Hopf Algebra Deformation Research Articles

Non-standard quantum algebras and finite dimensional -symmetric systems

In this work, -symmetric Hamiltonians defined on quantum algebras are presented. We study the spectrum of a family of non-Hermitian Hamiltonians written in terms of the generators of the non-standard Hopf algebra deformation of . By making use of a particular boson representation of the generators of , both the co-product and the commutation relations of the quantum algebra are shown to be invariant under the -transformation. In terms of these operators, we construct several finite dimensional -symmetry Hamiltonians, whose spectrum is analytically obtained for any arbitrary dimension. In particular, we show the appearance of Exceptional Points in the space of model parameters and we discuss the behaviour of the spectrum both in the exact -symmetry and the broken -symmetry dynamical phases. As an application, we show that this non-standard quantum algebra can be used to define an effective model Hamiltonian describing accurately the experimental spectra of three-electron hybrid qubits based on asymmetric double quantum dots. Remarkably enough, in this effective model, the deformation parameter z has to be identified with the detuning parameter of the system.

Cocycle deformations for weak Hopf algebras

In this paper we introduce a theory of multiplication alteration by two-cocycles for weak Hopf algebras. We show that, just like it happens for Hopf algebras, if H a weak Hopf algebra and H? its weak Hopf algebra deformation by a 2-cocycle ?, there is a braided monoidal category equivalence between the categories of left-right Yetter-Drinfel?d modules HYDH and H?YDH?. As a consequence we get in this context that the category Rep(D(H)) of left modules over the Drinfel?d double D(H) can be identified with the category Rep(D(H?)) of left modules over the Drinfel?d double D(H?).

Born’s Reciprocal Relativity theory, curved phase space, Finsler geometry and the cosmological constant

A very brief introduction of the history of Born’s Reciprocal Relativity Theory, Hopf algebraic deformations of the Poincare algebra, de Sitter algebra, and noncommutative spacetimes paves the road for the exploration of gravity in curved phase spaces within the context of the Finsler geometry of the cotangent bundle T∗M of spacetime. A scalar-gravity model is duly studied, and exact nontrivial analytical solutions for the metric and nonlinear connection are found that obey the generalized gravitational field equations, in addition to satisfying the zero torsion conditions for all of the torsion components. The curved base spacetime manifold and internal momentum space both turn out to be (Anti) de Sitter type. The most salient feature is that the solutions capture the very early inflationary and very-late-time de Sitter phases of the Universe. A regularization of the 8-dim phase space action leads naturally to an extremely small effective cosmological constant Λeff, and which in turn, furnishes an extremely small value for the underlying four-dim spacetime cosmological constant Λ, as a direct result of a correlation between Λeff and Λ resulting from the field equations. The rich structure of Finsler geometry deserves to be explored further since it can shine some light into Quantum Gravity, and lead to interesting cosmological phenomenology.

Canonical and Lie-algebraic twist deformations of Carroll, para-Galilei and Static Hopf algebras

In this paper, we provide five canonical and three Lie-algebraic twist-deformations of Carroll, para-Galilei and Static Hopf algebras. Particularly, in the case of Carroll Hopf structure, we get the same result with the use of contraction schemes of corresponding canonically and Lie-algebraically twisted Poincaré Hopf algebras as well.

Is dark matter and black hole cosmology an effect of Born’s reciprocal relativity theory?

Born’s reciprocal relativity theory (BRRT), based on a maximal proper-force, maximal speed of light, and inertial and non-inertial observers, is re-examined in full detail. Relativity of locality and chronology are natural consequences of this theory, even in flat phase space. The advantage of BRRT is that Lorentz invariance is preserved and there is no need to introduce Hopf algebraic deformations of the Poincaré algebra, de Sitter algebra, nor non-commutative space–times. After a detailed study of the notion of generalized force, momentum, and mass in phase space, we explain that what one may interpret as “dark matter” in galaxies, for example, is just an effect of observing ordinary galactic matter in different accelerating frames of reference than ours. Explicit calculations are provided that explain these novel relativistic effects due to the accelerated expansion of the Universe, and which may generate the present-day density parameter value ΩDM ∼ 0.25 of dark matter. The physical origins behind the numerical coincidences in black hole cosmology are also explored. We finalize with a rigorous study of the curved geometry of (co)tangent bundles (phase space) within the formalism of Finsler geometry, and provide a short discussion on Hamilton spaces.

