Related papers: Based morphisms for characters of quantum symmetri…
We introduce the notion of dual perfect bases and dual perfect graphs. We show that every integrable highest weight module $V_q(\lambda)$ over a quantum generalized Kac-Moody algebra $U_{q}(\mathcal{g})$ has a dual perfect basis and its…
We consider a $q$-analogue of abstract simplicial complexes, called $q$-complexes, and discuss the notion of shellability for such complexes. It is shown that $q$-complexes formed by independent subspaces of a $q$-matroid are shellable.…
Creation operators are given for the three distinguished bases of the type BCD universal character ring of Koike and Terada yielding an elegant way of treating computations for all three types in a unified manner. Deformed versions of these…
A $\mathbb Q$-conic bundle is a proper morphism from a threefold with only terminal singularities to a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. We study the structure of $\mathbb…
Given any quantum cluster algebra arising from a quantum unipotent subgroup of symmetrizable Kac-Moody type, we verify the quantization conjecture in full generality that the quantum cluster monomials are contained in the dual canonical…
Let $\mathcal{O}^{int}_q(m|n)$ be a semisimple tensor category of modules over a quantum ortho-symplectic superalgebra of type $B, C, D$ introduced in the author's previous work. It is a natural counterpart of the category of finitely…
In order to see the behavior of $\imath$canonical bases at $q = \infty$, we introduce the notion of $\imath$crystals associated to an $\imath$quantum group of certain quasi-split type. The theory of $\imath$crystals clarifies why…
We introduce a semisimple tensor category $\mc{O}^{int}_q(m|n)$ of modules over an quantum ortho-symplectic superalgebra. It is a natural counterpart of the category of finitely dominated integrable modules over the quantum classical…
This article investigates the two-parameter quantum matrix algebra at roots of unity. In the roots of unity setting, this algebra becomes a Polynomial Identity (PI) algebra and it is known that simple modules over such algebra are…
Using the formalism of discrete quantum group gauge theory, one can construct the quantum algebras of observables for the Hamiltonian Chern-Simons model. The resulting moduli algebras provide quantizations of the algebra of functions on the…
Let ${\mathbf U}^-_q$ be the negative part of the quantum enveloping algebra associated to a simply laced Kac-Moody Lie algebra ${\mathfrak g}$, and $\underline{\mathbf U}^-_q$ the algebra corresponding to the fixed point subalgebra of…
We use the theory of $\textbf{U}_q$-tilting modules to construct cellular bases for centralizer algebras. Our methods are quite general and work for any quantum group $\textbf{U}_q$ attached to a Cartan matrix and include the non-semisimple…
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 computational complexity of certain integrable quantum theories in 1+1 dimensions. We formalize a model of quantum computation based on these theories. In this model, distinguishable particles start out with known momenta and…
We show that there exists a unique crystal base of a parabolic Verma module over a quantum orthosymplectic superalgebra, which is induced from a $q$-analogue of a polynomial representation of a general linear Lie superalgebra.
We define two finite q-analogs of certain multiple harmonic series with an arbitrary number of free parameters, and prove identities for these q-analogs, expressing them in terms of multiply nested sums involving the Gaussian binomial…
Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship between these different notions is studied.…
We study BPS $q$-series associated to 3-manifolds decorated by a line defect along an embedded link. We prove that these $q$-series depend only on the class of the link in the skein module, thereby defining a homomorphism from the skein…
We construct a crystal base of the negative half of a quantum orthosymplectic superalgebra. It can be viewed as a limit of the crystal bases of $q$-deformed irreducible oscillator representations. We also give a combinatorial description of…
A theory of principal bundles possessing quantum structure groups and classical base manifolds is presented. Structural analysis of such quantum principal bundles is performed. A differential calculus is constructed, combining differential…