Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.1175

Green's formula with $\bbc^{*}$-action and Caldero-Keller's formula for cluster algebras

Jie Xiao·Fan Xu
View original source ↗Read source document ↗Export metadata ↓

Abstract

It is known that Green's formula over finite fields gives rise to the comultiplications of Ringel-Hall algebras and quantum groups (see\cite{Green}, also see \cite{Lusztig}). In this paper, we deduce the projective version of Green's formula in a geometric way. Then following the method of Hubery in \cite{Hubery2005}, we apply this formula to proving Caldero-Keller's multiplication formula for acyclic cluster algebras of arbitrary type.

Explore related subjects

math.QAmath.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jie Xiao, Fan Xu. 2008-01-10. Green's formula with $\bbc^{*}$-action and Caldero-Keller's formula for cluster algebras. https://arxiv.org/abs/0707.1175

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Approximation of graded bialgebras

Motivated by an equivalence of categories established by Kapranov and Schechtman, we introduce, for each non-negative integer d, the category of connected bialgebras modulo d+1. We show that these categories fit into an inverse system of categories whose inverse limit category is equivalent to the category of connected bialgebras. In addition, we extend the notion of approximation of connected bialgebras to those that are not necessarily generated in degree 1 and show that, for connected bialgebras in the category of Yetter-Drinfeld modules over a Hopf algebra, approximation is compatible with cocycle twisting.

math.QA↗
arXiv2026

Integral forms for Yangians of classical type

Let $\mathfrak g$ be a simple Lie algebra of classical type. We construct a $\mathbb Z[1/2]$-integral form of the Yangian $Y(\mathfrak g)$ generated by divided powers of the positive and negative Drinfeld generators, and establish its triangular decomposition and PBW basis. Under the loop filtration, its associated graded algebra is the current algebra part of Garland's integral form. As applications, over an algebraically closed field of characteristic $p>3$, we identify the small Yangian with the restricted Yangian and classify their finite dimensional simple modules in terms of restricted Drinfeld polynomials.

math.QA↗
arXiv2026

Quantum Color Newton Rings

We study integral Newton rings attached to symmetric quantum colors. We show that their boundary values at a deleted node are exactly the elements of Habiro's cyclotomic completion, and we describe their quantum neighborhoods at $q=1$ uniformly in the rank. The symmetric-color $\mathfrak{sl}_n$ invariants of a knot are the values of an element of such a ring. In its expansion at $q=1$, the coefficient of $(q-1)^m$ has denominator dividing the $(n-1+2m)$th power of the Alexander polynomial, with integral numerator, and this exponent is sharp. For $5_2$ and $6_1$ at $q=i$, the period matrices of the rank-two coefficient recurrences have spectral modules of exact order three in the Picard group of an affine elliptic curve.

math.QA↗