We study a presentation of the Khovanov - Lauda Rouquier's candidate 2-categorification quantum group using algebraic rewriting methods. use computational approach based on modulo isotopy axioms its pivotal structure to compute family linear bases for all vector spaces 2-cells this 2-category. show that these correspond and Lauda's conjectured generating sets, proving non-degeneracy their diagr...

The analog of the principal SO(3) subalgebra of a finite dimensional simple Lie algebra can be defined for any hyperbolic Kac Moody algebra g(A) associated with a symmetrizable Cartan matrix A, and coincides with the non-compact algebra SO(1, 2). We exhibit the decomposition of g(A) into representations of SO(1, 2); with the exception of the adjoint SO(1, 2) algebra itself, all of these represe...

For symmetrizable Kac-Moody Lie algebra g, Lusztig introduced the modified quantized enveloping algebra U̇(g) and its canonical basis in [12]. In this paper, for finite and affine type symmetric Lie algebra g we define a set which depend only on the root category and prove that there is a bijection between the set and the canonical basis of U̇(g), where the root category is the T -orbit category ...

This paper studies connections between the preprojective representations of a valued quiver, the (+)-admissible sequences of vertices, and the Weyl group by associating to each preprojective representation a canonical (+)-admissible sequence. A (+)-admissible sequence is the canonical sequence of some preprojective representation if and only if the product of simple reflections associated to th...

We study a class of models of compressible two-phase flows. This class, which includes the Baer–Nunziato model, is based on the assumption that each phase is described by its own pressure, velocity and temperature and on the use of void fractions obtained from averaging process. These models are nonconservative and non-strictly hyperbolic. We prove that the mixture entropy is non-strictly conve...

In this paper, we build the unfolding approach from acyclic sign-skew-symmetric matrices of finite rank to skew-symmetric matrices of infinite rank, which can be regard as an improvement of that in the skew-symmetrizable case. Using this approach, we give a positive answer to the problem by Berenstein, Fomin and Zelevinsky in [6] which asks whether an acyclic signskew-symmetric matrix is always...

The tetrad-based equations for vacuum gravity published by Estabrook, Robinson, and Wahlquist are simplified and adapted for numerical relativity. We show that the evolution equations as partial differential equations for the Ricci rotation coefficients constitute a rather simple first-order symmetrizable hyperbolic system, not only for the Nester gauge condition on the acceleration and angular...

we construct a family of two parameter quantum grou-ps‎ ‎$u_{r,s}(mathfrak{g})$ associated with a generalized kac-moody‎ ‎algebra corresponding to symmetrizable admissible borcherds cartan‎ ‎matrix‎. ‎we also construct the $textbf{a}$-form $u_{textbf{a}}$ and‎ ‎the classical limit of $u_{r,s}(mathfrak{g})$‎. ‎furthermore‎, ‎we‎ ‎display the equitable presentation for a subalgebra‎ ‎$u_{r,s}^{b-...

Abstract This article is dedicated to the long time behavior of a finite volume approximation general symmetrizable linear hyperbolic system on bounded domain. In continuous case this problem very difficult, and $\omega $–limit set (namely all possible limits) may be large complicated depict if no dissipation introduced. we prove that in general, with stable scheme, discrete solution converges ...