Quantum group structure of the q-deformed $W$ algebra $\WW_q$

Towards Deformed Chiral Algebras

We describe a new algebraic structure of "deformed chiral algebra" motivated by the study of the deformed W-algebras. We use it to gain some insights into the deformed Virasoro algebra.

Operator Representations of a q-Deformed Heisenberg Algebra

A class of well-behaved *-representations of a q-deformed Heisenberg algebra is studied and classified.

Some Remarks on the Representation of the Generalized Deformed Oscillator Algebra

The classification of the representations of the generalized deformed oscillator algebra is given together with several comments about possibility of introducing a coproduct structure in some type of deformed oscillator algebra.

Deformed boson algebras and the quantum double construction

The quantum double construction of a $q$-deformed boson algebra possessing a Hopf algebra structure is carried out explicitly. The $R$-matrix thus obtained is compared with the existing literature.

Algebra of screening operators for the deformed $W_n$ algebra

We construct a family of intertwining operators (screening operators) between various Fock space modules over the deformed $W_n$ algebra. They are given as integrals involving a product of screening currents and elliptic theta functions. We derive a set of quadratic relations among the screening operators, and use them to construct a Felder-type complex in the case of the deformed $W_3$ algebra.

The Polar Decomposition of q-Deformed Boson Algebra

In this paper we used the finite Fourier transformation to obtain the polar decomposition of the q-deformed boson algebra with $q$ a root of unity.

Is it possible to extend the deformed Weyl algebra $W_q(n)$ to a Hopf algebra?

We argue that the algebra $W_q(n)$, generated by $n$ pairs of deformed $q$-bosons, does not allow a Hopfalgebra structure. To this end we show that it is impossible to define a comultiplication even for the usual, nondeformed case. We indicate how the comultiplication on $U_q[osp(1/2n)]$ can be used in order to construct representations of deformed (not necessarily Hopf) algebras in tensor products of Fock spaces.

Spin in the q-Deformed Poincare Algebra

We investigate spin as algebraic structure within the q-deformed Poincare algebra, proceeding in the same manner as in the undeformed case. The q-Pauli-Lubanski vector, the q-spin Casimir, and the q-little algebras for the massless and the massive case are constructed explicitly.

A coset-type construction for the deformed Virasoro algebra

An analog of the minimal unitary series representations for the deformed Virasoro algebra is constructed using vertex operators of the quantum affine algebra $U_q(\hat{sl}_2)$. A similar construction is proposed for the elliptic algebra $A_{q,p}(\hat{sl}_2)$.

On the representation theory of non-semisimple (graded) deformed Fomin-Kirillov algebras

This work is motivated to study the representation theory of the non-semisimple deformed Fomin-Kirillov algebras $\mathcal{D}_4(α_1, α_2)$. In particular, we consider Gabriel's theorem applications in regard of constructing algebraic presentations.

Generalized Deformed su(2) Algebras, Deformed Parafermionic Oscillators and Finite W Algebras

Several physical systems (two identical particles in two dimensions, isotropic oscillator and Kepler system in a 2-dim curved space) and mathematical structures (quadratic algebra QH(3), finite W algebra $\bar {\rm W}_0$) are shown to posses the structure of a generalized deformed su(2) algebra, the representation theory of which is known. Furthermore, the generalized deformed parafermionic oscillator is identified with the algebra of several physical systems (isotropic oscillator and Kepler system in 2-dim curved space, Fokas--Lagerstrom, Smorodinsky--Winternitz and Holt potentials) and mathe…

Note on the Algebra of Screening Currents for the Quantum Deformed W-Algebra

With slight modifications in the zero modes contributions, the positive and negative screening currents for the quantum deformed W-algebra W_{q,p}(g) can be put together to form a single algebra which can be regarded as an elliptic deformation of the universal enveloping algebra of \hat{g}, where g is any classical simply-laced Lie algebra.

Heisenberg Double Description of kappa-Poincare Algebra and kappa-deformed Phase Space

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.

Geometry of Deformed Boson Algebras

Phase-space realisations of an infinite parameter family of quantum deformations of the boson algebra in which the $q$-- and the $qp$--deformed algebras arise as special cases are studied. Quantum and classical models for the corresponding deformed oscillators are provided. The deformation parameters are identified with coefficients of non-linear terms in the normal forms expansion of a family of classical Hamiltonian systems. These quantum deformations are trivial in the sense that they correspond to non-unitary transformations of the Weyl algebra. They are non-trivial in the sense that the d…

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 us…

Deformed Double Current Algebras via Deligne Categories

In this paper we give an alternative construction of a certain class of Deformed Double Current Algebras. These algebras are deformations of $U({\rm End}(\Bbbk^r)[x,y])$ and they were initially defined and studied by N.Guay in his papers. Here we construct them as algebras of endomorphisms in Deligne category. We do this by taking an ultraproduct of spherical subalgebras of the extended Cherednik algebras of finite rank.

Embedding q-deformed Heisenberg Algebras into Undeformed Ones

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.

Tail algebras for monotone and $q$-deformed exchangeable stochastic processes

We compute the tail algebras of exchangeable monotone stochastic processes. This allows us to prove the analogue of de Finetti's theorem for this type of processes. In addition, since the vacuum state on the $q$-deformed $C^*$-algebra is the only exchangeable state when $|q|<1$, we draw our attention to its tail algebra, which turns out to obey a zero-one law.

On deformed W-algebras and quantum affine algebras

We discuss some aspects of the deformed W-algebras W_{q,t}[g]. In particular, we derive an explicit formula for the Kac determinant, and discuss the center when t^2 is a primitive k-th root of unity. The relation of the structure of W_{q,t}[g] to the representation ring of the quantum affine algebra U_q(\hat g), as discovered recently by Frenkel and Reshetikhin, is further elucidated in some examples.

Deformed and extended Galilei group Hopf algebras

The $\hatκ$-deformed extended Galilei Hopf group algebra, ${\rm Fun}_{\hatκ}(\tilde G_{(m)})$, is introduced. It provides an example of a cocycle bicrossproduct structure, and is shown to be the contraction limit of a pseudoextension of the $κ$-Poincaré group algebra. The possibility of obtaining another deformed extended Galilei groups is discussed, including one obtained from a non-standard Poincaré deformation.

Beyond CFT : Deformed Virasoro and Elliptic Algebras

In this lecture we discuss `beyond CFT' from symmetry point of view. After reviewing the Virasoro algebra, we introduce deformed Virasoro algebras and elliptic algebras. These algebras appear in solvable lattice models and we study them by free field approach.

A $p$-DG deformed Webster algebra of type $A_1$

We define a $p$-DG structure on a deformation of Webster algebra of type $A_1$ and its splitter bimodules.

Unitary operator bases and q-deformed algebras

Starting from the Schwinger unitary operator bases formalism constructed out of a finite dimensional state space, the well-known q-deformed commutation relation is shown to emerge in a natural way, when the deformation parameter is a root of unity.

Algebraic curves for commuting elements in the q-deformed Heisenberg algebra

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.

Hopf Structure and Green Ansatz of Deformed Parastatistics Algebras

Deformed parabose and parafermi algebras are revised and endowed with Hopf structure in a natural way. The noncocommutative coproduct allows for construction of parastatistics Fock-like representations, built out of the simplest deformed bose and fermi representations. The construction gives rise to quadratic algebras of deformed anomalous commutation relations which define the generalized Green ansatz.

q-Deformed Minkowski Space based on a q-Lorentz Algebra

The Hilbert space representations of a non-commutative q-deformed Minkowski space, its momenta and its Lorentz boosts are constructed. The spectrum of the diagonalizable space elements shows a lattice-like structure with accumulation points on the light-cone.

与辫子相关的形变量子顶点代数模

Deformed quantum vertex algebra modules associated with braidings

该研究引入形变量子顶点代数模的概念，构造两类对应经典型有理R矩阵的辫子映射，探究其性质并展示相关形变模在杨子代数与反射代数中的应用。

Irreducible representations of deformed oscillator algebra and q-special functions

Different generators of a deformed oscillator algebra give rise to one-parameter families of $q$-exponential functions and $q$-Hermite polynomials related by generating functions. Connections of the Stieltjes and Hamburger classical moment problems with the corresponding resolution of unity for the $q$-coherent states and with 'coordinate' operators - Jacobi matrices, are also pointed out.

Crystal graphs of higher level q-deformed Fock spaces, Lusztig a-values and Ariki-Koike algebras

We show that the different labelings of the crystal graph for irreducible highest weight $\mathcal{U}\_q (\hat{\mathfrak{sl}}\_e)$-modules lead to different labelings of the simple modules for non semisimple Ariki-Koike algebras by using Lusztig $a$-values.

Graded contractions and bicrossproduct structure of deformed inhomogeneous algebras

A family of deformed Hopf algebras corresponding to the classical maximal isometry algebras of zero-curvature N-dimensional spaces (the inhomogeneous algebras iso(p,q), p+q=N, as well as some of their contractions) are shown to have a bicrossproduct structure. This is done for both the algebra and, in a low-dimensional example, for the (dual) group aspects of the deformation.

