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December 1, 2014

Xingting Wang (UCSD)

Isomorphism classes of small quantum groups in positive
characteristic

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**Abstract**:

Unipotent quantum groups are non-commutative versions of unipotent group
schemes in algebraic geometry. Finite unipotent quantum groups only appear
in positive characteristic. In this talk, we will provide a complete
classification of such objects up to prime-cube dimension, which can be
thought as a generalization of the well-known fact about the structure of
p-groups of small orders. In contrast to the finiteness of isomorphism
classes for each fixed order in group theory, we obtain nine infinite
families among these prime-cube ones, which are all naturally parameterized
by finite group quotients of the affine line. Further topics regarding
representations, cohomology and invariant theory of small unipotent quantum
groups will also be discussed during the talk.