κ-Poincaré–Hopf algebra and Hopf algebroid structure of phase space from twist

We unify κ-Poincaré algebra and κ-Minkowski spacetime by embedding them into quantum phase space. The quantum phase space has Hopf algebroid structure to which we apply the twist in order to get κ-deformed Hopf algebroid structure and κ-deformed Heisenberg algebra. We explicitly construct κ-Poincaré–Hopf algebra and κ-Minkowski spacetime from twist. It is outlined how this construction can be extended to κ-deformed super-algebra including exterior derivative and forms. Our results are relevant for constructing physical theories on noncommutative spacetime by twisting Hopf algebroid phase space structure.     

Differential structure on the κ-Minkowski spacetime from twist

We study four-dimensional κ  -Minkowski spacetime constructed by the twist deformation of U(igl(4,R))$U(\mathit{igl}(4,R))$. We demonstrate that the differential structure of such twist-deformed κ-Minkowski spacetime is closed in four dimensions contrary to the construction of κ-Poincaré bicovariant calculus which needs an extra fifth dimension. Our construction holds in arbitrary dimensional spacetimes.     

Braided algebras and the <Emphasis Type="Italic">κ</Emphasis>-deformed oscillators

Recently there were presented several proposals how to formulate the binary relations describing κ-deformed oscillator algebras. In this paper we shall consider multilinear products of κ-deformed oscillators consistent with the axioms of braided algebras. In general case the braided triple products are quasi-associative and satisfy the hexagon condition depending on the coassociator \({\Phi \in A\otimes A\otimes A}\) . We shall consider only the products of κ-oscillators consistent with co-associative braided algebra, with \({\Phi =1}\) . We shall consider three explicit examples of binary κ-deformed oscillator algebra relations and describe briefly their multilinear coassociative extensions satisfying the postulates of braided algebras. The third example, describing κ-deformed oscillators in group manifold approach to κ-deformed fourmomenta, is a new result.     

Some comments on q-deformed oscillators and q-deformed su(2) algebras

The various relations between q-deformed oscillators algebras and the q-deformed su(2) algebras are discussed. In particular, we exhibit the similarity of the q-deformed su(2) algebra obtained from q- oscillators via Schwinger construction and those obtained from q-Holstein-Primakoff transformation and show how the relation between $su_{\sqrt q } (2)$ and Hong Yan q-oscillator can be regarded as an special case of Inöuë- Wigner contraction. This latter observation and the imposition of positive norm requirement suggest that Hong Yan q-oscillator algebra is different from the usual $su_{\sqrt q } (2)$ algebra, contrary to current belief in the literature.     

Differential structure on κ-Minkowski spacetime realized as module of twisted Weyl algebra

The differential structure on the κ-Minkowski spacetime from Jordanian twist of Weyl algebra is constructed, and it is shown to be closed in 4-dimensions in contrast to the conventional formulation. Based on this differential structure, we have formulated a scalar field theory in this κ-Minkowski spacetime.     

Poisson Algebras of Block-Upper-Triangular Bilinear Forms and Braid Group Action

In this paper we study a quadratic Poisson algebra structure on the space of bilinear forms on ${\mathbb{C}^{N}}$ C N with the property that for any ${n, m \in \mathbb{N}}$ n , m ∈ N such that n m = N, the restriction of the Poisson algebra to the space of bilinear forms with a block-upper-triangular matrix composed from blocks of size ${m \times m}$ m × m is Poisson. We classify all central elements and characterise the Lie algebroid structure compatible with the Poisson algebra. We integrate this algebroid obtaining the corresponding groupoid of morphisms of block-upper-triangular bilinear forms. The groupoid elements automatically preserve the Poisson algebra. We then obtain the braid group action on the Poisson algebra as elementary generators within the groupoid. We discuss the affinisation and quantisation of this Poisson algebra, showing that in the case m = 1 the quantum affine algebra is the twisted q-Yangian for ${\mathfrak{o}_{n}}$ o n and for m = 2 is the twisted q-Yangian for ${(\mathfrak{sp}_{2n})}$ ( sp 2 n ) . We describe the quantum braid group action in these two examples and conjecture the form of this action for any m > 2. Finally, we give an R-matrix interpretation of our results and discuss the relation with Poisson–Lie groups.     

Effective hamiltonian of Bloch electrons in a homogeneous electrical field

In order to construct a band mechanics of Bloch electrons in a homogeneous electrical field E with the interband interaction taken into account, a method of determining the exact single-band Hamiltonian $$H_q = \varepsilon _q^F (\kappa ) + Fi\frac{\partial }{{\partial \kappa }}$$ is proposed, where ε _q ^F (κ) is the renormalized (effective) electron dispersion law for R = 0 and the q-th Bloch band,F= ¦e¦·E. The function ε _q ^F (κ) is expressed in terms of the interband element coordinates as well as in terms of periodic solutions of the system of ordinary differential equations which degenerateinto a common Riccati equation in a two-band approximation. A solution of the system and the form of ε _q ^F (κ), in agreement with the Wanhier result, is found in the quasiclassical approximation.     

