What are the relations among canonical basis, dual canonical basis, Semicanonical Basis, dual semicanonical bases?

Question

I am reading the lecture notes and would like to know more about canonical basis.

What are the relations among canonical basis, dual canonical basis, Semicanonical Basis, dual semicanonical bases?

Thank you very much.

Answer 1

I am not an expert (far from), but have done some reading on this myself a while back. So, I will share what I have found. I will summarize a few things below. Though the one of the best resources I have found is the webpage for a workshop at the University of Oregon on Cluster Algebras and Lusztig's Semicanonical Basis led by David Speyer from 2011. In particular the references page gives a great outline/overview/history along with specific references if one wants more detail. Reading the above link is probably best, but I'll highlight a few points to make the answer a bit more self-contained.

As usual we have a Lie group and Lie algebra $G$ and $\mathfrak{g}$ with maximal unipotent $N$ and $\mathfrak{n}$. The canonical basis $\mathcal{B}$ is a basis of $U(\mathcal{n}).$ The semicanonical basis $\mathcal{S}$ is another basis of $U(\mathfrak{n}).$ They are constructed in different ways and $\mathcal{B}$ comes from a $q \to 1$ specialization of a basis of $U_q(\mathfrak{n})$ while $\mathcal{S}$ does not come from a quantized version. The graded Hopf dual of $U(\mathfrak{n})$ is the coordinated ring $\mathbb{C}[N].$ The dual canonical basis $\mathcal{B}^*$ and dual semicanonical basis $\mathcal{S}^*$ are bases of $\mathbb{C}[N]$ obtained by this duality. These bases have many nice properties I will omit (but can be read about them in the link). The bases $\mathcal{B}^*$ and $\mathcal{S}^*$ are different in general (though they have elements in common). In fact we have the following theorem [GSL, Theorem 1.2] which states that $\mathcal{B}^*$ and $\mathcal{S}^*$ are the same only when $\mathfrak{g}$ is of type $A_n$ with $n < 5.$

[GSL] Geiss, Leclerc, and Schröer, Semicanonical bases and preprojective algebras