scholarly journals DIFFERENTIAL STRUCTURE ON κ-MINKOWSKI SPACE, AND κ-POINCARÉ ALGEBRA

2011 ◽  
Vol 26 (20) ◽  
pp. 3385-3402 ◽  
Author(s):  
STJEPAN MELJANAC ◽  
SAŠA KREŠIĆ-JURIĆ
Keyword(s):  
Power Series ◽  
Hopf Algebra ◽  
Minkowski Space ◽  
Differential Calculus ◽  
Weyl Algebra ◽  
Algebra Structure ◽  
Exterior Derivative ◽  
Lorentz Algebra ◽  
Poincaré Algebra ◽  
Formal Power

We construct realizations of the generators of the κ-Minkowski space and κ-Poincaré algebra as formal power series in the h-adic extension of the Weyl algebra. The Hopf algebra structure of the κ-Poincaré algebra related to different realizations is given. We construct realizations of the exterior derivative and one-forms, and define a differential calculus on κ-Minkowski space which is compatible with the action of the Lorentz algebra. In contrast to the conventional bicovariant calculus, the space of one-forms has the same dimension as the κ-Minkowski space.

scholarly journals DIFFERENTIAL ALGEBRAS ON κ-MINKOWSKI SPACE AND ACTION OF THE LORENTZ ALGEBRA

2012 ◽  
Vol 27 (10) ◽  
pp. 1250057 ◽  
Author(s):  
STJEPAN MELJANAC ◽  
SAŠA KREŠIĆ-JURIĆ ◽  
RINA ŠTRAJN
Keyword(s):  
Power Series ◽  
Minkowski Space ◽  
Formal Power Series ◽  
Standard Approach ◽  
Lorentz Algebra ◽  
Differential Algebras ◽  
Formal Power

We propose two families of differential algebras of classical dimension on κ-Minkowski space. The algebras are constructed using realizations of the generators as formal power series in a Weyl superalgebra. We also propose a novel realization of the Lorentz algebra [Formula: see text] in terms of Grassmann-type variables. Using this realization we construct an action of [Formula: see text] on the two families of algebras. Restriction of the action to κ-Minkowski space is covariant. In contrast to the standard approach the action is not Lorentz covariant except on constant one-forms, but it does not require an extra cotangent direction.

scholarly journals KAPPA-MINKOWSKI SPACETIME, KAPPA-POINCARÉ HOPF ALGEBRA AND REALIZATIONS

2012 ◽  
Vol 09 (06) ◽  
pp. 1261009 ◽  
Author(s):  
DOMAGOJ KOVAČEVIĆ ◽  
STJEPAN MELJANAC
Keyword(s):  
Hopf Algebra ◽  
Star Product ◽  
Minkowski Spacetime ◽  
Inner Product ◽  
Translation Invariance ◽  
Dual Algebra ◽  
Lorentz Algebra ◽  
Heisenberg Algebras ◽  
The Matrix ◽  
Poincaré Algebra

The κ-Minkowski spacetime and Lorentz algebra are unified in unique Lie algebra. Introducing commutative momenta, a family of κ-deformed Heisenberg algebras and κ-deformed Poincaré algebras are defined. They are determined by the matrix depending on momenta. Realizations and star product are defined and analyzed in general. The relation among the coproduct of momenta, realization and the star product is pointed out. Hopf algebra of the Poincaré algebra, related to the covariant realization, is presented in unified covariant form. Left–right dual realizations and dual algebra are introduced and considered. The generalized involution and the star inner product are defined and analyzed. Partial integration and deformed trace property are obtained in general. The translation invariance of the star product is pointed out.

Bicovariant differential calculus on the quantum superspace ℝq(1|2)

2016 ◽  
Vol 15 (09) ◽  
pp. 1650172 ◽  
Author(s):  
Salih Celik
Keyword(s):  
Hopf Algebra ◽  
Differential Calculus ◽  
Vector Fields ◽  
Lie Superalgebra ◽  
Function Algebra ◽  
Algebra Structure ◽  
Hopf Algebra Structure ◽  
Bicovariant Differential Calculus ◽  
Quantum Supergroup ◽  
Dual Hopf Algebra

Super-Hopf algebra structure on the function algebra on the extended quantum superspace has been defined. It is given a bicovariant differential calculus on the superspace. The corresponding (quantum) Lie superalgebra of vector fields and its Hopf algebra structure are obtained. The dual Hopf algebra is explicitly constructed. A new quantum supergroup that is the symmetry group of the differential calculus is found.

scholarly journals A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra

2007 ◽  
Vol 309 (1) ◽  
pp. 318-359 ◽  
Author(s):  
Nikolai Durov ◽  
Stjepan Meljanac ◽  
Andjelo Samsarov ◽  
Zoran Škoda
Keyword(s):  
Power Series ◽  
Lie Algebra ◽  
Formal Power Series ◽  
Weyl Algebra ◽  
Universal Formula ◽  
Formal Power

A Two-parameter bicovariant differential calculus on the (1 + 2)-dimensional q-superspace