Hopf-algebraic Deformations of Products and Wick Polynomials

Abstract We present an approach to cumulant–moment relations and Wick polynomials based on extensive use of convolution products of linear functionals on a coalgebra. This allows, in particular, to understand the construction of Wick polynomials as the result of a Hopf algebra deformation under the action of linear automorphisms induced by multivariate moments associated to an arbitrary family of random variables with moments of all orders. We also generalize the notion of deformed product in order to discuss how these ideas appear in the recent theory of regularity structures.

Poisson–Hopf algebra deformations of Lie–Hamilton systems

Hopf algebra deformations are merged with a class of Lie systems of Hamiltonian type, the so-called Lie–Hamilton systems, to devise a novel formalism: the Poisson–Hopf algebra deformations of Lie–Hamilton systems. This approach applies to any Hopf algebra deformation of any Lie–Hamilton system. Remarkably, a Hopf algebra deformation transforms a Lie–Hamilton system, whose dynamic is governed by a finite-dimensional Lie algebra of functions, into a non-Lie–Hamilton system associated with a Poisson–Hopf algebra of functions that allows for the explicit description of its t-independent constants of the motion from deformed Casimir functions. We illustrate our approach by considering the Poisson–Hopf algebra analogue of the non-standard quantum deformation of and its applications to deform well-known Lie–Hamilton systems describing oscillator systems, Milne–Pinney equations, and several types of Riccati equations. In particular, we obtain a new position-dependent mass oscillator system with a time-dependent frequency.

Basic quantizations of D = 4 Euclidean, Lorentz, Kleinian and quaternionic {mathfrak{o}}^{star }(4) symmetries

We construct firstly the complete list of five quantum deformations of D = 4 complex homogeneous orthogonal Lie algebra mathfrak{o}left(4;mathbb{C}right)cong mathfrak{o}left(3;mathbb{C}right)oplus mathfrak{o}left(3;mathbb{C}right) , describing quantum rotational symmetries of four-dimensional complex space-time, in particular we provide the corresponding universal quantum R-matrices. Further applying four possible reality conditions we obtain all sixteen Hopf-algebraic quantum deformations for the real forms of mathfrak{o}left(4;mathbb{C}right) : Euclidean mathfrak{o}(4) , Lorentz mathfrak{o}left(3, 1right) , Kleinian mathfrak{o}left(2, 2right) and quaternionic {mathfrak{o}}^{star }(4) . For mathfrak{o}left(3, 1right) we only recall well-known results obtained previously by the authors, but for other real Lie algebras (Euclidean, Kleinian, quaternionic) as well as for the complex Lie algebra mathfrak{o}left(4;mathbb{C}right) we present new results.

Deformations of spacetime and internal symmetries

Algebraic deformations provide a systematic approach to generalizing the symmetries of a physical theory through the introduction of new fundamental constants. The applications of deformations of Lie algebras and Hopf algebras to both spacetime and internal symmetries are discussed. As a specific example we demonstrate how deforming the classical flavor group $SU(3)$ to the quantum group $SU_q(3)\equiv U_q(su(3))$ (a Hopf algebra) and taking into account electromagnetic mass splitting within isospin multiplets leads to new and exceptionally accurate baryon mass sum rules that agree perfectly with experimental data.

Hopf algebras, distributive (Laplace) pairings and hash products: a unified approach to tensor product decompositions of group characters

We show for bicommutative graded connected Hopf algebras that a certain distributive (Laplace) subgroup of the convolution monoid of 2-cochains parameterizes certain well behaved Hopf algebra deformations. Using the Laplace group, or its Frobenius subgroup, we define higher derived hash products, and develop a general theory to study their main properties. Applying our results to the (universal) bicommutative graded connected Hopf algebra of symmetric functions, we show that classical tensor product and character decompositions, such as those for the general linear group, mixed co- and contravariant or rational characters, orthogonal and symplectic group characters, Thibon and reduced symmetric group characters, are special cases of higher derived hash products. In the appendix we discuss a relation to formal group laws.

