The aim of this paper is to give a complete classification of irreducible finite dimensional representations of the nonstandard q-deformation U'_q(so(n)) (which does not coincide with the Drinfeld-Jimbo quantum algebra U_q(so(n)) of the universal enveloping algebra U(so(n,C)) of the Lie algebra so(n,C) when q is not a root of unity. These representations are exhausted by irreducible representations of the classical type and of the nonclassical type. Theorem on complete reducibility of finite dimensional representations of U'_q(so(n)) is proved.
A classification of finite dimensional irreducible representations of the nonstandard $q$-deformation $U'_q(so_n)$ of the universal enveloping algebra $U(so(n, C))$ of the Lie algebra $so(n, C)$ (which does not coincides with the Drinfeld--Jimbo quantized universal enveloping algebra $U_q(so_n)$) is given for the case when $q$ is not a root of unity. It is shown that such representations are exhausted by representations of the classical and nonclassical types. Examples of the algebras $U'_q(so_3)$ and $U'_q(so_4)$ are considered in detail. The notions of weights, highest weights, highest weigh…
The nonstandard q-deformation $U'_q({\rm so}_n)$ of the universal enveloping algebra $U({\rm so}_n)$ has irreducible finite dimensional representations which are a q-deformation of the well-known irreducible finite dimensional representations of $U({\rm so}_n)$. But $U'_q({\rm so}_n)$ also has irreducible finite dimensional representations which have no classical analogue. The aim of this paper is to give these representations which are called nonclassical type representations. They are given by explicit formulas for operators of the representations corresponding to the generators of $U'_q({\r…
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.
We introduce the notion of a crystal base of a finite dimensional q-deformed Kac module over the quantum superalgebra $U_q(\gl(m|n))$, and prove its existence and uniqueness. In particular, we obtain the crystal base of a finite dimensional irreducible $U_q(\gl(m|n))$-module with typical highest weight. We also show that the crystal base of a q-deformed Kac module is compatible with that of its irreducible quotient $V(λ)$ given by Benkart, Kang and Kashiwara when $V(λ)$ is an irreducible polynomial representation.
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.
An algebra homomorphism $ψ$ from the nonstandard q-deformed (cyclically symmetric) algebra $U_q(so_3)$ to the extension ${\hat U}_q(sl_2)$ of the Hopf algebra $U_q(sl_2)$ is constructed. Not all irreducible representations of $U_q(sl_2)$ can be extended to representations of ${\hat U}_q(sl_2)$. Composing the homomorphism $ψ$ with irreducible representations of ${\hat U}_q(sl_2)$ we obtain representations of $U_q(so_3)$. Not all of these representations of $U_q(so_3)$ are irreducible. Reducible representations of $U_q(so_3)$ are decomposed into irreducible components. In this way we obtain all …
We study the nonstandard $q$-deformation $U'_q({\rm so}_4)$ of the universal enveloping algebra $U({\rm so}_4)$ obtained by deforming the defining relations for skew-symmetric generators of $U({\rm so}_4)$. This algebra is used in quantum gravity and algebraic topology. We construct a homomorphism $ϕ$ of $U'_q({\rm so}_4)$ to the certain nontrivial extension of the Drinfeld--Jimbo quantum algebra $U_q({\rm sl}_2)^{\otimes 2}$ and show that this homomorphism is an isomorphism. By using this homomorphism we construct irreducible finite dimensional representations of the classical type and of the…
The main aim of this paper is to give classes of irreducible infinite dimensional representations and of irreducible $*$-representations of the q-deformed algebra $U'_q(so_{2,2})$ which is a real form of the non-standard deformation $U'_q(so_4)$ of the universal enveloping algebra $U(so(4,C))$. These representations are described by two complex parameters and are obtained by "analytical continuation" of the irreducible finite dimensional representations of the algebra $U'_q(so_4)$ in the basis corresponding to the reduction from $U'_q(so_4)$ to $U(so_2\oplus so_2)$.
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.
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.
