Quantum Serre Relations Research Articles

Bivariate continuous q-Hermite polynomials and deformed quantum Serre relations

To Nicolás Andruskiewitsch on his 60th birthday, with admiration We introduce bivariate versions of the continuous [Formula: see text]-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence relations) and analytic properties (determining the orthogonality measure). We find a direct link between bivariate continuous [Formula: see text]-Hermite polynomials and the star product method of [S. Kolb and M. Yakimov, Symmetric pairs for Nichols algebras of diagonal type via star products, Adv. Math. 365 (2020), Article ID: 107042, 69 pp.] for quantum symmetric pairs to establish deformed quantum Serre relations for quasi-split quantum symmetric pairs of Kac–Moody type. We prove that these defining relations are obtained from the usual quantum Serre relations by replacing all monomials by multivariate orthogonal polynomials.

Local coordinate systems on quantum flag manifolds

This paper consists of 3 sections. In the first section, we will give a brief introduction to the ''Feigin's homomorphisms'' and will see how they will help us to prove our main and fundamental theorems related to quantum Serre relations and screening operators. In the second section, we will introduce Local integral of motions as the space of invariants of nilpotent part of quantum affine Lie algebras and will find two and three-point invariants in the case of $U_q(\hat{sl_2}) $ by using Volkov's scheme. In the third section, we will introduce lattice Virasoro algebras as the space of invariants of Borel part $U_q(B_{+})$ of $U_q(g)$ for simple Lie algebra $g$ and will find the set of generators of Lattice Virasoro algebra connected to $sl_2$ and $U_q(sl_2)$ And as a new result, we found the set of some generators of lattice Virasoro algebra.

Minimal Generators of Hall Algebras of 1-cyclic Perfect Complexes

AbstractLet $A$ be the path algebra of a Dynkin quiver over a finite field, and let $C_1(\mathscr{P})$ be the category of 1-cyclic complexes of projective $A$-modules. In the present paper, we give a PBW-basis and a minimal set of generators for the Hall algebra ${\mathcal{H}}\,(C_1(\mathscr{P}))$ of $C_1(\mathscr{P})$. Using this PBW-basis, we firstly prove the degenerate Hall algebra of $C_1(\mathscr{P})$ is the universal enveloping algebra of the Lie algebra spanned by all indecomposable objects. Secondly, we calculate the relations among the generators in ${\mathcal{H}}\,(C_1(\mathscr{P}))$, and obtain quantum Serre relations in a quotient of certain twisted version of ${\mathcal{H}}\,(C_1(\mathscr{P}))$. Moreover, we establish relations between the degenerate Hall algebra, twisted Hall algebra of $A$ and those of $C_1(\mathscr{P})$, respectively.

Liftings of Nichols algebras of diagonal type III. Cartan type G2

We complete the classification of Hopf algebras whose infinitesimal braiding is a principal Yetter–Drinfeld realization of a braided vector space of Cartan type G2 over a cosemisimple Hopf algebra.We develop a general formula for a class of liftings in which the quantum Serre relations hold. We give a detailed explanation of the procedure for finding the relations, based on the recent work of Andruskiewitsch, Angiono and Rossi Bertone.

Twisted Quantum Toroidal Algebras $${T_q^-(\mathfrak g)}$$ T q - ( g )

We construct a principally graded quantum loop algebra for the Kac-Moody algebra. As a special case a twisted analog of the quantum toroidal algebra is obtained together with the quantum Serre relations.

On Nichols algebras of diagonal type

We give an explicit and essentially minimal list of defining relations of a Nichols algebra of diagonal type with finite root system. This list contains the well-known quantum Serre relations but also many new variations. A conjecture by Andruskiewitsch and Schneider states that any finite-dimensional pointed Hopf algebra over an algebraically closed field of characteristic zero is generated as an algebra by its group-like and skew-primitive elements. As an application of our main result, we prove the conjecture when the group of group-like elements is abelian.

Twisted Generalized Weyl Algebras, Polynomial Cartan Matrices and Serre-Type Relations

A twisted generalized Weyl algebra (TGWA) is defined as the quotient of a certain graded algebra by the maximal graded ideal I with trivial zero component, analogous to how Kac–Moody algebras can be defined. In this article we introduce the class of locally finite TGWAs and show that one can associate to such an algebra a polynomial Cartan matrix (a notion extending the usual generalized Cartan matrices appearing in Kac–Moody algebra theory) and that the corresponding generalized Serre relations hold in the TGWA. We also give an explicit construction of a family of locally finite TGWAs depending on a symmetric generalized Cartan matrix C and some scalars. The polynomial Cartan matrix of an algebra in this family may be regarded as a deformation of the original matrix C and gives rise to quantum Serre relations in the TGWA. We conjecture that these relations generate the graded ideal I for these algebras, and prove it in type A 2.