Deformed Harmonic Oscillator Algebras defined by their Bargmann representations

Deformed Harmonic Oscillator Algebras are generated by four operators, two mutually adjoint $a$ and $a^\dagger$, and two self-adjoint $N$ and the unity $1$ such as: $[a,N] = a, [a^\dagger, N]= -a^\dagger, a^\dagger a = ψ(N)$ and $aa^\dagger =ψ(N+1)$. The Bargmann Hilbert space is defined as a space of functions, holomorphic in a ring of the complex plane, equipped with a scalar product involving a true integral. In a Bargmann representation, the operators of a Deformed Harmonic Oscillator Algebra act on a Bargmann Hilbert space and the creation (or the annihilation operator) is the multiplicat…

On a possible algebra morphism of U$_q$[OSP(1/2N)] onto the deformed oscillator algebra W$_q$(N)

We formulate a conjecture, stating that the algebra of $n$ pairs of deformed Bose creation and annihilation operators is a factor-algebra of $U_q[osp(1/2n)]$, considered as a Hopf algebra, and prove it for $n=2$ case. To this end we show that for any value of $q$ $U_q[osp(1/4)]$ can be viewed as a superalgebra, freely generated by two pairs $B_1^\pm$, $B_2^\pm$ of deformed para-Bose operators. We write down all Hopf algebra relations, an analogue of the Cartan-Weyl basis, the "commutation" relations between the generators and a basis in $U_q[osp(1/2n)]$ entirely in terms of $B_1^\pm$, $B_2^\pm…

变形W代数的迹公式

Trace Formulas for Deformed W-Algebras

本文研究了变形W代数的迹公式，揭示了其与q变形W代数模双的联系，并提出了一种通过迹公式统一变形W代数的图景。

Braided Chains of q-Deformed Heisenberg Algebrae

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…

Matrix elements of vertex operators of deformed W-algebra and Harish Chandra Solutions to Macdonald's difference equations

In this paper we prove that certain matrix elements of vertex operators of deformed W-algebra satisfy Macdonald difference equations and form n! -dimensional space of solutions. These solutions are the analogues of Harish Chandra solutions with prescribed asymptotic behavior. We obtain formulas for analytic continuation as a consequence of braiding properties of vertex operators of deformed W-algebra.

Extensions of deformed $W$-algebras via $qq$-characters

We use combinatorics of $qq$-characters to study extensions of deformed $W$-algebras. We describe additional currents and part of the relations in the cases of $\mathfrak{gl}(n|m)$ and $\mathfrak{osp}(2|2n)$.

Deformed preprojective algebras and symplectic reflection algebras for wreath products

We determine the PBW deformations of the wreath product of a symmetric group with a deformed preprojective algebra of an affine Dynkin quiver. In particular, we show that there is precisely one parameter which does not come from deformation of the preprojective algebra. We prove that the PBW deformation is Morita equivalent to a corresponding symplectic reflection algebra for wreath product.

A q-Lorentz Algebra From q-Deformed Harmonic Oscillators

A mapping between the operators of the bosonic oscillator and the Lorentz rotation and boost generators is presented. The analog of this map in the $q$-deformed regime is then applied to $q$-deformed bosonic oscillators to generate a $q$-deformed Lorentz algebra, via an inverse of the standard chiral decomposition. A fundamental representation, and the co-algebra structure, are given, and the generators are reformulated into $q$-deformed rotations and boosts. Finally, a relation between the $q$-boson operators and a basis of $q$-deformed Minkowski coordinates is noted.

Quadratic Relations of the Deformed $W$-Algebra for the Twisted Affine Lie Algebra of Type $A_{2N}^{(2)}$

We revisit the free field construction of the deformed $W$-algebra by Frenkel and Reshetikhin [Comm. Math. Phys. 197 (1998), 1-32], where the basic $W$-current has been identified. Herein, we establish a free field construction of higher $W$-currents of the deformed $W$-algebra associated with the twisted affine Lie algebra $A_{2N}^{(2)}$. We obtain a closed set of quadratic relations and duality, which allows us to define deformed $W$-algebra ${\mathcal W}_{x,r}\big(A_{2N}^{(2)}\big)$ using generators and relations.

$q$-形变 Witt 代数与 $q$-形变 $W(2,2)$ 代数上的平均算子

Averaging operators on $q$-deformed Witt and $q$-deformed $W(2,2)$ algebras

本文研究了 Hom-Lie 代数上的平均算子，分类了 $q$-形变 Witt 和 $q$-形变 $W(2,2)$ Hom-代数上的齐次平均算子，并推导出诱导的 Hom-Leibniz 代数结构及其乘法性条件。

Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra

We study an analog of the AGT relation in five dimensions. We conjecture that the instanton partition function of 5D N=1 pure SU(2) gauge theory coincides with the inner product of the Gaiotto-like state in the deformed Virasoro algebra. In four dimensional case, a relation between the Gaiotto construction and the theory of Braverman and Etingof is also discussed.

