This award supports research on the analysis of the algebraic structure of non-standard quantum groups. The specific aims of the project are: (1) To analyze in detail the algebraic structure of the Cremmer-Gervais quantum groups along the lines of previous work of Hodges and Levasseur on the standard and multi-parameter cases; (2) To study the structure of the associated Poisson groups and Lie bialgebras and to compare this with the structure of the associated quantum group; (3) To construct further families of non-standard quantum groups. This research is in the general area of ring theory. A ring is an algebraic object having both an addition and a multiplication defined on it. Although the additive operation satisfies the commutative law, the multiplicative operation is not required to do so. An example of a ring for which multiplication in not commutative is the collection of nxn matrices over the integers. The study of noncommutative rings has become an important part of algebra because of its increasing significance to other branches of mathematics and physics.