The q-deformed algebra ${\rm so}'_q(r,s)$ is a real form of the q-deformed algebra $U'_q({\rm so}(n,\mathbb{C}))$, $n=r+s$, which differs from the quantum algebra $U_q({\rm so}(n,\mathbb{C}))$ of Drinfeld and Jimbo. We study representations of the most degenerate series of the algebra ${\rm so}'_q(r,s)$. The formulas of action of operators of these representations upon the basis corresponding to restriction of representations onto the subalgebra ${\rm so}'_q(r)\times {\rm so}'_q(s)$ are given. Most of these representations are irreducible. Reducible representations appear under some conditions…
Deformed quantum vertex algebra modules associated with braidings
Dipartimento di Matematica, Sapienza Università di Roma（罗马大学数学系）；INFN sezione di Roma（罗马国家核物理研究所）；University of Zagreb Faculty of Science, Department of Mathematics（萨格勒布大学理学院数学系）
An algebra homomorphism $ψ$ from the q-deformed algebra $U_q({\rm iso}_2)$ with generating elements $I$, $T_1$, $T_2$ and defining relations $[I,T_2]_q=T_1$, $[T_1,I]_q=T_2$, $[T_2,T_1]_q=0$ (where $[A,B]_q=q^{1/2}AB-q^{-1/2}BA$) to the extension ${\hat U}_q({\rm m}_2)$ of the Hopf algebra $U_q({\rm m}_2)$ is constructed. The algebra $U_q({\rm iso}_2)$ at $q=1$ leads to the Lie algebra ${\rm iso}_2 \sim {\rm m}_2$ of the group ISO(2) of motions of the Euclidean plane. The Hopf algebra $U_q({\rm m}_2)$ is treated as a Hopf $q$-deformation of the universal enveloping algebra of ${\rm iso}_2$ and…
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.
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 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.
We prove the equivalence of two presentations of deformed double current algebras associated to a complex simple Lie algebra, the first one obtained via a degeneration of affine Yangians while the other one naturally appeared in the construction of the elliptic Casimir connection. We also construct a specific central element of these algebras and, in type A, show that they contain a very large center for certain values of their parameters.
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.
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.
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.
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.
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.
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…
The boson representation of the sp(4,R) algebra and two distinct deformations of it, are considered, as well as the compact and noncompact subalgebras of each. The initial as well as the deformed representations act in the same Fock space. One of the deformed representation is based on the standard q-deformation of the boson creation and annihilation operators. The subalgebras of sp(4,R) (compact u(2) and three representations of the noncompact u(1,1) are also deformed and are contained in this deformed algebra. They are reducible in the action spaces of sp(4,R) and decompose into irreducible …
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.
The elliptic algebra A_{q,p}(sl(N)_{c}) at the critical level c=-N has an extended center containing trace-like operators t(z). Families of Poisson structures, defining q-deformations of the W_N algebra, are constructed. The operators t(z) also close an exchange algebra when (-p^{1/2})^{NM} = q^{-c-N} for M \in Z. It becomes Abelian when in addition p=q^{Nh} where h is a non-zero integer. The Poisson structures obtained in these classical limits contain different q-deformed W_N algebras depending on the parity of h, characterizing the exchange structures at p =/ q^{Nh} as new W_{q,p}(sl(N)) al…
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.
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 $A_{q,p}(\hat{sl}_2)$.
For the nonstandard $q$-deformed algebras $U_q(so_n)$, defined recently in terms of trilinear relations for generating elements, most general finite dimensional irreducible representations directly corresponding to those of nondeformed algebras $so(n)$ (i.e., characterized by the same sets of only integers or only half-integers as in highest weights of the latter) are given explicitly in a $q$-analogue of Gel'fand-Tsetlin basis. Detailed proof, for $q$ not equal to a root of unity, that representation operators indeed satisfy relevant (trilinear) relations and define finite dimensional irreduc…
The representation theory of deformed oscillator algebras, defined in terms of an arbitrary function of the number operator~$N$, is developed in terms of the eigenvalues of a Casimir operator~$C$. It is shown that according to the nature of the $N$ spectrum, their unitary irreducible representations may fall into one out of four classes, some of which contain bosonic, fermionic or parafermionic Fock-space representations as special cases. The general theory is illustrated by classifying the unitary irreducible representations of the Arik-Coon, Chaturvedi-Srinivasan, and Tamm-Dancoff oscillator…
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 orthogonal and pseudo-orthogonal Lie algebras are constructed which differ from deformations of Lie algebras in terms of Cartan subalgebra and root vectors and which make it possible to construct representations by operators acting according to Gel'fand--Tsetlin-type formulas. Unitary representations of the q-deformed algebras U_q(so_{n,1}) are found.
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.
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.
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.
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…
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…
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.
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.
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 $\mathfrak{osp}(2|2n)$.
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.
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.
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.
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.
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.
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.
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.
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.
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 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…
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…
