Kyungyong Lee

Local syzygies of multiplier ideals

In recent years, multiplier ideals have found many applications in local and global algebraic geometry. Because of their importance, there has been some interest in the question of which ideals on a smooth complex variety can be realized as multiplier ideals. Other than integral closure no local obstructions have been known up to now, and in dimension two it was established by Favre-Jonsson and Lipman-Watanabe that any integrally closed ideal is locally a multiplier ideal. We prove the somewhat unexpected result that multiplier ideals in fact satisfy some rather strong algebraic properties involving higher syzygies. It follows that in dimensions three and higher, multiplier ideals are very special among all integrally closed ideals.

Proof of a positivity conjecture of M. Kontsevich on non-commutative cluster variables

We prove a conjecture of Kontsevich, which asserts that the iterations of the noncommutative rational map

Greedy bases in rank 2 quantum cluster algebras

F_r:(x,y)-->(xyx^{-1},(1+y^r)x^{-1})

A Combinatorial Formula for Certain Elements of Upper Cluster Algebras

are given by noncommutative Laurent polynomials with nonnegative integer coefficients.

The existence of greedy bases in rank 2 quantum cluster algebras

Significance The quantum cluster algebras are a family of noncommutative rings introduced by Berenstein and Zelevinsky as the quantum deformation of the commutative cluster algebras. At the heart of their definition is a desire to understand bases of quantum algebras arising from the representation theory of nonassociative algebras. Thus a natural and important problem in the study of quantum cluster algebras is to study their bases with good properties. In this paper, we lay out a framework for understanding the interrelationships between various bases of rank two quantum cluster algebras. We identify a quantum lift of the greedy basis for rank 2 coefficient-free cluster algebras. Our main result is that our construction does not depend on the choice of initial cluster, that it builds all cluster monomials, and that it produces bar-invariant elements. We also present several conjectures related to this quantum greedy basis and the triangular basis of Berenstein and Zelevinsky.

Positivity for Cluster Algebras of Rank 3

We develop an elementary formula for certain non-trivial elements of upper cluster algebras. These elements have positive coefficients. We show that when the cluster algebra is acyclic these elements form a basis. Using this formula, we show that each non- acyclic skew-symmetric cluster algebra of rank 3 is properly contained in its upper cluster algebra.

A Short Note on Containment of Cores

At the heart of the definition of quantum cluster algebras is a desire to understand nice bases in quantum algebras arising from the representation theory of non-associative algebras. Of particular interest is the dual canonical basis in the quantized coordinate ring of a unipotent group, or more generally in the quantized coordinate ring of a double Bruhat cell. Through a meticulous study of these algebras and their bases the notion of a cluster algebra was discovered by Fomin and Zelevinsky [8] with the notion of a quantum cluster algebra following in the work [1] of Berenstein and Zelevinsky. Underlying the definition of quantum cluster algebras is a deep conjecture that the quantized coordinate rings described above in fact have the structure of a quantum cluster algebra and that the cluster monomials arising from these cluster structures belong to the dual canonical bases of the quantum algebras. The most pressing questions in the theory are thus related to understanding bases of a (quantum) cluster algebra. Several bases are already known for both classical and quantum cluster algebras. Our main interest in bases of classical cluster algebras is related to the positivity phenomenon observed in the dual canonical basis [16]. The constructions of interest originated with the atomic bases in finite types and affine type A (in the sense of [9]) which were discovered and constructed explicitly through the works of several authors. These atomic bases consist of all indecomposable positive elements of the cluster algebra, they are “atomic” in the sense that an indecomposable positive element cannot be written as a sum of two nonzero positive elements. This line of investigation was initiated by Sherman and Zelevinsky in [21] where atomic bases were constructed in type A (1) 1 , the so called Kronecker type. In the works [3, 4] atomic bases were constructed by Cerulli Irelli for finite types and type A (1) 2 respectively. The construction of atomic bases for affine type A was completed by Dupont and Thomas in [6] using triangulations of surfaces with marked points on the boundary. The representation theory of quivers also played prominently in these works but since we will not pursue this direction here we refer the reader to the previously cited works for more details. Pushing beyond affine types it was shown by Lee, Li, and Zelevinsky in [14] that for wild types the set of all indecomposable positive elements can be linearly dependent and therefore do not form a basis. In contrast, these authors in [13] constructed for rank 2 coefficient-free cluster algebras a combinatorially defined “greedy basis” which consists of a certain subset of the indecomposable positive elements. Our main goal in this note is to establish the existence of a quantum lift of the greedy basis.

A combinatorial approach to root multiplicities of rank 2 hyperbolic Kac–Moody algebras

We prove the positivity conjecture for skew-symmetric coefficient-free cluster algebras of rank 3.

Positivity for cluster algebras

We show that the operation of taking the core of an ideal does not preserve inclusions.

Greedy elements in rank 2 cluster algebras

ABSTRACT In this paper we study root multiplicities of rank 2 hyperbolic Kac–Moody algebras using the combinatorics of Dyck paths.