|Oct 16, 2014
For an algebraic group, the universal enveloping algebra of its Lie algebra and its coordinate ring are more or less dual to each other.
Similarly, their q-analogue, the quantum groups and the quantum coordinate rings are dual to each other.
The quantum group (more precisely modified quantum groups) have a nice basis, called the lower global basis.
Its dual basis is called the upper global basis of the quantum coordinate rings.
In this series of lectures, starting from the review of crystal basis theory, I explain the lower and the upper global basis of modified quantum groups and the quantum coordinate rings.