Hopf Algebras and Their Actions on Rings

本研究涉及Hopf代数H对代数r的作用。这些作用推广了更常见的自同构群的作用，李代数的导子作用，以及给出群阶代数的群的协作用。首席研究员将研究交叉产品；涉及Hopf代数作用的对偶问题；一般点Hopf代数的结构；结合代数的广义李氏结构，其中这个李氏结构意味着李氏括号被取为合适双代数的一组模；Hopf代数在范畴中的推广。本研究涉及一种称为Hopf代数的代数对象。虽然这个领域是相当技术性的，但霍普夫代数正变得越来越有兴趣，甚至对非数学家来说也是如此。Hopf代数在分子化学家感兴趣的结理论中很有用。某些类型的霍普夫代数现在被称为量子群，以表彰它们在物理学中的应用。Hopf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.