On q-Deformed {\mathfrak{gl}_{\ell+1}}-Whittaker Function II

A representation of a specialization of a q-deformed class one lattice ${\mathfrak{gl}_{\ell+1}}$ -Whittaker function in terms of cohomology groups of line bundles on the space ${\mathcal{QM}_d(\mathbb{P}^{\ell})}$ of quasi-maps ${\mathbb{P}^1 \to \mathbb{P}^{\ell}}$ of degree d is proposed. For ? = 1, this provides an interpretation of the non-specialized q-deformed ${\mathfrak{gl}_{2}}$ -Whittaker function in terms of ${\mathcal{QM}_d(\mathbb{P}^1)}$ . In particular the (q-version of the) Mellin-Barnes representation of the ${\mathfrak{gl}_2}$ -Whittaker function is realized as a semi-infinite period map. The explicit form of the period map manifests an important role of q-version of Γ-function as a topological genus in semi-infinite geometry. A relation with the Givental-Lee universal solution (J-function) of q-deformed ${\mathfrak{gl}_2}$ -Toda chain is also discussed.     

On heavy baryon decay form factors

An(n?1)/2+1 parameter, solution of the Yang Baxter equation is presented giving rise to the quantum Group \(GL_{X;q_{ij} } (n)\) . Determinant and inverse are constructed. The group acts covariantly on a quantum vector space of non-commutative coordinates. The associated exterior space can be identified with the differentials exhibiting a multiparameter deformed differential calculus following the construction of Wess and Zumino.     

The Dual Stream Function Representation of an Ideal Steady Fluid Flow and its Local Geometric Structure

Using the methodology of Lie groups and Lie algebras we determine new symmetry and equivalence classes of the stationary three-dimensional Euler equations by introducing potential functions that are based on the so-called dual stream function representation of the steady state velocity field u(x, y, z) = ?λ(x, y, z) × ?μ(x, y, z), which itself can only be defined locally. In particular an infinite dimensional Lie algebra for Beltrami fields is gained. We show that this Lie algebra generates canonical transformations of a Hamiltonian flow for the dual pair of variables \(\lambda \) and \(\mu \) . It enables us to make the classification of a two-dimensional Riemannian manifold \(M^{2}\) wherein \((\lambda ,\mu )\) presents the local coordinates of \(M^{2}\) . Furthermore the local geometry of this manifold is explored in detail. As a result an infinite set of locally conserved currents and charges in the context of a conformal field theory is finally observed.     

Extended gravity theories from dynamical noncommutativity

In this paper we couple noncommutative vielbein gravity to scalar fields. Noncommutativity is encoded in a $\star $ -product between forms, given by an abelian twist (a twist with commuting vector fields). A geometric generalization of the Seiberg–Witten map for abelian twists yields an extended theory of gravity coupled to scalars, where all fields are ordinary (commutative) fields. The vectors defining the twist can be related to the scalar fields and their derivatives, and hence acquire dynamics. Higher derivative corrections to the classical Einstein–Hilbert and Klein–Gordon actions are organized in successive powers of the noncommutativity parameter $\theta ^{AB}$ .     

The Weil Algebra of a Hopf Algebra I: A Noncommutative Framework

We generalize the notion, introduced by Henri Cartan, of an operation of a Lie algebra ${\mathfrak{g}}$ in a graded differential algebra Ω. We define the notion of an operation of a Hopf algebra ${\mathcal{H}}$ in a graded differential algebra Ω which is referred to as a ${\mathcal{H}}$ -operation. We then generalize for such an operation the notion of algebraic connection. Finally we discuss the corresponding noncommutative version of the Weil algebra: The Weil algebra ${W(\mathcal{H})}$ of the Hopf algebra ${\mathcal{H}}$ is the universal initial object of the category of ${\mathcal{H}}$ -operations with connections.     

Extensions of Ordering Sets of States from Effect Algebras onto Their MacNeille Completions

In (Rie?anová and Zajac in Rep. Math. Phys. 70(2):283–290, 2012) it was shown that an effect algebra E with an ordering set $\mathcal{M}$ of states can by embedded into a Hilbert space effect algebra $\mathcal{E}(l_{2}(\mathcal{M}))$ . We consider the problem when its effect algebraic MacNeille completion $\hat{E}$ can be also embedded into the same Hilbert space effect algebra $\mathcal {E}(l_{2}(\mathcal{M}))$ . That is when the ordering set $\mathcal{M}$ of states on E can be extended to an ordering set of states on $\hat{E}$ . We give an answer for all Archimedean MV-effect algebras and Archimedean atomic lattice effect algebras.     

