Quantum Algebra Research Articles

Algebraic deformation quantization of Leibniz algebras

ABSTRACTIn this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. We construct canonical rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra.Our motivation is deformation quantization of Leibniz algebras in the sense of [6]. Namely, the canonical rack bialgebras we have constructed for any Leibniz algebra lead to a simple explicit formula of the rack-star-product on the dual of a Leibniz algebra recently constructed by Dherin and Wagemann in [6]. We clarify this framework setting up a general deformation theory for rack bialgebras and show that the rack-star-product turns out to be a deformation of the trivial rack bialgebra product.

Drinfeld–Sokolov reduction in quantum algebras: canonical form of generating matrices

We define the second canonical forms for the generating matrices of the Reflection Equation algebras and the braided Yangians, associated with all even skew-invertible involutive and Hecke symmetries. By using the Cayley–Hamilton identities for these matrices, we show that they are similar to their canonical forms in the sense of Chervov and Talalaev (J Math Sci (NY) 158:904–911, 2008).

Nearly associative deformation quantization

We study several classes of non-associative algebras as possible candidates for deformation quantization in the direction of a Poisson bracket that does not satisfy Jacobi identities. We show that in fact alternative deformation quantization algebras require the Jacobi identities on the Poisson bracket and, under very general assumptions, are associative. At the same time, flexible deformation quantization algebras exist for any Poisson bracket.

Quantum deformation of planar amplitudes

In maximally supersymmetric four-dimensional gauge theories planar on-shell diagrams are closely related to the positive Grassmannian and the cell decomposition of it into the union of so called positroid cells. (This was proven by N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, A. Postnikov, and J. Trnka.) We establish that volume forms on positroids used to express scattering amplitudes can be q-deformed to Hochschild homology classes of corresponding quantum algebras. The planar amplitudes are represented as sums of contributions of some set of positroid cells; we quantize these contributions. In classical limit our considerations allow us to obtain explicit formulas for contributions of positroid cells to scattering amplitudes.

Generalized Ehrenfest Relations, Deformation Quantization, and the Geometry of Inter-model Reduction

This study attempts to spell out more explicitly than has been done previously the connection between two types of formal correspondence that arise in the study of quantum–classical relations: one the one hand, deformation quantization and the associated continuity between quantum and classical algebras of observables in the limit $$\hbar \rightarrow 0$$ , and, on the other, a certain generalization of Ehrenfest’s Theorem and the result that expectation values of position and momentum evolve approximately classically for narrow wave packet states. While deformation quantization establishes a direct continuity between the abstract algebras of quantum and classical observables, the latter result makes in-eliminable reference to the quantum and classical state spaces on which these structures act—specifically, via restriction to narrow wave packet states. Here, we describe a certain geometrical re-formulation and extension of the result that expectation values evolve approximately classically for narrow wave packet states, which relies essentially on the postulates of deformation quantization, but describes a relationship between the actions of quantum and classical algebras and groups over their respective state spaces that is non-trivially distinct from deformation quantization. The goals of the discussion are partly pedagogical in that it aims to provide a clear, explicit synthesis of known results; however, the particular synthesis offered aspires to some novelty in its emphasis on a certain general type of mathematical and physical relationship between the state spaces of different models that represent the same physical system, and in the explicitness with which it details the above-mentioned connection between quantum and classical models.

A Cohomological Perspective on Algebraic Quantum Field Theory

Algebraic quantum field theory is considered from the perspective of the Hochschild cohomology bicomplex. This is a framework for studying deformations and symmetries. Deformation is a possible approach to the fundamental challenge of constructing interacting QFT models. Symmetry is the primary tool for understanding the structure and properties of a QFT model.This perspective leads to a generalization of the algebraic quantum field theory framework, as well as a more general definition of symmetry. This means that some models may have symmetries that were not previously recognized or exploited.To first order, a deformation of a QFT model is described by a Hochschild cohomology class. A deformation could, for example, correspond to adding an interaction term to a Lagrangian. The cohomology class for such an interaction is computed here. However, the result is more general and does not require the undeformed model to be constructed from a Lagrangian. This computation leads to a more concrete version of the construction of perturbative algebraic quantum field theory.

Algebraic quantum kk-theory

ABSTRACTLet 𝒢 be an algebraic quantum group. We introduce an equivariant algebraic kk-theory for 𝒢-module algebras. We study an adjointness theorem related with smash product and trivial action. We also discuss a duality property.

