In 1990 Beilinson, Lusztig and MacPherson provided a geometric realization of modified quantum $mathfrak{gl}_n$ and its canonical basis. A key step of their work is a construction of a monomial basis. Recently, Du and Fu provided an algebraic constru
ction of the canonical basis for modified quantum affine $mathfrak{gl}_n$, which among other results used an earlier construction of monomial bases using Ringel-Hall algebra of the cyclic quiver. In this paper, we give an elementary algebraic construction of a monomial basis for affine Schur algebras and modified quantum affine $mathfrak{gl}_n$.
LAMOST has released more than two million spectra, which provide the opportunity to search for double-peaked narrow emission line (NEL) galaxies and AGNs. The double-peaked narrow-line profiles can be well modeled by two velocity components, respecti
vely blueshifted and redshifted with respect to the systemic recession velocity. This paper presents 20 double-peaked NEL galaxies and AGNs found from LAMOST DR1 using a search method based on multi-gaussian fit of the narrow emission lines. Among them, 10 have already been published by other authors, either listed as genuine double-peaked NEL objects or as asymmetric NEL objects, the remaining 10 being first discoveries. We discuss some possible origins for double-peaked narrow-line features, as interaction between jet and narrow line regions, interaction with companion galaxies and black hole binaries. Spatially resolved optical imaging and/or follow-up observations in other spectral bands are needed to further discuss the physical mechanisms at work.
A well-known Petersons theorem says that the number of abelian ideals in a Borel subalgebra of a rank-$r$ finite dimensional simple Lie algebra is exactly $2^r$. In this paper, we determine the dimensional distribution of abelian ideals in a Borel su
balgebra of finite dimensional simple Lie algebras, which is a refinement of the Petersons theorem capturing more Lie algebra invariants.
For symplectic Lie algebras $mathfrak{sp}(2n,mathbb{C})$, denote by $mathfrak{b}$ and $mathfrak{n}$ its Borel subalgebra and maximal nilpotent subalgebra, respectively. We construct a relationship between the abelian ideals of $mathfrak{b}$ and the c
ohomology of $mathfrak{n}$ with trivial coefficients. By this relationship, we can enumerate the number of abelian ideals of $mathfrak{b}$ with certain dimension via the Poincare polynomials of Weyl groups of type $A_{n-1}$ and $C_n$.
Xu introduced a family of root-tree-diagram nilpotent Lie algebras of differential operators, in connection with evolution partial differential equations. We generalized his notion to more general oriented tree diagrams. These algebras are natural an
alogues of the maximal nilpotent Lie subalgebras of finite-dimensional simple Lie algebras. In this paper, we use Hodge Laplacian to study the cohomology of these Lie algebras. The total rank conjecture and $b_2$-conjecture for the algebras are proved. Moreover, we find the generating functions of the Betti numbers by means of Young tableaux for the Lie algebras associated with certain tree diagrams of single branch point. By these functions and Euler-Poincare principle, we obtain analogues of the denominator identity for finite-dimensional simple Lie algebras. The result is a natural generalization of the Botts classical result in the case of special linear Lie algebras.