Unified description for $$\kappa $$ κ -deformations of orthogonal groups

In this paper we provide universal formulas describing Drinfeld-type quantization of inhomogeneous orthogonal groups determined by a metric tensor of an arbitrary signature living in a spacetime of arbitrary dimension. The metric tensor does not need to be in diagonal form and \(\kappa \)-deformed coproducts are presented in terms of classical generators. It opens the possibility for future applications in deformed general relativity. The formulas depend on the choice of an additional vector field which parametrizes classical \(r\)-matrices. Non-equivalent deformations are then labeled by the corresponding type of stability subgroups. For the Lorentzian signature it covers three (non-equivalent) Hopf-algebraic deformations: time-like, space-like (a.k.a. tachyonic) and light-like (a.k.a. light-cone) quantizations of the Poincaré algebra. Finally the existence of the so-called Majid–Ruegg (non-classical) basis is reconsidered.

Additive deformations of Hopf algebras

Additive deformations of bialgebras in the sense of Wirth are deformations of the multiplication map of the bialgebra fulfilling a compatibility condition with the coalgebra structure and a continuity condition. Two problems concerning additive deformations are considered. With a deformation theory a cohomology theory should be developed. Here a variant of the Hochschild cohomology is used. The main result in the first part of this paper is the characterization of the trivial deformations, i.e. deformations generated by a coboundary. Starting with a Hopf algebra, one would expect the deformed multiplications to have some analogue of the antipode, which we call deformed antipodes. We prove, that deformed antipodes always exist, explore their properties, give a formula to calculate them given the deformation and the antipode of the original Hopf algebra and show in the cocommutative case, that each deformation splits into a trivial part and into a part with constant antipodes.

Hopf algebra deformations of binary polyhedral groups

We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a strengthening of a result of Nichols and Richmond on cosemisimple Hopf algebras with a two-dimensional irreducible comodule in the finite-dimensional context. Finally, we give some applications to the classification of certain classes of semisimple Hopf algebras.

Additive deformations of braided Hopf algebras

Additive deformations of bialgebras in the sense of J. Wirth, i.e. deformations of the multiplication map fulfilling a certain compatibility condition w.r.t. the coalgebra structure, can be generalized to braided bialgebras. The theorems for additive deformations of Hopf algebras can also be carried over to that case. We consider *-structures and prove a general Schoenberg correspondence in this context. Finally we give some examples.

On n-ary algebras, branes and poly-vector gauge theories in noncommutative Clifford spaces

In this paper, poly-vector-valued gauge field theories in noncommutative Clifford spaces are presented. They are based on noncommutative (but associative) star products that require the use of the Baker–Campbell–Hausdorff formula. Using these star products allows the construction of actions for noncommutative p-branes (branes moving in noncommutative spaces). Noncommutative Clifford-space gravity as a poly-vector-valued gauge theory of twisted diffeomorphisms in Clifford spaces would require quantum Hopf algebraic deformations of Clifford algebras. We proceed with the study of n-ary algebras and find an important relationship among the n-ary commutators of the noncommuting spacetime coordinates [X1, X2, …, Xn] with the poly-vector-valued coordinates X123⋅⋅⋅n in noncommutative Clifford spaces given by [X1, X2, …, Xn] = n!X123⋅⋅⋅n. The large N limit of n-ary commutators of n hyper-matrices leads to Eguchi–Schild p-brane actions for p + 1 = n. A noncomutative n-ary • product of n functions is constructed which is a generalization of the binary star product * of two functions and is associated with the deformation quantization of n-ary structures and deformations of the Nambu–Poisson brackets.

