在上个世纪八十年代早期, Drinfeld 和Jimbo 各自独立地引入了量子群. 自那之后, 由于量子群在数学和物理的重要作用人们对量子群开展了广泛而深入的研究. 最重要的一类量子群是李代数的量子包络代数. 在上个世纪七十年代末, Kac给出了单李超代数的分类, 这项工作推广了Cartan-Killing关于单李代数分类的经典工作, 而后Yamane在上世纪九十年代中期研究了A-G 型李超代数的量子包络超代数. 利用 Gabriel 的著名定理, Ringel上个世纪九十年代初期得到下面的深刻结果: 一个任意箭向的 Dynkin 箭图的有限域上的表示范畴的 Hall 代数给出该箭图的半单李代数的包络代数的正部. 我们自然希望是否可用 Ringel-Hall 方法实现A-G 型李超代数的(量子)包络超代数. 更具体地说是否能由有限维分次遗传超代数来实现 A-G 型李超代数的 (量子) 包络超代数. 据我们所知, 尽管国内外很多专家都在关注这个问题, 但并没有得到任何进展. 我们认为应首先考察有限维超代数的结构及其表示理论. 本论文由两部分组成, 第一部分 (第二章--第五章) 主要研究 有限维超代数的 Gabriel 形式构造和有限维分次遗传超代数的分类. 为此我们首先研究了有限群分次代数及其表示理论, 并定义了超species 和分次表示型, 引入了有限群分次代数的分次 Morita 等价理论, 从而给出有限维超代数的 Gabriel 形式构造并得到分次遗传超代数或等价的无循环超species 的表示分类. 在第二部分 (第六章, 第七章) 我们研究超代数的分次等价理论, 计算所有Clifford 超代数的分次 Morita 等价约化, 并证明超代数的 Hochschild 和循环(上)同调是分次等价不变量, 由此得到超代数的分次平移扭中心是分次等价不变量.我们在第一部分得到的结果表明: 只是简单修正Ringel-Hall 方法并不能实现A-G 李超代数的(量子) 包络代数, 最主要的困难在于我们不能得到 A-G 型李超代数根的表示刻画. 尽管如此, 若引入定义在一个任意箭向的 Dykin 箭图的有限域上的表示范畴的 Grothendieck 群的ccolor 映射, 则用 Ringel-Hall 方法, 我们仍可得到一个Hopf 超代数和李超代数并可证明其满足量子 Serre 关系, 遗憾地是我们不知道这些关系是否完备.

Quantum groups were introduced independently by Drinfeld and Jimbo around 1984, since then they are increasingly interesting and studied extensively due to their connection with various branches of mathematics and physics. One of the most important examples of quantum groups are quantized deformations of universal enveloping algebras of Lie algebras. As a generalization of the case of simple Lie algebras, Kac classified the simple Lie superalgebras and Yamane developed the theory of quantized enveloping superalgebras of Lie superalgebras of type A-G. Motivated by the Ringel's groundbreaking discovery that the Hall algebra of the category of Fq-representations of a Dynkin quiver (equipped with an arbitrary orientation) provides a realization of the positive part of the enveloping algebra of the simple complex Lie algebra associated to the same Dynkin diagram, it is natural to ask whether we can realize similarly the quantized enveloping superalgebras of Lie superalgebras, more precisely, whether we can realize the quantized enveloping superalgebras of Lie superalgebras by the graded module category of finite-dimensional gr-hereditary superalgebras. As far as we know, even many experts thought or are thinking about this problem, there seems no any results on this projection. But we think, before this, we should learn much more about the structure and representation theory of finite-dimensional superalgebras. The dissertation is consisting of two parts. In the first part (Chapter 2-5), we introduce the superspecies to provide the Gabriel's formalized presentations of all finite-dimensional superalgebras. All acyclic superspecies, or equivalently all finite-dimensional (gr-basic) gr-hereditary superalgebras, are classified according to their graded representation types. To this end, graded Morita equivalence, graded representation type, and graded species are introduced for finite group graded algebras. In the second part (Chapter 6--7), we consider the graded equivalent theory of superalgebras, calculate the graded Morita reductions of Clifford superalgebras over real numbers field and complex numbers field respectively, and show that the Hochschild and cyclic (co)homology of superalgebras are graded equivalences invariants, in particular, the shift-twisted graded centers of superalgebras are graded equivalent invariants.The results in the first part implicate that it seems unlikely that any minor modification of the Ringel-Hall approach can achieve the quantized enveloping superalgebras of Lie superalgebras of type A-G, the main obstacles are that we can not obtain the super version of Gabriel's theorem, i.e., a representation interpretation of the roots of Lie superalgebras of type A-G. Nevertheless, we can construct the Ringel-Hall superalgebras by introduce the color maps on the Grothendieck group of the category of Fq-representations of a Dynkin quiver (equiped with an arbitrary orientation) provides a Hopf superalgebra and a Lie superalgebra, which satisfies the quantized Serre-relations. Unfortunately, we do not know whether these quantized Serre-relations are complete.