Dualities and Representations of Lie Superalgebras
Dualities and Representations of Lie Superalgebras
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DOI：
10.1090/gsm/144
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发表时间：
2012-12
期刊：
影响因子：
--
通讯作者：
Shun-Jen Cheng;Weiqiang Wang
中科院分区：
文献类型：
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作者：
Shun-Jen Cheng;Weiqiang Wang
这本书给出了一个系统的结构和代表性理论的有限维复杂李超代数的经典类型，并作为一个很好的介绍代表性理论的李超代数。一些民间传说的结果被严格证明（有时在细节上被纠正），有时有新的证明。三个重要的对偶在书中提出，与确定李超代数的不可约字符的统一主题。按照复杂程度的顺序，它们是舒尔二元性、豪二元性和超级二元性。对称函数的组合学是根据需要发展的，与Harish-Chandra同态以及李超代数的不可约特征有关。从零开始给出了奇异李超代数的Schur-Sergeev对偶。首次以书的形式给出了李超代数的Howe对偶。超对偶是近几年来发展起来的一种新的研究方法，用来理解经典李超代数的Bernstein-Gelfand-Gelfand模范畴。超对偶将经典李超代数的表示理论与经典李代数的表示理论直接联系起来，从而通过经典李代数的Kazhdan-Lusztig多项式解决了李超代数的不可约特征标问题。
This book gives a systematic account of the structure and representation theory of finite-dimensional complex Lie superalgebras of classical type and serves as a good introduction to representation theory of Lie superalgebras. Several folklore results are rigorously proved (and occasionally corrected in detail), sometimes with new proofs. Three important dualities are presented in the book, with the unifying theme of determining irreducible characters of Lie superalgebras. In order of increasing sophistication, they are Schur duality, Howe duality, and super duality. The combinatorics of symmetric functions is developed as needed in connections to Harish-Chandra homomorphism as well as irreducible characters for Lie superalgebras. Schur-Sergeev duality for the queer Lie superalgebra is presented from scratch with complete detail. Howe duality for Lie superalgebras is presented in book form for the first time. Super duality is a new approach developed in the past few years toward understanding the Bernstein-Gelfand-Gelfand category of modules for classical Lie superalgebras. Super duality relates the representation theory of classical Lie superalgebras directly to the representation theory of classical Lie algebras and thus gives a solution to the irreducible character problem of Lie superalgebras via the Kazhdan-Lusztig polynomials of classical Lie algebras.