Let C 0 g be the category of finite-dimensional integrable modules over the quantum affine algebra U ′ q (g) and let R A∞ -gmod denote the category of finitedimensional graded modules over the quiver Hecke algebra of type A ∞ . In this paper, we investigate the relationship between the categories C 02010 Mathematics Subject Classification. 81R50, 16G, 16T25,17B37.

Abstract. We give a definition 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 of a quantum cluster algebra, 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. In the course of the study, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.

Reactive oxygen species (ROS) contribute to the development of non-alcoholic fatty liver disease. ROS generation by infiltrating macrophages involves multiple mechanisms, including Toll-like receptor 4 (TLR4)-mediated NADPH oxidase (NOX) activation. Here, we show that palmitate-stimulated CD11b+F4/80low hepatic infiltrating macrophages, but not CD11b+F4/80high Kupffer cells, generate ROS via dynamin-mediated endocytosis of TLR4 and NOX2, independently from MyD88 and TRIF. We demonstrate that differently from LPS-mediated dimerization of the TLR4–MD2 complex, palmitate binds a monomeric TLR4–MD2 complex that triggers endocytosis, ROS generation and increases pro-interleukin-1β expression in macrophages. Palmitate-induced ROS generation in human CD68lowCD14high macrophages is strongly suppressed by inhibition of dynamin. Furthermore, Nox2-deficient mice are protected against high-fat diet-induced hepatic steatosis and insulin resistance. Therefore, endocytosis of TLR4 and NOX2 into macrophages might be a novel therapeutic target for non-alcoholic fatty liver disease.

We prove that, for simple modules $M$ and $N$ over a quantum affine algebra, their tensor product $M \otimes N$ has a simple head and a simple socle if $M \otimes M$ is simple. A similar result is proved for the convolution product of simple modules over quiver Hecke algebras. In the second version, the statement (1.11) (in the revised version) is modified and its proof is given in Section 4.Comment: 21 pages (the first version), 23 pages (the second version

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