Related papers: The exponential nature and positivity
An integral PBW-basis of type $A_1^{(1)}$ has been constructed by Zhang [Z] and Chen [C] using the Auslander-Reiten quiver of the Kronecker quiver. We associate a geometric order to elements in this basis following an idea of Lusztig [L1]…
A geometric construction of Lusztig's modified quantum algebra of symmetric type is presented by using certain localized equivariant derived categories of double framed representation varieties of quivers.
A theorem, usually attributed to Barr, yields that (A) geometric implications deduced in classical L_{\infty\omega} logic from geometric theories also have intuitionistic proofs. Barr's theorem is of a topos-theoretic nature and its proof…
We construct a monomial basis of a quantum affine algebra of simply-laced type, associated to the PBW basis of Beck-Nakajima. We show that there exists a simple algorithm of computing canonical basis in terms of the monomial basis. We…
We develop a bar involution and canonical basis for every morphism space of the oriented skein category through a diagrammatic approach. In particular, our construction gives rise to Kazhdan-Lusztig type bases on quantized walled Brauer…
Let U be the quantum group associated to a Lie algebra g of rank n. The negative part U^- of U has a canonical basis B with favourable properties, introduced by Kashiwara and Lusztig. The approaches of Kashiwara and Lusztig lead to a set of…
Some filtrations of the tensor product of a highest weight module and a lowest weight module over quantum group $U_q(\mathfrak g)$ are constructed in \cite{LZ:2009} and one can use them to define some ideals of the modified quantized…
We prove the existence of canonical bases in the K-theory of quiver varieties. This existence was conjectured by Lusztig.
Let $n$ be a maximal nilpotent subalgebra of a complex symmetric Kac-Moody Lie algebra. Lusztig has introduced a basis of U(n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of…
It is known that the set of irreducible components of nilpotent varieties provides a geometric realization of the crystal basis for quantum groups. For each reduced expression of a Weyl group element, Gei{\ss}, Leclerc and Schr\"{o}er has…
Let $U^-_q = U^-_q(\mathfrak g)$ be the negative part of the quantum group associated to a finite dimensional simple Lie algebra $\mathfrak g$, and $\sigma : \mathfrak g \to \mathfrak g$ be the automorphism obtained from the diagram…
In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we…
In 2012 Raghavan, Samuel, and Subrahmanyam showed that the Kazhdan--Lusztig basis for the Iwahori--Hecke algebra in type A provides a ``canonical'' basis for the centraliser algebra of the Schur algebra acting on tensor space. In 2022 the…
Let U_q be the quantum group associated to a Lie algebra g of rank n. The negative part U^- of U has a canonical basis B with favourable properties, introduced by Kashiwara and Lusztig. The approaches of Kashiwara and Lusztig lead to a set…
The first goal of this paper is to study the amount of compatibility between two important constructions in the theory of quantized enveloping algebras, namely the canonical basis and the quantum Frobenius morphism. The second goal is to…
We study the Kostant-Lusztig $\mathbb A$-base of the multiparameter quantum groups. To simplify calculations, especially for $G_2$-type, we utilize the duality of the pairing of the universal $R$-matrix.
We study the canonical basis for the negative part of the quantum generalized Kac-Moody algebra associated to a symmetric Borcherds-Cartan matrix. The algebras associated to two different matrices satisfying certain conditions may coincide.…
Let $G$ be a connected simply-connected simple complex algebraic group and $\mathfrak{g}$ the corresponding simple Lie algebra. In the first half of the present paper, we study the relation between the positive part $U_q(\mathfrak{n^+})$ of…
We study factorization algebras on configuration spaces of points on the curved, colored by elements of the root lattice. We show that the factorization algebra attached to Lusztig's quantum group can be obtained as a direct image of a…
A unipotent triangular relationship is established between the dual standard monomial theoretic basis and canonical basis for the negative part of the quantized universal enveloping algebra of type A.