Invariance quantum groups of the deformed oscillator algebra

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.

Deformed Virasoro algebras from elliptic quantum algebras

We revisit the construction of deformed Virasoro algebras from elliptic quantum algebras of vertex type, generalizing the bilinear trace procedure proposed in the 90's. It allows us to make contact with the vertex operator techniques that were introduced separately at the same period. As a by-product, the method pinpoints two critical values of the central charge for which the center of the algebra is extended, as well as (in the $gl(2)$ case) a Liouville formula.

Non Abelian Sugawara Construction and the q-deformed N=2 Superconformal Algebra

The construction of a q-deformed N=2 superconformal algebra is proposed in terms of level 1 currents of ${\cal{U}}_{q} ({\widehat{su}}(2))$ quantum affine Lie algebra and a single real Fermi field. In particular, it suggests the expression for the q-deformed Energy-Momentum tensor in the Sugawara form. Its constituents generate two isomorphic quadratic algebraic structures. The generalization to ${\cal{U}}_{q} ({\widehat{su}}(N+1))$ is also proposed.

Generalized Deformed Oscillators and Algebras

The generalized deformed oscillator schemes introduced as unified frameworks of various deformed oscillators are proved to be equivalent, their unified representation leading to a correspondence between the deformed oscillator and the N=2 supersymmetric quantum mechanics (SUSY-QM) scheme. In addition, several physical systems (two identical particles in two dimensions, isotropic oscillator and Kepler system in a 2-dim curved space) and mathematical structures (quadratic algebra QH(3), finite W algebra $\bar {\rm W}_0$) are shown to possess the structure of a generalized deformed su(2) algebra,…

The inhomogeneous invariance quantum group of q-deformed boson algebra with continuous parameters

We present a q-deformed boson algebra using continuous momentum parameters and investigate its inhomogeneous invariance quantum group.

Norm of the Whittaker vector of the deformed Virasoro algebra

We give a proof of the recursive formula on the norm of Whittaker vector of the deformed Virasoro algebra, which is an analog of the one for the Virasoro Lie algebra proposed by Al. Zamolodchikov. Our formula gives a proof of the pure SU(2) 5d AGT relation proposed by Awata and Yamada. We also give a summary of the structures of the deformed Virasoro algebra and the fundamental properties of the Verma module.

Twisted and Non-Twisted Deformed Virasoro Algebra via Vertex Operators of $U_q(\widehat{\mathfrak{sl}}_2)$

The work is devoted to a probably new connection between deformed Virasoro algebra and quantum $\widehat{\mathfrak{sl}}_2$. We give an explicit realization of Virasoro current via vertex operators of level 1 integrable representation of $\widehat{\mathfrak{sl}}_2$. The same is done for a twisted version of deformed Virasoro algebra.

Deformed commutators on quantum group module-algebras

We construct quantum commutators on module-algebras of quasi-triangular Hopf algebras. These are quantum-group covariant, and have generalized antisymmetry and Leibniz properties. If the Hopf algebra is triangular they additionally satisfy a generalized Jacobi identity, turning the module-algebra into a quantum-Lie algebra.

Deformed Heisenberg algebra with reflection

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…

变形 $W$-代数的二次关系

Quadratic relations of the deformed $W$-algebra

本文通过自由场构造获得变形 $W$-代数的 $W$-流的封闭二次关系，从而用生成元和关系定义该代数，并综述了两类变形 $W$-代数（${\cal W}_{x,r}(A_{2N}^{(2)})$ 和 $q$-$Y_{L_1,L_2,L_3}$）及其推广。

The q-Deformed Oscillator Representations and Their Coherent States of the su(1,1) Algebra

We present various oscillator representations of the q-deformed su(1,1) algebra such as the Holstein-Primakoff, the Dyson, the Fock-Bargmann, the anyonic, and the parabose oscillator representations and discuss their coherent states with the resolution of unity.

R-deformed Heisenberg algebra

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 …

无穷小模群：$q$-形变的 $\mathfrak{sl}_2$ 与 Witt 代数

Infinitesimal Modular Group: $q$-Deformed $\mathfrak{sl}_2$ and Witt Algebra

该研究构造了三维Heisenberg代数、$\mathfrak{sl}_2$与Witt代数的新型$q$-形变，借助微分算子实现，关联模群与已有的$q$-形变有理数，导出双曲平面上的$q$-形变Möbius变换。