Graded algebras and applications

代数的群分次在数学中起着重要的作用。经典的例子是经典李代数的Cartan分解和合成代数的Cayley-Dickson分次。这两种方法都被广泛应用于李理论，这是许多从事纯数学和应用数学以及数学物理工作的工具。分级还提供了代数的变形，这在搜索描述物理模型的对象时很重要。我们关于简单代数的分级分类的工作始于二十多年前，现在世界上至少有10个国家的专家参与其中。在代数闭域的情况下，在有限维代数的情况下，分类基本上是完全的。这些方法包括经典的代数群，但也有我和我的同事提出的方法，涉及Hopf代数和代数群方案。与M.Bresar共同开发的所谓函数恒等式的技术使我们能够处理无限维代数。在任意域的情况下，人们需要知道分次除代数，因为分次单代数可以由分次除代数上的分次向量空间的线性算子来表示。在实数最“实用”的情况下，我们最近与M.Kochetov，A.Rodrigo-Escudero和M.Zaicev的论文为实分次单代数理论提供了一个很好的基础，并可能应用于微分几何。我们已经开始了关于任意域上代数的分次的工作，以及域本身；丰富的域理论和除法代数，将使我们能够用新的工具和有趣的例子来丰富理论。最近，A.Elduque和M.Kotchetov发表了关于经典单李代数的有限维不可约模的分次可能性的结果。现在，与M.Kochetov和A.Shihadeh一起，我们致力于为这些表示和无限维表示提供明确的分级。提供分级将澄清这些模块的结构，并将对在这一流行领域工作的人有用。另外一个方向是PI-代数理论。对于大多数简单代数，它们的普通恒等式是未知的。分级身份要容易得多，但它们定义了普通身份。在与F.Yasumura的工作中，我们证明了代数闭域上具有相同分次恒等式的分次单有限维代数与分次代数同构。我们将致力于将其扩展到其他情况，以及代数的嵌入，以取代同构。在我们与Susan Montgomery的合作中，我们利用我们对简单代数上的分级的广泛知识来研究具有足够大的类群元素群的Hopf代数的作用。从Taft代数及其Drinfeld倍数开始，我们将探索这种方法有效的新情况。

Group gradings of algebras play an important role in mathematics. Classical example are Cartan decomposition of classical Lie algebras, and Cayley-Dickson gradings of composition algebras.  Both are widely used in Lie Theory, a tool of many working in Pure and Applied Mathematics and in Mathematical Physics. Gradings also provide deformations of algebras, important in searches of objects describing physical models. Our work on the classification of gradings on simple algebras started more than two decades ago, and now involves specialists in at least 10 countries of the world. In the case of algebraically closed fields, the classification is essentially complete in the case of finite-dimensional algebras. The methods include classical algebraic groups but also approaches suggested  my colleagues and myself, involving Hopf algebras and algebraic group schemes. The technique of so called functional identities, developed jointly with M. Bresar, allowed us to handle infinite-dimensional algebras. In the case of arbitrary fields, one needs to know graded-division algebras, since graded-simple algebras can be represented by linear operators of graded vector spaces over graded-division algebras. In the most "practical" case of real numbers, our recent papers with M. Kochetov, A. Rodrigo-Escudero and M. Zaicev provide a good basis for the theory of real graded-simple algebras, with possible applications to Differential Geometry. We have already started work on the gradings of algebras over arbitrary fields, and fields themselves; rich theory of fields and division algebras, will enable us to enrich the theory with new tools and interesting examples.  An important application is the graded modules over Lie algebras. Recently, A. Elduque and M. Kotchetov have published results on the possibility of grading finite-dimensional irreducible modules of classical simple Lie algebras. Now, with M. Kochetov and A. Shihadeh, we work on providing explicit grading to those representations and on infinite-dimensional representations. Providing gradings will clarify the structure of these modules and will be useful  for those working in this popular area.  One more directions is the theory of PI-algebras. For most simple algebras, their ordinary identities are not known. Graded identities are much easier, yet they define ordinary identities. In a work with F. Yasumura, we proved that graded-simple finite-dimensional algebras over an algebraically closed field, with the same graded identities, are isomorphic as graded algebras. We will be working on the extension of this to other situations and to the embeddings of algebras, in place of isomorphisms. In our work with Susan Montgomery, we use our extensive knowledge of the gradings on simple algebras to the study of actions of Hopf algebras, with sufficiently large groups of group-like elements. Starting with Taft algebras and their Drinfeld doubles, we will explore new situations where this approach works.