*Just some notes on the pre-print. I am looking at this paper to better understand this pre-print. In particular I am hoping to learn more about the support of a probability on a quantum group. Flags and notes are added but mistakes are mine alone.*

#### Abstract

From this paper I will look at:

- lattice operations on , for a LCQG (analogues of intersection and generation)

## 1. Introduction

Idempotent states on quantum groups correspond with “subgroup-like” objects. In this work, on LCQG, the correspondence is with *quasi-subgroups *(the work of Franz & Skalski the correspondence was with *pre-subgroups *and *group-like projections*).

Let us show the kind of thing I am trying to understand better.

Let be the algebra of function on a finite quantum group. Let be concentrated on a pre-subgroup . We can associate to a group like projection .

Let, and this is another thing I am trying to understand better, this support, the support of be ‘the smallest’ (?) projection such that . Denote this projection by . Define similarly. That are concentrated on is to say that and .

Define a map by

(or should this be or ?)

We can decompose, in the finite case, .

**Claim: **If is concentrated on , … I don’t have a proof but it should fall out of something like together with the decomposition of above. It may also require that is a trace, I don’t know. Something very similar in the preprint.

From here we can do the following. That is a group-like projection means that:

Hit both sides with to get:

.

By the fact that are supported on , the right-hand side equals one, and by the as-yet-unproven claim, we have

.

However this is the same as

,

in other words , that is remains supported on . As a corollary, a random walk driven by a probability concentrated on a pre-subgroup remains concentrated on .

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