Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Monoidal categorification of cluster algebras (merged version)   We prove that the quantum cluster algebra structure of a unipotent quantum
coordinate ring $A_q(\mathfrak{n}(w))$, associated with a symmetric Kac-Moody
algebra and its Weyl group element $w$, admits a monoidal categorification via
the representations of symmetric Khovanov-Lauda- Rouquier algebras. In order to
achieve this goal, we give a formulation of monoidal categorifications of
quantum cluster algebras and provide a criterion for a monoidal category of
finite-dimensional graded $R$-modules to become a monoidal categorification,
where $R$ is a symmetric Khovanov-Lauda-Rouquier algebra. Roughly speaking,
this criterion asserts that a quantum monoidal seed can be mutated successively
in all the directions, once the first-step mutations are possible. Then, we
show the existence of a quantum monoidal seed of $A_q(\mathfrak{n}(w))$ which
admits the first-step mutations in all the directions. As a consequence, we
prove the conjecture that any cluster monomial is a member of the upper global
basis up to a power of $q^{1/2}$. In the course of our investigation, we also
give a proof of a conjecture of Leclerc on the product of upper global basis
elements.

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The categories of this paper are: hep-lat
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
18
The categories of this paper are: econ
8
The categories of this paper are: hep-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg