The correspondence between the canonical and semicanonical bases

TL;DR

This paper proves the equivalence of two inductive algorithms for constructing bases in quantum groups, establishing an isomorphism between associated graphs and analyzing the transition matrix between canonical and semicanonical bases.

Contribution

It demonstrates that Lusztig's two key algorithms coincide and establishes the isomorphism of their associated graphs, clarifying the relationship between different bases in quantum groups.

Findings

The two Lusztig algorithms are equivalent.

The associated colored graphs are isomorphic.

The transition matrix between bases is upper triangular with ones on the diagonal.

Abstract

Given any symmetric Cartan datum, Lusztig has provided a pair of key lemmas to construct the perverse sheaves over the corresponding quiver and the functions of irreducible components over the corresponding preprojective algebra respectively. In the present article, we prove that these two inductive algorithms of Lusztig coincide. Consequently we can define two colored graphs and prove that they are isomorhic. This result finishes the statement that Lusztig's functions of irreducible components are basis of the enveloping algebra and deduces the crystal structure (in the sense of Kashiwara-Saito) from the semicanonical basis directly inside Lusztig's convolution algebra of the preprojective algebra. As an application, we prove that the transition matrix between the canonical basis and the semicanonical basis is upper triangular with all diagonal entries equal to 1.