Nguyen Anh Ky
Some one- and two-parametric deformations of U[sl(2)] and their representations are considered. Interestingly, a newly introduced two-parametric deformation admits a class of infinite - dimensional representations which have no classical (non-deformed) and one-parametric deformation analogues, even at generic deformation parameters.
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Some introductory concepts and basic definitions of the Lie superalgebras and their quantum deformations are exposed. Especially the induced representation methods in both cases are described. Based on the Kac representation theory we have succeeded in constructing representations of several higher rank superalgebras. When representations of quantum superalgebras are concerned, we have developed a method which can be applied not only to the one-parametric quantum deformations but also to the multi-parametric ones. As an intermediate step the Gel'fand-Zetlin basis description is extended to the…
The two-parametric quantum superalgebra $U_{p,q}[gl(2/1)]$ is consistently defined. A construction procedure for induced representations of $U_{p,q}[gl(2/1)]$ is described and allows us to construct explicitly all (typical and nontypical) finite-dimensional representations of this quantum superalgebra. In spite of some specific features, the present approach is similar to a previously developed method [1] which, as shown here, is applicable not only to the one-parametric quantum deformations but also to the multi-parametric ones.
A two-parametric deformation of U[sl(2)] and its representations are considered. This newly introduced two-parametric quantum group denoted as $U_{pq}[sl(2)]$ admits a class of infinite-dimensional representations which have no classical (non-deformed) and one-parametric deformation analogues, even at generic deformation parameters. Interestingly that finite-dimensional representations of $U_{pq}[sl(2)]$ allow arbitrary complex "spins" (i.e., not necessary they to be integral or half-integral numbers), unlike those in the classical and one-parametric deformation cases.
In this contribution to the proceedings of the 68eme Rencontre entre Physiciens Theoriciens et Mathematiciens on Deformation Quantization I shall report on some recent joint work with Henrique Bursztyn on the representation theory of *-algebras arising from deformation quantization as I presented this in my talk.
We discuss the procedure of Rieffel induction of representations in the framework of formal deformation quantization of Poisson manifolds. We focus on the central role played by algebraic notions of complete positivity.
In this review we discuss various aspects of representation theory in deformation quantization starting with a detailed introduction to the concepts of states as positive functionals and the GNS construction. But also Rieffel induction of representations as well as strong Morita equivalence, the Dirac monopole and the strong Picard groupoid are discussed.
Motivated by deformation quantization we investigate the algebraic GNS construction of *-representations of deformed *-algebras over ordered rings and compute their classical limit. The question if a GNS representation can be deformed leads to the deformation of positive linear functionals. Various physical examples from deformation quantization like the Bargmann-Fock and the Schr{ö}dinger representation as well as KMS functionals are discussed.
The two-parametric quantum superalgebra $U_{p,q}[gl(2/2)]$ and its induced representations are considered. A method for constructing all finite-dimensional irreducible representations of this quantum superalgebra is also described in detail. It turns out that finite-dimensional representations of the two-parametric $U_{p,q}[gl(2/2)]$, even at generic deformation parameters, are not simply trivial deformations from those of the classical superalgebra $gl(2/2)$, unlike the one-parametric cases.
We recover the holomorphic discrete series representations of $SU(1,n)$ as well as some unitary irreducible representations of $SU(n+1)$ by deformation of a minimal realization of $sl(n+1, {\mathbb C})$.
We describe a new technique to obtain representations of the braid group B_n from the R-matrix of a quantum deformed algebra of the one dimensional harmonic oscillator. We consider the action of the R-matrix not on the tensor product of representations of the algebra, that in the harmonic oscillator case are infinite dimensional, but on the subspace of the tensor product corresponding to the lowest weight vectors.
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 …
In the framework of deformation quantization we apply the formal GNS construction to find representations of the deformed algebras in pre-Hilbert spaces over $\mathbb C[[λ]]$ and establish the notion of local operators in these pre-Hilbert spaces. The commutant within the local operators is used to distinguish `thermal' from `pure' representations. The computation of the local commutant is exemplified in various situations leading to the physically reasonable distinction between thermal representations and pure ones. Moreover, an analogue of von Neumann's double commutant theorem is proved in …
In these lecture notes I give an introduction to deformation quantization. The quantization problem is discussed in some detail thereby motivating the notion of star products. Starting from a deformed observable algebra, i.e. the star product algebra, physical applications require to study representations of this algebra. I review the recent development of a representation theory including techniques like Rieffel induction and Morita equivalence. Applications beyond quantization theory are found in noncommutative field theories.