2016 ◽  
Vol 13 (03) ◽  
pp. 1650029
Author(s):  
Ergün Yasar
Keyword(s):  
Hopf Algebra ◽  
Differential Calculus ◽  
Lie Superalgebra ◽  
Point Of View ◽  
Algebra Structure ◽  
Hopf Algebra Structure ◽  
Bicovariant Differential Calculus ◽  
Two Parameter

We construct a two-parameter bicovariant differential calculus on [Formula: see text] with the help of the covariance point of view using the Hopf algebra structure of [Formula: see text]. To achieve this, we first use the consistency of calculus and the approach of [Formula: see text]-matrix which satisfies both ungraded and graded Yang–Baxter equations. In particular, based on this differential calculus, we investigate Cartan–Maurer forms for this [Formula: see text]-superspace. Finally, we obtain the quantum Lie superalgebra corresponding the Cartan–Maurer forms.

scholarly journals Hopf Algebra Structure of Generalized Quasi-Symmetric Functions in Partially Commutative Variables

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Adam Doliwa
Keyword(s):  
Power Series ◽  
Hopf Algebra ◽  
Hopf Algebras ◽  
Symmetric Functions ◽  
Algebra Structure ◽  
Natural Basis ◽  
Dual Algebra ◽  
Power Sum ◽  
Hopf Algebra Structure

We introduce a coloured generalization  $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions  described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the set of sentences over alphabet $A$ (the set of colours). We present also its graded dual algebra $\mathrm{QSym}_A$ of coloured quasi-symmetric functions together with its realization in terms of power series in partially commutative variables.  We provide formulas expressing multiplication, comultiplication and the antipode for these Hopf algebras in various bases — the corresponding generalizations of the complete homogeneous, elementary, ribbon Schur and power sum bases of $\mathrm{NSym}$, and the monomial and fundamental bases of $\mathrm{QSym}$. We study also certain distinguished series of trees in the setting of restricted duals to Hopf algebras.

Algebraic Theory of Causal Double Products

2010 ◽  
Vol 60 (5) ◽  
Author(s):  
R. Hudson
Keyword(s):  
Power Series ◽  
Hopf Algebra ◽  
Tensor Product ◽  
Formal Power Series ◽  
Fock Space ◽  
Algebraic Theory ◽  
Stochastic Integrals ◽  
Double Product ◽  
Triangular Elements ◽  
Formal Power

AbstractCorresponding to each “rectangular” double product in the form of a formal power series R[h] with coefficients in the tensor product 풯(ℒ)⊙ 풯 (ℒ) with itself of the Itô Hopf algebra, we construct “triangular” elements T[h] of 풯(ℒ) satisfying ΔT[h] = T[h](1) R[h]T{h](2). In Fock space representations of 풯(ℒ) by iterated quantum stochastic integrals when ℒ is the algebra of Itô differentials of the calculus, these correspond to “causal” double product integrals in a single Fock space.

scholarly journals κ-DEFORMED POINCARÉ ALGEBRAS AND QUANTUM CLIFFORD–HOPF ALGEBRAS

2010 ◽  
Vol 07 (05) ◽  
pp. 821-836 ◽  
Author(s):  
ROLDÃO DA ROCHA ◽  
ALEX E. BERNARDINI ◽  
JAYME VAZ
Keyword(s):  
Hopf Algebra ◽  
Clifford Algebra ◽  
Hopf Algebras ◽  
Conformal Group ◽  
Algebra Structure ◽  
De Sitter ◽  
Minkowski Space Time ◽  
Quantum Clifford Algebra ◽  
Poincaré Algebra ◽  
Time Quantum

The Minkowski space–time quantum Clifford algebra structure associated with the conformal group and the Clifford–Hopf alternative κ-deformed quantum Poincaré algebra is investigated in the Atiyah–Bott–Shapiro mod 8 theorem context. The resulting algebra is equivalent to the deformed anti-de Sitter algebra [Formula: see text], when the associated Clifford–Hopf algebra is taken into account, together with the associated quantum Clifford algebra and a (not braided) deformation of the periodicity Atiyah–Bott–Shapiro theorem.

HOPF ALGEBRA STRUCTURE OF (2+1)-DIMENSIONAL QUANTUM SUPERSPACE, ITS DUAL AND ITS DIFFERENTIAL CALCULUS

2010 ◽  
Vol 25 (26) ◽  
pp. 2241-2253 ◽  
Author(s):  
MUTTALIP OZAVSAR
Keyword(s):  
Hopf Algebra ◽  
Differential Calculus ◽  
Lie Superalgebra ◽  
Algebra Structure ◽  
Hopf Superalgebra ◽  
Hopf Algebra Structure ◽  
Quantum Supergroup ◽  
Noncommuting Coordinates

We consider a (2+1)-dimensional quantum superspace which has noncommuting coordinates in Manin sense and it was shown that this space has a Hopf algebra structure, i.e. the quantum supergroup, when it is extended by the inverse of the bosonic variable. Differential structures on this space were given by constructing the differential calculus in the sense of Woronowicz. Thus, we deduce that the corresponding quantum Lie superalgebra which as a commutation superalgebra appears classical, and as Hopf structure is non-cocommutative q-deformed. Finally, dual Hopf superalgebra was given.

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