本次計劃的目的在於將超楊子之拋物型表示法應用在有限W-超代數的研究上。更精確地說， 我們希望能將有限W-超代數利用超楊子給出一個明確的刻劃，透過此刻劃我們能將有限W- 超代數內的元素表達成超楊子的某個子代數的商代數。更進一步的應用，我們可以利用研究 超楊子的表現理論來研究有限W-超代數的表現理論。目前在Ａ型的超楊子方面已經有了初步 的進展(在某些特殊情形之下已經完整地證明)，估計能在此計畫期間內將最一般的情形下給出 此刻畫，並且嘗試著發展超楊子的表現理論(在此方面目前僅有非常少的已知結果)。另外在Ｑ 型超楊子方面近年來亦有一些進展:由E. Poletaeva 及V. Serganova 在2015 證明，在最簡單的 情形(principal nilpotent case)之下，此種刻劃方式在Ｑ型超楊子亦可被構造出來，但是其技術 手法與上述的Ａ型不同，其結果也還有一些改善的空間。我們希望在此計畫期間內能夠將此 Ｑ型的結果以我們的手法給出一個新的證明並改善其結果，更進一步能此為基礎將此刻劃推 廣到更一般的(rectangular case)Ｑ型有限Ｗ-超代數上。 ;The first main goal of this project is to give a realization of finite W-superalgebra (of type A) in terms of a certain quotient of the super Yanigan Y_{M|N} associated to the general linear Lie superalgebra gl_{M|N}. There are already some partial results in literature, but the most general case is still open. In our last project, we proved an important result that for any given 01-sequence and any given composition of M+N, we obtain a parabolic presentation of Y_{M|N}. This result can help us to go though some obstacles that we were not able to handle, and we are confident that this will lead us to build up the connection for the most general case in the period of our upcoming research project. Moreover, we believe that the above connection between finite W-superalgebras and super Yangians can also be applied to other types, in particular, the Q-type case. In fact, in 2015 E. Poletaeva and V. Serganova give a surjective homomorphism from the super Yangian Y(q_1) to the finite W-superalgebra when the nilpotent element is principal, which is the easiest non-trivial case. The philosophy is very similar to ours, but there are some technical differences. We believe that our method can also be applied to the Q-type case as well, and our approach can be applied to deal with some more general cases, at least it is very possible to workout the rectangular case. The second goal of our research project is to give a new proof of the result by Poletaeva and Serganova, and then extend the result to the rectangular cases.
