In this series of two talks, I am going to discuss a precise sense in which the algebra A generated by two elements x, y with the relation yx = qxy is commutative. The algebra A is an example of a "quantum symmetric algebra," which is an analogue of a polynomial ring in the category of modules over a quantized universal enveloping algebra.

I will recall the relevant background about quantum groups, their representations, and the braidings. Then I'll define quantum symmetric algebras, and formulate the appropriate notion of commutativity. I will illustrate with simple examples throughout, so the talks should be accessible to a wide audience.