On the algebraic quantization of a massive scalar field in anti-de Sitter spacetime

We discuss the algebraic quantization of a real, massive scalar field in the Poincaré patch of the [Formula: see text]-dimensional anti-de Sitter spacetime, with arbitrary boundary conditions. By using the functional formalism, we show that it is always possible to associate to such system an algebra of observables enjoying the standard properties of causality, time-slice axiom and F-locality. In addition, we characterize the wavefront set of the ground state associated to the system under investigation. As a consequence, we are able to generalize the definition of Hadamard states and construct a global algebra of Wick polynomials.

Relationships between quantized algebras and their semiclassical limits

A Poisson ℂ-algebra R appears in classical mechanical system and its quantized algebra appearing in quantum mechanical system is a ℂ[[ħ]]-algebra Q = R[[ħ]] with star product * such that for any a,b Є R ⊆ Q, a*b = ab + B1(a,b)ħ + B2(a,b)ħ2 + … subject to {a,b}= ħ-1(a * b ‒ b * a)|ħ=0, … (**) where Bi : R ⨯ R → R are bilinear products. The given Poisson algebra R is recovered from its quantized algebra Q by R = Q/ħQ with Poisson bracket (**), which is called its semiclassical limit. But it seems that the star product in Q is complicate and that Q is difficult to understand at an algebraic point of view since it is too big. For instance, if λ is a nonzero element of ℂ then ħ - λ is a unit in Q and thus a so-called deformation of R, Q/(ħ - λ)Q, is trivial. Hence it seems that we need an appropriate 픽-subalgebra A of Q such that A contains all generators of Q, ħ є A and A is understandable at an algebraic point of view, where 픽 is a subring of C[[ħ]]. Here we discuss how to find nontrivial deformations from quantized algebras and the natural map in [6] from a class of infinite deformations onto its semiclassical limit. The results are illustrated by examples.

Hardy–Littlewood inequalities on compact quantum groups of Kac type

The Hardy-Littlewood inequality on $\mathbb{T}$ compares the $L^p$-norm of a function with a weighted $\ell^p$-norm of its Fourier coefficients. The approach has recently been studied for compact homogeneous spaces and we study a natural analogue in the framework of compact quantum groups. Especially, in the case of the reduced group $C^*$-algebras and free quantum groups, we establish explicit $L^p-\ell^p$ inequalities through inherent information of underlying quantum group, such as growth rate and rapid decay property. Moreover, we show sharpness of the inequalities in a large class, including $C(G)$ with compact Lie group, $C_r^*(G)$ with polynomially growing discrete group and free quantum groups $O_N^+$, $S_N^+$.

A simple method for obtaining PBW-basis for some small quantum algebras

A simple method for obtaining PBW-basis for some small quantum algebras

Contextuality in Algebraic Quantum Theory

Klyachko, Can, Binicioğlu, and Shumovsky (KCBS) have introduced an inequality which holds if there is a noncontextual hidden variable theory. It is called KCBS inequality. Its violation shows a contradiction between predictions of quantum theory and noncontextual hidden variable theories. Bell-Kochen-Specker (BKS) theorem also shows that there are no noncontextual hidden variable theories in quantum theory by showing that there is no value assignment on a set of all observables. Thus, its approach is different from KCBS inequality. In the present paper, we show a logical relation between KCBS inequality and a value assignment, and examine a consequence of this logical relation.

Coupling coefficients of suq(1,1) and multivariate q-Racah polynomials

Gasper & Rahman's multivariate q-Racah polynomials are shown to arise as connection coefficients between families of multivariate q-Hahn or q-Jacobi polynomials. The families of q-Hahn polynomials are constructed as nested Clebsch–Gordan coefficients for the positive-discrete series representations of the quantum algebra suq(1,1). This gives an interpretation of the multivariate q-Racah polynomials in terms of 3nj symbols. It is shown that the families of q-Hahn polynomials also arise in wavefunctions of q-deformed quantum Calogero–Gaudin superintegrable systems.

Yang–Baxter maps, discrete integrable equations and quantum groups

For every quantized Lie algebra there exists a map from the tensor square of the algebra to itself, which by construction satisfies the set-theoretic Yang–Baxter equation. This map allows one to define an integrable discrete quantum evolution system on quadrilateral lattices, where local degrees of freedom (dynamical variables) take values in a tensor power of the quantized Lie algebra. The corresponding equations of motion admit the zero curvature representation. The commuting Integrals of Motion are defined in the standard way via the Quantum Inverse Problem Method, utilizing Baxter's famous commuting transfer matrix approach. All elements of the above construction have a meaningful quasi-classical limit. As a result one obtains an integrable discrete Hamiltonian evolution system, where the local equation of motion are determined by a classical Yang–Baxter map and the action functional is determined by the quasi-classical asymptotics of the universal R-matrix of the underlying quantum algebra. In this paper we present detailed considerations of the above scheme on the example of the algebra Uq(sl(2)) leading to discrete Liouville equations, however the approach is rather general and can be applied to any quantized Lie algebra.

Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver

Reflection states are introduced in the vertical and horizontal modules of the Ding-Iohara-Miki (DIM) algebra (quantum toroidal mathfrak{g}{mathfrak{l}}_1 ). Webs of DIM representations are in correspondence with (p, q)-web diagrams of type IIB string theory, under the identification of the algebraic intertwiner of Awata, Feigin and Shiraishi with the refined topological vertex. Extending the correspondence to the vertical reflection states, it is possible to engineer the mathcal{N}=1 quiver gauge theory of D-type (with unitary gauge groups). In this way, the Nekrasov instanton partition function is reproduced from the evaluation of expectation values of intertwiners. This computation leads to the identification of the vertical reflection state with the orientifold plane of string theory. We also provide a translation of this construction in the Iqbal-Kozcaz-Vafa refined topological vertex formalism.

The Lusztig automorphism of Uq(𝔰𝔩2) from the equitable point of view

We consider the quantum algebra [Formula: see text] in the equitable presentation. From this point of view, we describe the Lusztig automorphism and the corresponding Lusztig operator.

Noncommutative discriminants via Poisson primes

We present a general method for computing discriminants of noncommutative algebras. It builds a connection with Poisson geometry and expresses the discriminants as products of Poisson primes. The method is applicable to algebras obtained by specialization from families, such as quantum algebras at roots of unity. It is illustrated with the specializations of the algebras of quantum matrices at roots of unity and more generally all quantum Schubert cell algebras.

Quantum groups, Yang–Baxter maps and quasi-determinants

For any quasi-triangular Hopf algebra, there exists the universal R-matrix, which satisfies the Yang–Baxter equation. It is known that the adjoint action of the universal R-matrix on the elements of the tensor square of the algebra constitutes a quantum Yang–Baxter map, which satisfies the set-theoretic Yang–Baxter equation. The map has a zero curvature representation among L-operators defined as images of the universal R-matrix. We find that the zero curvature representation can be solved by the Gauss decomposition of a product of L-operators. Thereby obtained a quasi-determinant expression of the quantum Yang–Baxter map associated with the quantum algebra Uq(gl(n)). Moreover, the map is identified with products of quasi-Plücker coordinates over a matrix composed of the L-operators. We also consider the quasi-classical limit, where the underlying quantum algebra reduces to a Poisson algebra. The quasi-determinant expression of the quantum Yang–Baxter map reduces to ratios of determinants, which give a new expression of a classical Yang–Baxter map.

Cohomological Hall algebras and affine quantum groups

We study the preprojective cohomological Hall algebra (CoHA) introduced by the authors in Yang and Zhao (The cohomological Hall algebra of a preprojective algebra. arXiv: 1407.7994v5 , 2015) for any quiver Q and any one-parameter formal group $${\mathbb {G}}$$ . In this paper, we construct a comultiplication on the CoHA, making it a bialgebra. We also construct the Drinfeld double of the CoHA. The Drinfeld double is a quantum affine algebra of the Lie algebra $$\mathfrak {g}_Q$$ associated to Q, whose quantization comes from the formal group $${\mathbb G}$$ . We prove, when the group $${\mathbb G}$$ is the additive group, the Drinfeld double of the CoHA is isomorphic to the Yangian.

Geometric Schur Duality of Classical Type

This is a generalization of the classic work of Beilinson, Lusztig and MacPherson. In this paper (and an Appendix) we show that the quantum algebras obtained via a BLM-type stabilization procedure in the setting of partial Flag varieties of type B/C are two (modified) coideal subalgebras of the quantum general linear Lie algebra, $$ \overset{.}{\mathbf{U}} $$ ℐ and $$ \overset{.}{\mathbf{U}} $$ ʅ . We provide a geometric realization of the Schur-type duality of Bao–Wang between such a coideal algebra and Iwahori–Hecke algebra of type B. The monomial bases and canonical bases of the Schur algebras and the modified coideal algebra $$ \overset{.}{\mathbf{U}} $$ ℐ are constructed. In an Appendix by three authors, a more subtle 2-step stabilization procedure leading to $$ \overset{.}{\mathbf{U}} $$ ʅ is developed, and then monomial and canonical bases of $$ \overset{.}{\mathbf{U}} $$ ʅ are constructed. It is shown that $$ \overset{.}{\mathbf{U}} $$ ʅ is a subquotient of $$ \overset{.}{\mathbf{U}} $$ ℐ with compatible canonical bases. Moreover, a compatibility between canonical bases for modified coideal algebras and Schur algebras is established.