Graded algebras and applications

代数的群评分在数学中起着重要作用。经典的例子是经典李代数的 Cartan 分解和复合代数的 Cayley-Dickson 分级。 两者都广泛应用于李理论，李理论是许多从事纯粹数学和应用数学以及数学物理工作的工具。分级还提供代数的变形，这对于搜索描述物理模型的对象很重要。 我们对简单代数分级的分类工作始于二十多年前，现在涉及世界上至少 10 个国家的专家。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整的。这些方法包括经典代数群，但也包括我的同事和我自己建议的方法，包括 Hopf 代数和代数群方案。与 M. Bresar 联合开发的所谓函数恒等式技术使我们能够处理无限维代数。 在任意域的情况下，我们需要知道分级除代数，因为分级简单代数可以由分级向量空间在分级除代数上的线性算子来表示。在最“实用”的实数案例中，我们最近与 M. Kochetov、A. Rodrigo-Escudero 和 M. Zaicev 合作的论文为实分级简单代数理论提供了良好的基础，并可能应用于微分几何。我们已经开始对任意域上的代数以及域本身进行分级的工作；丰富的域和除法代数理论将使我们能够用新的工具和有趣的例子来丰富理论。 一个重要的应用是李代数的分级模块。最近，A. Elduque 和 M. Kotchetov 发表了关于对经典简单李代数的有限维不可约模进行分级的可能性的结果。现在，我们与 M. Kochetov 和 A. Shihadeh 一起致力于为这些表示和无限维表示提供明确的分级。提供评分将阐明这些模块的结构，并且对于在这个流行领域工作的人来说将很有用。 另一个方向是 PI 代数理论。对于大多数简单代数来说，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 的合作中，我们证明了代数闭域上的分级简单有限维代数，具有相同的分级恒等式，与分级代数同构。我们将致力于将其扩展到其他情况和代数嵌入，以代替同构。 在与 Susan Montgomery 的合作中，我们利用对简单代数分级的广泛知识来研究 Hopf 代数的作用，以及足够大的类群元素群。从塔夫脱代数及其德林菲尔德双数开始，我们将探索这种方法适用的新情况。

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.