Related papers: Quantum loop groups and critical convolution algeb…
We provide a generalized definition for the quantized Clifford algebra introduced by Hayashi using another parameter $k$ that we call the twist. For a field of characteristic not equal to $2$, we provide a basis for our quantized Clifford…
Categorified quantum groups play an increasing role in quantum topology and representation theory. The Steenrod algebra is a fundamental component of algebraic topology. In this paper we show that categorified quantum groups can be extended…
This is a short, self-contained expository survey, focused on algebraic and analytic aspects of quantum groups. Topics covered include the definition of ``quantum group,'' the Yang-Baxter equation, quantized universal enveloping algebras,…
From a quantum $K$-matrix of the fundamental representation, we construct one for the Kirillov-Reshetikhin module by fusion construction. Using the $\imath$crystal theory by the last author, we also obtain combinatorial $K$-matrices…
Nakajima introduced a $t$-deformation of $q$-characters, $(q,t)$-characters for short, and their twisted multiplication through the geometry of quiver varieties. The Nakajima $(q,t)$-characters of Kirillov-Reshetikhin modules satisfy a…
In this paper we describe a multiparameter deformation of the function algebra of a semisimple coadjoint orbit. In the first section we use the representation of the Lie algebra on a generalized Verma module to quantize the Kirillov bracket…
We generalize the Caldero-Chapoton formula for cluster algebras of finite type to the skew-symmetrizable case. This is done by replacing representation categories of Dynkin quivers by categories of locally free modules over certain…
Symmetry groups are projectively represented in quantum mechanics, and crystalline symmetries are fundamental in condensed matter physics. Here, we systematically present a unified theory of quantum mechanical space groups from two…
Homology theories categorifying quantum group link invariants are known to be governed by the representation theory of quiver Hecke algebras, also called KLRW algebras. Here we show that certain cylindrical KLRW algebras, relevant in…
We construct a large collection of "quantum projective spaces", in the form of Koszul, Calabi-Yau algebras with the Hilbert series of a polynomial ring. We do so by starting with the toric ones (the q-symmetric algebras), and then deforming…
This paper is concerned with a non-compact GIT quotient of a vector space, in the presence of an abelian group action and an equivariant regular function (potential) on the quotient. We define virtual counts of quasimaps from prestable…
We study a family of integrable systems of nonlinearly coupled harmonic oscillators on the classical and quantum levels. We show that the integrability of these systems follows from their symmetry characterized by algebras called here…
We show that for a finite-type Lie algebra $\mathfrak{g}$, the representation theory of quiver Hecke algebras provides a natural framework for the construction of Newton-Okounkov bodies associated to the quantum coordinate rings $\Aqnw$.…
We show that the shuffle algebra associated to a doubled quiver (determined by 3-variable wheel conditions) is generated by elements of minimal degree. Together with results of Varagnolo-Vasserot and Yu Zhao, this implies that the…
We consider a natural generalisation of symmetric Nakayama algebras, namely, symmetric special biserial algebras with at most one non-uniserial indecomposable projective module. We describe the basic algebras explicitly by quiver and…
Let $G = GL(n)$ and $K = GL(p) \times GL(q)$ with $p+q=n$, where the groups are taken over $\C$. In this paper we study a certain family of $K$-orbit closures on the flag variety $X$ of $G$. The geometry of these orbit closures plays a…
This paper considers an invariant of modules over a finite-dimensional Hopf algebra, called the critical group. This generalizes the critical groups of complex finite group representations studied by Benkart, Klivans, Reiner and Gaetz. A…
We define a class of quantum linear Galois algebras which include the universal enveloping algebra Uq(gln), the quantum Heisenberg Lie algebra and other quantum orthogonal Gelfand-Zetlin algebras of type A, the subalgebras of G-invariants…
We give a new construction of the algebraic $K$-theory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and…
Adapting a recent work of Brannan et al., on extending graph $C^*$-algebras to Quantum graphs, we introduce "Quantum Quivers" as an analogue of quivers where the edge and vertex set has been replaced by a $C^*$-algebra and the maps between…