# What is the motivation behind the naming of quantum double theory?

I am reading Kitaev's famous toric code paper. In it, Kitaev gave an algebraic construction of his model. This algebraic construction is called quantum double. After reading the reference therein on the quantum double, I still don't quite understand why it is called "quantum double." Could someone briefly enlighten me on why people call it "quantum double"? In what sense it is a double?

## 1 Answer

The terminology is a bit like a "Quantum Goup" which is neither "quantum" nor a "group."

A quantum double is a standard way (due to Drinfeld I think) of constructing a Hopf algebra (a bialgebra with antipode etc) from a group algebra. It's a "double" in the sense that the double $$D(H)$$ combines both the group algebra $$H(G)$$ of a finite group $$G$$ and its dual $$H^*(G)$$. An element $$g\in G$$ has a projector function $$P_g\in H^*(g)$$ assigned to it so that $$P_gP_{g'}=\delta_{gg'}P_g$$ and, just as an element of the group algebra is a linear combination of $$g$$'s, a general element of $$D(H)$$ is alinear combination of the $$P_g {h}$$. There there is a product $$\nabla:(D\otimes D)\to D$$ defined by $$\nabla(P_g h \otimes P_{g'}h') = \delta_{g, hg'h^{-1}} P_g hh'$$ and a coproduct $$\Delta: D\to D\otimes D$$ given by $$\Delta(P_hg)= \sum_{h=h'h''} P_{h'}g\otimes P_{h''} g$$

If this does not seem very helpful, I am sympathetic. It took me a long time to figure out Hopf algebras, and I am still not comfortable with them.

Unless you are into category theory, the Toric Code is best studied as it is presented in the physics literature.

• I see! Thank you so much for your advice! This is an answer I was hoping for! Thank you so much for such a detailed answer! Jan 29, 2021 at 19:15