Related papers: PBW bases and KLR algebras

A lot of recent activity has been directed towards various constructions of "natural" bases in cluster algebras. We develop a new approach to this problem which is close in spirit to Lusztig's construction of a canonical basis, and the…

Let $\g$ be an affine Kac-Moody Lie algebra. Let $\U^+$ be the positive part of the Drinfeld-Jimbo quantum enveloping algebra associated to $\g$. We construct a basis of $\U^+$ which is related to the Kashiwara-Lusztig global crystal basis…

For quantum group of affine type, Lusztig gave an explicit construction of the affine canonical basis by simple perverse sheaves. In this paper, we construct a bar-invariant basis by using a PBW basis arising from representations of the…

For symmetrizable Kac-Moody Lie algebra $\textbf{g}$, Lusztig introduced the modified quantized enveloping algebra $\dot{\textbf{U}}(\textbf{g})$ and its canonical basis in [12]. In this paper, for finite and affine type symmetric Lie…

We consider the finite W-superalgebras for a basic classical Lie superalgebra g associated with an even nilpotent element in g both over the field of complex numbers field and and over a filed of positive characteristic. We present the PBW…

A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large…

This paper constructs homogeneous affine sandwich cellular bases of weighted KLRW algebras in types $B$, $A^{(2)}$, $D^{(2)}$. Our construction immediately gives homogeneous sandwich cellular bases for the finite dimensional quotients of…

The article concerns the subalgebra U_v^+(w) of the quantized universal enveloping algebra of the complex Lie algebra sl_{n+1} associated with a particular Weyl group element of length 2n. We verify that U_v^+(w) can be endowed with the…

The quantum Grothendieck ring of a certain category of finite-dimensional modules over a quantum loop algebra associated with a complex finite-dimensional simple Lie algebra $\mathfrak{g}$ has a quantum cluster algebra structure of…

We present a proof of an upper bound for the lengths of finite dimensional representations of algebras obeying a modified PBW property, including Lie algebras and quantum groups. The sharpness of the bound is proved and discussed.

We classify PBW-deformations of quadratic-constant type of certain quantizations of exterior algebras. These correspond to the fundamental modules of quantum $\mathfrak{sl}_N$, their duals, and their direct sums. We show that the first two…

We classify simple representations of Khovanov-Lauda-Rouquier algebras in finite type. The classification is in terms of a standard family of representations that is shown to yield the dual PBW basis in the Grothendieck group. Finally, we…

We construct common triangular bases for almost all the known (quantum) cluster algebras from Lie theory. These bases provide analogs of the dual canonical bases, long anticipated in cluster theory. In cases where the generalized Cartan…

Let ${\mathbf U}_q^-$ be the negative half of a quantum group of finite type. We construct the canonical basis of ${\mathbf U}_q^-$ by applying the folding theory of quantum groups, and piecewise linear parametrization of canonical basis.…

We categorify a class of quantum groups associated with quivers, possibly with loops, by constructing the corresponding Khovanov-Lauda-Rouquier algebras (KLR) algebras $R$. We prove that the indecomposable projective $R$-modules realize the…

The canonical basis for quantized universal enveloping algebras associated to the finite--dimensional simple Lie algebras, was introduced by Lusztig. The principal technique is the explicit construction (via the braid group action) of a…

We study quadratic algebras over a field $\textbf{k}$. We show that an $n$-generated PBW algebra $A$ has finite global dimension and polynomial growth \emph{iff} its Hilbert series is $H_A(z)= 1 /(1-z)^n$. Surprising amount can be said when…

Many rings and algebras arising in quantum mechanics, algebraic analysis, and non-commutative algebraic geometry can be interpreted as skew PBW (Poincar\'e-Birkhoff-Witt) extensions. In the present paper we study two aspects of these…

We give an algebraic construction of standard modules (infinite dimensional modules categorifying the PBW basis of the underlying quantized enveloping algebra) for Khovanov-Lauda-Rouquier algebras in all finite types. This allows us to…

The goal of this work is to provide an elementary construction of the canonical basis $\mathbf B(w)$ in each quantum Schubert cell~$U_q(w)$ and to establish its invariance under modified Lusztig's symmetries. To that effect, we obtain a…