We show how one can construct a differential calculus over an algebra where position variables x and momentum variables p have be defined. As the simplest example we consider the one-dimensional q-deformed Heisenberg algebra. This algebra has a subalgebra generated by x and its inverse which we call the coordinate algebra. A physical field is considered to be an element of the completion of this algebra. We can construct a derivative which leaves invariant the coordinate algebra and so takes physical fields into physical fields. A generalized Leibniz rule for this algebra can be found. Based on this derivative differential forms and an exterior differential calculus can be constructed.
Any deformation of a Weyl or Clifford algebra can be realized through some change of generators in the undeformed algebra. Here we briefly describe and motivate our systematic procedure for constructing all such changes of generators for those particular deformations where the original algebra is covariant undersome Lie group and the deformed algebra is covariant under the corresponding quantum group.
Derivation of $κ$-Poincare bicovariant commutation relations between coordinates and 1-forms on $κ$-Minkowski space is given using Dirac operator and Allain Connes formula. The deformed U(1) gauge theory and appearance of an additional spin 0 gauge field is discussed.
The differential calculus on `non-standard' $h$-Minkowski spaces is given. In particular it is shown that, for them, it is possible to introduce coordinates and derivatives which are simultaneously hermitian.
In this work, Z$_3$-graded quantum $(h,j)$-superplane is introduced with a help of proper singular $g$ matrix and a Z$_3$-graded calculus is constructed over this new $h$-superplane. A new Z$_3$-graded $(h,j)$-deformed quantum (super)group is constructed via the obtained calculus.
Given M copies of a q-deformed Weyl or Clifford algebra in the defining representation of a quantum group $G_q$, we determine a prescription to embed them into a unique, inclusive $G_q$-covariant algebra. The different copies are "coupled" to each other and are naturally ordered into a "chain". In the case $G_q=SL_q(N)$ a modified prescription yields an inclusive algebra which is even explicitly $SL_q(M) X SL_q(N)$-covariant, where $SL_q(M)$ is a symmetry relating the different copies. By the introduction of these inclusive algebrae we significantly enlarge the class of $G_q$-covariant deforme…
We propose the ``short'' version of q-deformed differential calculus on the light-cone using twistor representation. The commutation relations between coordinates and momenta are obtained. The quasi-classical limit introduced gives an exact shape of the off-shell shifting.
The differential calculus on the quantum Heisenberg group is conlinebreak structed. The duality between quantum Heisenberg group and algebra is proved.
It is shown that the deformed Heisenberg algebra involving the reflection operator R (R-deformed Heisenberg algebra) has finite-dimensional representations which are equivalent to representations of paragrassmann algebra with a special differentiation operator. Guon-like form of the algebra, related to the generalized statistics, is found. Some applications of revealed representations of the R-deformed Heisenberg algebra are discussed in the context of OSp(2|2) supersymmetry. It is shown that these representations can be employed for realizing (2+1)-dimensional supersymmetry. They give also a …
A universality of deformed Heisenberg algebra involving the reflection operator is revealed. It is shown that in addition to the well-known infinite-dimensional representations related to parabosons, the algebra has also finite-dimensional representations of the parafermionic nature. We demonstrate that finite-dimensional representations are representations of deformed parafermionic algebra with internal Z_2-grading structure. On the other hand, any finite- or infinite-dimensional representation of the algebra supply us with irreducible representation of osp(1|2) superalgebra. We show that the…
The kappa-deformed dual pair of Poincare algebra and Poincare group is formulated in the framework of Heisenberg doubles. The covariant kappa-deformed phase space is described in detail as a subalgebra.The realizations of proposed algebraic scheme are considered.
Two approaches to the tangent space of a noncommutative space whose coordinate algebra is the enveloping algebra of a Lie algebra are known: the Heisenberg double construction and the approach via deformed derivatives, usually defined by procedures involving orderings among noncommutative coordinates or equivalently involving realizations via formal differential operators. In an earlier work, we rephrased the deformed derivative approach introducing certain smash product algebra twisting a semicompleted Weyl algebra. We show here that the Heisenberg double in the Lie algebra case, is isomorphi…
In this paper we investigate a quantum stochastic calculus build of creation, annihilation and number of particles operators which fulfill some deformed commutation relations. Namely, we introduce a deformation of a number of particles operator which has simple commutation relations with well known q-deformed creation and annihilation operators. Since all operators considered in this theory are bounded we do not deal with some difficulties of a non deformed theory of Hudson and Parthasarathy. We define stochastic integrals and estimate them in the operator norm. We prove Ito's formula as well.
This lecture consists of two sections. In section 1 we consider the simplest version of a q-deformed Heisenberg algebra as an example of a noncommutative structure. We first derive a calculus entirely based on the algebra and then formulate laws of physics based on this calculus. Then we realize that an interpretation of these laws is only possible if we study representations of the algebra and adopt the quantum mechanical scheme. It turns out that observables like position or momentum have discrete eigenvalues and thus space gets a lattice-like structure. In section 2 we study a framework for…
We discuss the left-covariant 3-dimensional differential calculus on the quantum sphere $SU_q (2)/U(1) $. The $SU_q (2)$-spinor harmonics are treated as coordinates of the quantum sphere. We consider the gauge theory for the quantum group $SU_q (2)\times U(1) $ on the deformed Euclidean space $E_q (4)$. A $q$-generalization of the harmonic-gauge-field formalism is suggested . This formalism is applied for the harmonic (twistor) interpretation of the quantum-group self-duality equation (QGSDE). We consider the zero-curvature representation and the general construction of QGSDE-solutions in term…
In this paper we extend the eliminant construction of Burchnall and Chaundy for commuting differential operators in the Heisenberg algebra to the q-deformed Heisenberg algebra and show that it again provides annihilating curves for commuting elements, provided q satisfies a natural condition. As a side result we obtain estimates on the dimensions of the eigenspaces of elements of this algebra in its faithful module of Laurent series.
We construct the rings of generalized differential operators on the ${\bf h}$-deformed vector space of ${\bf gl}$-type. In contrast to the $q$-deformed vector space, where the ring of differential operators is unique up to an isomorphism, the general ring of ${\bf h}$-deformed differential operators $\operatorname{Diff}_{{\bf h},σ}(n)$ is labeled by a rational function $σ$ in $n$ variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system and describe some properties of the rings $\operatorname{Diff}_{{\bf h},σ}(n)$.
A decomposition of the level-one $q$-deformed Fock representations of $\uqn$ is given. It is found that the action of $\upqn$ on these Fock spaces is centralized by a Heisenberg algebra, which arises from the center of the affine Hecke algebra $\widehat{H}_N$ in the limit $N \rightarrow \infty$. The $q$-deformed Fock space is shown to be isomorphic as a $\upqn$-Heisenberg-bimodule to the tensor product of a level-one irreducible highest weight representation of $\upqn$ and the Fock representation of the Heisenberg algebra. The isomorphism is used to decompose the $q$-wedging operators, which a…
The $(4+4)$-dimensional $κ$-deformed quantum phase space as well as its $(10+10)$-dimensional covariant extension by the Lorentz sector can be described as Heisenberg doubles: the $(10+10)$-dimensional quantum phase space is the double of $D=4$ $κ$-deformed Poincaré Hopf algebra $\mathbb{H}$ and the standard $(4+4)$-dimensional space is its subalgebra generated by $κ$-Minkowski coordinates $\hat{x}_μ$ and corresponding commuting momenta $\hat{p}_μ$. Every Heisenberg double appears as the total algebra of a Hopf algebroid over a base algebra which is in our case the coordinate sector. We exhibi…
We define a three-parameter deformation of the Weyl-Heisenberg algebra that generalizes the q-oscillator algebra. By a purely algebraical procedure, we set up on this quantum space two differential calculi that are shown to be invariant on the same quantum group, extended to a ten-generator Hopf-star-algebra. We prove that, when the values of the parameters are related, the two differential calculi reduce to one that is invariant under two quantum groups.
Based on the vanishing of the second Hochschild cohomology group of the enveloping algebra of the Heisenberg algebra it is shown that differential algebras coming from quantum groups do not provide a non-trivial deformation of quantum mechanics. For the case of a q-oscillator there exists a deforming map to the classical algebra. It is shown that the differential calculus on quantum planes with involution, i.e. if one works in position-momentum realization, can be mapped on a q-difference calculus on a commutative real space. Although this calculus leads to an interesting discretization it is …
A differential calculus is set up on a deformation of the oscillator algebra. It is uniquely determined by the requirement of invariance under a seven-dimensional quantum group. The quantum space and its associated differential calculus are also shown to be invariant under a nine generator quantum group containing the previous one.
We describe the deformed covariant phase space corresponding to the so-called kappa-deformation of D=4 relativistic symmetries, with quantum ``time'' coordinate and Heisenberg algebra obtained according to the Heisenberg double construction. The associated modified uncertainty relations are analyzed, and in particular it is shown that these relations are consistent with independent estimates of quantum-gravity limitations on the measurability of space-time distances.
