Hopf Algebras and Their Actions on Rings

研究了Hopf代数H在代数R上的作用. 这些行动推广了更熟悉的行动的群体作为自同构，李代数作为导子，和coactions的群体，使集团分次代数。 主要研究人员将研究交叉产品;对偶问题涉及的行动的一个霍普夫代数;结构一般指出霍普夫代数;广义李结构的结合代数，其中这种李结构意味着李括号采取一类comodules的一个合适的双代数;和扩展的霍普夫代数的类别。 这项研究涉及一个代数对象称为一个霍普夫代数。 虽然该领域是相当技术性的，霍普夫代数正成为越来越多的兴趣，甚至非数学家。 霍普夫代数在分子化学家感兴趣的纽结理论中很有用。 某些类型的霍普夫代数现在被称为量子群，以表彰其在物理学中的应用。 霍普夫代数用于研究微分算子，使组合学更像微积分。

This research concerns the action of a Hopf algebra H on an algebra R. These actions generalize the more familiar actions of groups as automorphisms, of Lie algebras as derivations, and of coactions of groups which give group-graded algebras. The principal investigator will study crossed products; duality problems involving the action of a Hopf algebra; the structure of general pointed Hopf algebras; the generalized Lie structure of associative algebras, where this Lie structure means that the Lie bracket is taken in a category of comodules of a suitable bialgebra; and extensions of Hopf algebras in categories. This research is concerned with an algebraic object called a Hopf algebra. While the area is quite technical, Hopf algebras are becoming of increasing interest, even to non mathematicians. Hopf algebras are useful in the type of knot theory that is of interest to molecular chemists. Certain types of Hopf algebras are now referred to as quantum groups in recognition of their applications in physics. Hopf algebras are used to study differential operators and to make combinatorics more like calculus.