufeng Zhao;Xiaoping Xu
中科院分区：
文献类型：
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作者：
Yufeng Zhao;Xiaoping Xu
众所周知，n维射影群在射影空间的多项式函数上得到李代数sl（n+1）的非齐次表示。利用Witt代数的Shen混合积（也称为Larsson函子），我们将sl（n+1）的上述表示推广到任意有限维不可约gl(n)-模的张量空间上具有多项式空间的非齐次表示。此外，这种表示的结构完全由投影算子技术和著名的Kostant特征恒等式来确定，这些恒等式的元素在包络代数中。特别地，对于任意给定的有限维不可约sl(n)-模，我们得到了一般不是最高权值型的无限维不可约sl(n+1)-模的一个新的单参数族。该结果也可用于以射影群为对称性的量子场论的研究。
It is well known that n-dimensional projective group gives rise to a non-homogeneous representation of the Lie algebra sl(n+1) on the polynomial functions of the projective space. Using Shen's mixed product for Witt algebras (also known as Larsson functor), we generalize the above representation of sl(n+1) to a non-homogeneous representation on the tensor space of any finite-dimensional irreducible gl(n)-module with the polynomial space. Moreover, the structure of such a representation is completely determined by employing projection operator techniques and well-known Kostant's characteristic identities for certain matrices with entries in the universal enveloping algebra. In particular, we obtain a new one parameter family of infinite-dimensional irreducible sl(n+1)-modules, which are in general not highest-weight type, for any given finite-dimensional irreducible sl(n)-module. The results could also be used to study the quantum field theory with the projective group as the symmetry.