Coset Constructions of Logarithmic (1, p) Models

One of the best understood families of logarithmic onformal field theories consists of the (1, p) models (p =  2, 3, . . .) of central charge c _{1, p} =1 ? 6(p ? 1)²/p. This family includes the theories corresponding to the singlet algebras ${\mathcal{M}(p)}$ and the triplet algebras ${\mathcal{W}(p)}$ , as well as the ubiquitous symplectic fermions theory. In this work, these algebras are realised through a coset construction. The ${W^{(2)}_n}$ algebra of level k was introduced by Feigin and Semikhatov as a (conjectured) quantum hamiltonian reduction of ${\widehat{\mathfrak{sl}}(n)_k}$ , generalising the Bershadsky–Polyakov algebra ${W^{(2)}_3}$ . Inspired by work of Adamovi? for p = 3, vertex algebras ${\mathcal{B}_p}$ are constructed as subalgebras of the kernel of certain screening charges acting on a rank 2 lattice vertex algebra of indefinite signature. It is shown that for p≤5, the algebra ${\mathcal{B}_p}$ is a quotient of ${W^{(2)}_{p-1}}$ at level ?(p ? 1)²/p and that the known part of the operator product algebra of the latter algebra is consistent with this holding for p> 5 as well. The triplet algebra ${\mathcal{W}(p)}$ is then realised as a coset inside the full kernel of the screening operator, while the singlet algebra ${\mathcal{M}(p)}$ is similarly realised inside ${\mathcal{B}_p}$ . As an application, and to illustrate these results, the coset character decompositions are explicitly worked out for p =  2 and 3.     

Lie Triple Derivations of CSL Algebras

Let be a commutative subspace lattice generated by finite many commuting independent nests on a complex separable Hilbert space with , and the associated CSL algebra. It is proved that every Lie triple derivation from into any σ-weakly closed algebra containing is of the form X→XT?TX+h(X)I, where and h is a linear mapping from into ? such that h([[A,B],C])=0 for all .     

Group Actions on DG-Manifolds and Exact Courant Algebroids

Let G be a Lie group acting by diffeomorphisms on a manifold M and consider the image of T[1]G and T[1]M, of G and M respectively, in the category of differential graded manifolds. We show that the obstruction to lift the action of T[1]G on T[1]M to an action on a ${\mathbb{R}[n]}$ -bundle over T[1]M is measured by the G equivariant cohomology of M. We explicitly calculate the differential graded Lie algebra of the symmetries of the ${\mathbb{R}[n]}$ -bundle over T[1]M and we use this differential graded Lie algebra to understand which actions are hamiltonian. We show how split Exact Courant algebroids could be obtained as the derived Leibniz algebra of the symmetries of ${\mathbb{R}[2]}$ -bundles over T[1]M, and we use this construction to propose that the infinitesimal symmetries of a split Exact Courant algebroid should be encoded in the differential graded Lie algebra of symmetries of a ${\mathbb{R}[2]}$ -bundle over T[1]M. With this setup at hand, we propose a definition for an action of a Lie group on an Exact Courant algebroid and we propose conditions for the action to be hamiltonian.     

Differential calculus on the quantum superspace and deformation of phase space

We investigate non-commutative differential calculus on the supersymmetric version of quantum space in which quantum supergroups are realized. Multiparametric quantum deformation of the general linear super-group,GL _q(m|n), is studied and the explicit form for the \(\hat R - matrix\) is presented. We apply these results to the quantum phase-space construction ofOSp _q(2n|2m) and calculate their \(\hat R - matrices\) .     

Electrodynamics of a generalized charged particle in doubly special relativity framework

In the present paper, dynamics of generalized charged particles are studied in the presence of external electromagnetic interactions. This particular extension of the free relativistic particle model lives in Non-Commutative κ$\kappa$-Minkowski space–time, compatible with Doubly Special Relativity, that is motivated to describe Quantum Gravity effects. Furthermore we have also considered the electromagnetic field to be dynamical and have derived the modified forms of Lienard–Wiechert like potentials for these extended charged particle models. In all the above cases we exploit the new and extended form of κ$\kappa$-Minkowski algebra where electromagnetic effects are incorporated in the lowest order, in the Dirac framework of Hamiltonian constraint analysis.     

A candidate for a four-quark vector-meson?

Previous claims for the existence of vector mesons with masses in the vicinity of 1100 MeV and 1300 MeV are reconsidered in the light of recent developments in the analysis and interpretation ofe ⁺ e ^? annihilation and diffractive photoproduction data. It is shown that these states are compatible with present evidence and can be incorporated in an extended mixing scheme of conventional \(q\bar q\) with unconventional \(qq\bar q \bar q\) states.