We show that R-matricies of all simple quantum groups have the properties which permit to present quantum group twists as transitions to other coordinate frames on quantum spaces. This implies physical equivalence of field theories invariant with respect to q-groups (considered as q-deformed space-time groups of transformations) connected with each other by the twists. Taking into account this freedom we study quantum spaces of the special type: with commuting coordinates but with q-deformed differential calculus and construct $GL_r(N)$ invariant multidimensional Jackson derivatives. We consid…
We give an explicit description of the trace, or Hochschild homology, of the quantum Heisenberg category defined by Licata and Savage. We also show that as an algebra, it is isomorphic to "half" of a central extension of the elliptic Hall algebra of Burban and Schiffmann, specialized at $σ= \barσ^{-1} = q$. A key step in the proof may be of independent interest: we show that the sum (over $n$) of the Hochschild homologies of the positive affine Hecke algebras $\mathrm{AH}_n^+$ is again an algebra, and that this algebra injects into both the elliptic Hall algebra and the trace of the $q$-Heisen…
The ring $\text{Diff}_{\mathbf{h}}(n)$ of $\mathbf{h}$-deformed differential operators appears in the theory of reduction algebras. In this thesis, we construct the rings of generalized differential operators on the $\mathbf{h}$-deformed vector spaces of $\mathfrak{gl}$-type. In contrast to the $q$-deformed vector spaces for which the ring of differential operators is unique up to an isomorphism, the general ring of $\mathbf{h}$-deformed differential operators $\text{Diff}_{\mathbf{h},σ}(n)$ is labeled by a rational function $σ$ in $n$ variables, satisfying an over-determined system of finite-…
We present a description of a new kind of the deformed canonical commutation relations, their representations and generated by them Heisenberg-Weyl algebra. This deformed algebra allows us to derive operations of the Hopf algebra structure: comultiplication, counit and antipode. We discuss properties of a discrete spectrum of the Hamiltonian of the deformed harmonic oscillator corresponding to this oscillator-like system.
We associate a deformation of Heisenberg algebra to the suitably normalized Yang $R$-matrix and we investigate its properties. Moreover, we construct new examples of quantum vertex algebras which possess the same representation theory as the aforementioned deformed Heisenberg algebra.
We re-examine all the contractions related with the ${\cal U}_q(su(2))$ deformed algebra and study the consequences that the contraction process has for their structure. We also show using ${\cal U}_q(su(2))\times{\cal U}(u(1))$ as an example that, as in the undeformed case, the contraction may generate Hopf algebra cohomology. We shall show that most of the different Hopf algebra deformations obtained have a bicrossproduct or a cocycle bicrossproduct structure, for which we shall also give their dual `group' versions. The bicovariant differential calculi on the deformed spaces associated with…
New deformations of the Poincare group $Fun(P(1+1))$ and its dual enveloping algebra $U(p(1+1))$ are obtained as a contraction of the $h$-deformed (Jordanian) quantum group $Fun(SL_h(2))$ and its dual. A nonstandard quantization of the Heisenberg algebra $U(h(1))$ is also investigated.
Some quantum algebras build from deformed oscillator algebras may be described in terms of a particular case of extended umbral calculus. We give here an example of a specific relation between such certain quantum algebras and generalized Clifford algebras also in the context of azimuthal quantization of angular momentum which was interpreted afterwards as the finite dimensional quantum mechanics .
In this paper, as a generalization of Kirillov's orbit theory, we explore the relationship between the dressing orbits and irreducible *-representations of the Hopf C*-algebras (A,Δ) and (\tilde{A}, \tildeΔ) we constructed earlier. We discuss the one-to-one correspondence between them, including their topological aspects. On each dressing orbit (which are symplectic leaves of the underlying Poisson structure), one can define a Moyal-type deformed product at the function level. The deformation is more or less modeled by the irreducible representation corresponding to the orbit. We point out tha…
A non-commutative differential calculus on the $h$-superplane is presented via a contraction of the $q$-superplane. An R-matrix which satisfies both ungraded and graded Yang-Baxter equations is obtained and a new deformation of the $(1+1)$ dimensional classical phase space (the super-Heisenberg algebra) is introduced.
Some properties of the $q$-Fourier-sine transform are studied and $q$-analogues of the Heisenberg uncertainty principle is derived for the $q$-Fourier-cosine transform studied in \cite{FB} and for the $q$-Fourier-sine transform.
