表現理論中的標準基底

Abstract

這篇論文的主題是如何構造量子群的canonical basis. 對於研究李代數或是量子群的表現, 找一組好的基底是非常重要且基本的問題 . 所謂好的意思是指對於這組基底我們的表現會非常的簡明 或是 tensor product法則非常簡單. 對於古典的例子如 sln, son, 我們已經有這樣的基底, 但對於一般的情況我們並不清楚. 在[2], Lusztig給了一個方法去構造一般的的情形的canonical basis . 事實上他構造的是量子群的canonical basis, 但我們知道量子群的表現總是可以對應到李代數的表現, 因此對於李代數的情形我們有了一般的答案. 值得一提的是這樣的構造方法只有在量子群上才行的通. 這篇論文中主要就是要去探討canonical basis 的構造方法, 進而了解 量子群對於李代數的影響.

Finding a canonical basis for a Lie algebra and its representation is an elementary and important problem. Canonical means it is unique (up to scalar) and has many remarkable properties. For example , the action on those basis should be simple, they should behave well under tensor product, etc. Such problems have been studied for a long time , for example Hodge in the case slm, Gelfand in the case of som, slm, but all of them seem to restrict to the case when the Lie algebra is of classical type. So it is natural to ask whether a g of non-classical type possess a canonical basis ? If it exsits how to constuct it ? In the 1990, Lusztig [2] constructed his canonical bases and solve this problem, more presicely, he constructed a bases B for the postive part of the quantized envoloping algebra U+q (g) (A,D,E type), in [4] for any symmetrizable Kac-Moody algebra. He showed that for any finte dimentional irreducible representation V , applying B to the lowest weight vector of V , one gets a basis of V . He shows that B has many remrakable properties and this is the canonical basis of V . Specializing q to 1, we obtain a solution of the corresponding classical problem. In fact, at the same time Kashiwara also announced a canonical bases, that he called crystal bases. Similarly, they have many nice properties, the most important of which is that the tensor product rule is very simple. In [3] Lusztig shows that those two basis coincide ! Surprisingly, the construction of Lusztig’s canonical bases, as well as Kashiwara’s crystal bases can only be carried out at the quantum level. So it seems that the q-deformation is essential for the classical problem , this interests me a lot, and this is the motivaton for me to choose this topic as my thesis. The main idea of the construction is as follows. First we already have many bases of PBW type. It follows from Gabriel’s theorem that [6] any basis of PBW type can be naturally parametrized by ismorphism classes of representations of the corresponding quiver (see chapter 4). Moreover Ringel [7] has showed that the multiplication of U+q (g) can be interpreted in terms of quivers. Now an isomorphism class of representatons of quiver of a fixed dimention can be viewed as an orbit in the corresponding algebraic group. The dimension of the orbit can be computed explicitly, which can be used to give the basis of PBW type a partial order. So by the standard argument of Kazhdan-Lusztig theory we get an unique bar invariant bases B and this is our canonical bases. Ringel’s interpretation of the multiplication in U+q (g) can be reformulated in terms of perverse sheaves, using convolution operation on complexes. The theory of perverse sheaves gives us more information on the mutiplication and using this we establish the positivity of the canonical bases B (see chapter 9). It should mention that in this thesis we only deal with the case when g is of A,D,E type (simple laced). For non simple laced it can be cover by simple laced case. For general symmetrizable Kac-Moody algebra our construction doesn’t work (we even don’t have basis of PBW type), however in [3] Lusztig used theory of perverse sheaves to overcome this problem and constructed the canonical bases for any symmetrizable Kac-Moddy algebra (this work is really amazing but complicated ).