AR quivers, exceptional sequences and algorithms in derived Hall algebras

Consider the canonical isomorphism between the positive part U + of the quantum group Uq(g) and the Hall algebra H(Λ), where the semisimple Lie algebra g and the finite-dimensional hereditary algebra Λ share a Dynkin diagram. Chen and Xiao have given two algorithms to decompose the root vectors into linear combinations of monomials of Chevalley generators of U + , respectively induced by the braid group action on the exceptional sequences of Λ-modules and the structure of the Auslander-Reiten quiver of Λ. In this paper, we obtain the corresponding algorithms for the derived Hall algebra DH(Λ), which was introduced by Toen. We show that both algorithms are applicable to the lattice algebra and Heisenberg double in the sense of Kapranov. All the new recursive formulae have the same flavor with the quantum Serre relations.

On Constructing Ideals of the Hall Algebra of Type B

Let Hv(An) and Hv(Bn) be the Hall algebras over ℚ(v) of the Dynkin quivers An and Bn (n ≥ 1), respectively, where v is an indeterminate and the quivers have linear orientation. By comparing the quantum Serre relations, we find a natural algebra epimorphism π : Hv(Bn) → Hv2(An). We determine the kernel of π by giving two sets of generators. Let φ be the natural algebra homomorphism from Hv(An) to the quantized Schur algebra Sv(n + 1, r)(r ≥ 1) and write [Formula: see text] for the induced map. We obtain several ideals of Hv(Bn) by lifting the kernel of φ to the kernel of the composition map [Formula: see text].

Fundamental relations in Ringel–Hall algebras

In the present paper, we show that the Ringel–Hall algebra of a finitary algebra over a finite field satisfies fundamental relations in a more general setting. By twisting the multiplication, we obtain the quantum Serre relations. Finally, certain relations between Ringel–Hall algebras of an algebra and its factor algebras are discussed.

Quantum superalgebras at roots of unity and non-Abelian symmetries of integrable models

We consider integrable vertex models whose Boltzmann weights (R-matrices) are trigonometric solutions to the graded Yang–Baxter equation. As is well known, the latter can be generically constructed from quantum affine superalgebras Uq(ĝ). These algebras do not form a symmetry algebra of the model for generic values of the deformation parameter q when periodic boundary conditions are imposed. If q is evaluated at a root of unity we demonstrate that in certain commensurate sectors one can construct non-Abelian subalgebras which are translation invariant and commute with the transfer matrix and therefore with all charges of the model. In the line of argument, we introduce the restricted quantum superalgebra Uresq(ĝ) and investigate its root of unity limit. We prove several new formulae involving supercommutators of arbitrary powers of the Chevalley–Serre generators and derive higher order quantum Serre relations as well as an analogue of Lustzig's quantum Frobenius theorem for superalgebras.

On correlation functions of drinfeld currents and shuffle algebras

We express the vanishing conditions satisfied by the correlation functions of Drinfeld currents of quantum affine algebras, imposed by the quantum Serre relations. We discuss the relation of these vanishing conditions with a shuffle algebra description of the algebra of Drinfeld currents.

Field of Fractions of Certain Quantum Algebras

In this paper, we show that the field of fractions of certain algebras appearing in the theory of quantum groups are isomorphic to the field of fractions of quantum n-space when the parameter q is not a root of 1; more precisely, the quantum Serre relations of type A n and the relations defining the quantum Weyl algebra induce q-communtation relations in the field of fractions. Also, we study the problem of separation of these skew fields using the standard embeddings in non commutative series.

Explicit quantum symmetries of WZNW theories

Following the recent proposal made by Gómez and Sierra for minimal models, we find the explicit quantum group structure of WZNW models, in their Coulomb gas representation. We construct the quantum group generators in the Chevalley basis as operators creating, destroying or counting screening charges, and show that they satisfy all the required properties, in particular the quantum Serre relations. Using duality under reflection about the background charge, we find the irreducible representations and their tensor product. Our results hold for any untwisted Kac-Moody algebra.