Theory Update 64

Carl Brannen now likes Hopf algebras. What are Hopf algebras, and why should we care? As categorical physicists, we will not worry so much about elements of an algebra, but will define the concept in terms of diagrams. An algebra is a set with a multiplication operation. A category theorist says that the multiplication is an arrow $m: A \otimes A \rightarrow A$ from two copies of the algebra back to the algebra, which we call $A$. A Hopf algebra requires maps as follows:
Now $A$ is both an algebra (with multiplication $m$) and a coalgebra (with comultiplication $\Delta$). An algebra must have an identity element, and this is the unit map $\eta: I \rightarrow A$ from the scalars $I$ into $A$. Similarly, a coalgebra has a counit. Observe that the duality between algebras and coalgebras is represented by a flip of diagrams across a horizontal axis on the page.

The one strange element that we need to add is the antipode map. Consider an example. The algebra of functions on a finite group is a Hopf algebra. The comultiplication is defined by $\Delta (f) (x,y) = f(xy)$, using group multiplication. Multiplication of functions $m(f,g)$ is given pointwise. The antipode satisfies $S(f)(x) = f(x^{-1})$, relying on the existence of inverse elements in the group. Thus the group structure naturally puts a Hopf algebra structure on its function space. The antipode obeys two laws of the form
Hopf algebras became important in physics in the late 1970s when a group of Russian physicists, who were studying algebras for quantized non linear partial differential equations, stumbled upon a set of commutation relations using sinh functions. The mathematician Drinfeld soon understood the significance of these rules, and the theory of Quantum Groups was born. A Quantum Group is not really a group. It is a Hopf algebra, namely a deformation of the universal enveloping algebra of a Lie algebra. The deformation parameter $q$ is like a quantum parameter $e^{\hbar}$.