Canonical Bases in Representation Theory

• Main Authors: Tsao-Hsien Chen, 陳朝銑
• Other Authors: Shun-Jen Cheng
• Format: Others
• Language: en_US
• Published: 2006
• Online Access: http://ndltd.ncl.edu.tw/handle/55359760773107284958

碩士 === 國立臺灣大學 === 數學研究所 === 94 === 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 ).