From quantum universal enveloping algebras to quantum algebras

The ‘local’ structure of a quantum group Gq is currently considered to be an infinite-dimensional object: the corresponding quantum universal enveloping algebra Uq(g), which is a Hopf algebra deformation of the universal enveloping algebra of an n-dimensional Lie algebra g = Lie(G). However, we show how, by starting from the generators of the underlying Lie bialgebra (g, δ), the analyticity in the deformation parameter(s) allows us to determine in a unique way a set of n ‘almost primitive’ basic objects in Uq(g), which could be properly called the ‘quantum algebra generators’. So, the analytical prolongation (gq, Δ) of the Lie bialgebra (g, δ) is proposed as the appropriate local structure of Gq. Besides, as in this way (g, δ) and Uq(g) are shown to be in one-to-one correspondence, the classification of quantum groups is reduced to the classification of Lie bialgebras. The suq(2) and suq(3) cases are explicitly elaborated.

On the noncommutative and nonassociative geometry of octonionic space time, modified dispersion relations and grand unification

The octonionic geometry (gravity) developed long ago by Oliveira and Marques, J. Math. Phys. 26, 3131 (1985) is extended to noncommutative and nonassociative space time coordinates associated with octonionic-valued coordinates and momenta. The octonionic metric Gμν already encompasses the ordinary space time metric gμν, in addition to the Maxwell U(1) and SU(2) Yang-Mills fields such that it implements the Kaluza-Klein Grand unification program without introducing extra space time dimensions. The color group SU(3) is a subgroup of the exceptional G2 group which is the automorphism group of the octonion algebra. It is shown that the flux of the SU(2) Yang-Mills field strength Fμν through the area-momentum Σμν in the internal isospin space yields corrections O(1∕MPlanck2) to the energy-momentum dispersion relations without violating Lorentz invariance as it occurs with Hopf algebraic deformations of the Poincare algebra. The known octonionic realizations of the Clifford Cl(8), Cl(4) algebras should permit the construction of octonionic string actions that should have a correspondence with ordinary string actions for strings moving in a curved Clifford-space target background associated with a Cl(3, 1) algebra.

Bicrossproduct approach to the Connes–Moscovici Hopf algebra

We give a rigorous proof that the (codimension one) Connes–Moscovici Hopf algebra H CM is isomorphic to a bicrossproduct Hopf algebra linked to a group factorisation of the diffeomorphism group Diff + ( R ) . We construct a second bicrossproduct U CM equipped with a nondegenerate dual pairing with H CM . We give a natural quotient Hopf algebra k λ [ Heis ] of H CM and Hopf subalgebra U λ ( heis ) of U CM which again are in duality. All these Hopf algebras arise as deformations of commutative or cocommutative Hopf algebras that we describe in each case. Finally we develop the noncommutative differential geometry of k λ [ Heis ] by studying first order differential calculi of small dimension.

Non-anticommutative supersymmetric field theory and quantum shift

Abstract Non-anticommutative Grassmann coordinates in four-dimensional twist-deformed N = 1 Euclidean superspace are decomposed into geometrical ones and quantum shift operators. This decomposition leads to the mapping from the commutative to the non-anticommutative supersymmetric field theory. We apply this mapping to the Wess–Zumino model in commutative field theory and derive the corresponding non-anticommutative Lagrangian. Based on the theory of twist deformations of Hopf algebras, we comment on the preservation of the (initial) N = 1 super-Poincare algebra and on the consequent super-Poincare invariant interpretation of the discussed model, but also provide a measure for the violation of the super-Poincare symmetry.

Integrable mixing of An−1 type vertex models

Given a family of monodromy matrices T0,T1,…,TK−1 corresponding to integrable anisotropic vertex models of Anμ−1 type, μ=0,1,…,K−1, we built up a related mixed vertex model by means of gluing the lattices on which they are defined, in such a way that integrability property is preserved. The gluing process is implemented through one-dimensional representations of rectangular quantum matrix algebras A(Rnμ−1:Rnμ), namely, the gluing matrices ζμ. Algebraic Bethe ansatz is applied on a pseudovacuum space with a selected basis and, for each element of this basis, it yields a set of nested Bethe ansatz equations matching up to the ones corresponding to an Am−1 quasiperiodic model, with m equal to minμ∈ZK{rank ζμ}.