Free field representation of level-$k (\not=0,-2)$ Yangian double ${\cal D}Y(sl_2)_k$ and a corresponding deformation of Wakimoto modules are presented. We also realize two types of vertex operators intertwining these modules.
Representations of $SO(5)_{q}$ and Non-Minimal $q$-Deformation
$SO(5)_{q}$的表示可以在这样的基础上构建，使得Chevalley三元组$(e_{1},\;f_{1},\;h_{1})$或$(e_{2},\;f_{2},\;h_{2})$具有标准$SU(2)_{q}$矩阵元素。每种情况下的另一个三联体都有更复杂的作用。这种表征的$q$变形呈现出显著的差异。在一种情况下，发现{ \bf非最小}变形是必要的。下文对此进行了解释和说明。更广泛的利益的两个基地的平行使用指出。
We study a deformation of the Cuntz-Toeplitz $C^*$-algebra determined by the relations $a_i^*a_i=1+q a_ia_i^*, a_i^*a_j=0$. We define well-behaved unbounded *-representations of the *-algebra defined by relations above and classify all such irreducible representations up to unitary equivalence.
In this paper, we use the unitary representation theory of $SL_2(\mathbb R)$ to understand the Rankin-Cohen brackets for modular forms. Then we use this interpretation to study the corresponding deformation problems that Paula Cohen, Yuri Manin and Don Zagier initiated. Two uniqueness results are established.
We constructed a multi-parametric deformation of the Brauer algebra representation related with the symplectic Lie algebras. The notion of Manin matrix of type C was generalised to the case of the multi-parametric deformation by using this representation and corresponding quadratic algebras. We derived pairing operators for these quadratic algebras and minors for the considered Manin matrices. The rank of pairing operators and dimensions of components of quadratic algebras were calculated.
The aim of this paper is to announce the results on irreducible nonclassical type representations of the nonstandard q-deformations U'_q(so_n), U_q(iso_n) and U'_q(so_{n,1}) of the universal enveloping algebras of the Lie algebras so(n,C), iso_n and so_{n,1} when q is a real number (the algebra U'_q(so_{n,1}) is a real form of the algebra U'_q(so_{n+1})). These representations are characterized by the properties that they are singular at the point q=1.
We demonstrate the main idea of constructing irreducible unitary representations of Lie groups by using Fedosov deformation quantization in the concrete case of the group Aff(R) of affine transformations of the real straight line. By an exact computation of the star-product and the operator $\hat{\ell}_Z$, we show that the resulting representations exhausted all the irreducible representations of this groups.
We propose the notion of q-characters for finite-dimensional representations of quantum affine algebras. It is motivated by our theory of deformed W-algebras. We show that the q-characters give rise to a homomorphism from the Grothendieck ring of representations of a quantum affine algebra to a polynomial ring. We conjecture that the image of this homomorphism is equal to the intersection of certain "screening operators". We also discuss the connection between q-characters and Bethe Ansatz.
In this letter we give an overview on recent developments in representation theory of star product algebras. In particular, we relate the *-representation theory of *-algebras over rings C = R(i) with an ordered ring R and i^2 = -1 to the *-representation theory of *-algebras over \mathbb{C} and point out some properties of the Picard groupoid corresponding to the notion of strong Morita equivalence. Some Morita invariants are interpreted as arising from actions of this groupoid.
In this paper Witten type deformation of osp(1/2) algbera is introduced and its realization and matrix representation are obtained. The matrix representation is shown to be possible only when the dimension is odd.
Given an associative multiplication in matrix algebra compatible with the usual one or, in other words, linear deformation of matrix algebra, we construct a solution to the classical Yang-Baxter equation. We also develop a theory of such deformations and construct numerous examples. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures. We also describe an important class of M-structures related to the affine Dynkin diagrams of A, D, E-type. These M-structures and their representations are describe…
We describe properties of the nonstandard q-deformation U'_q(so_n) of the universal enveloping algebra U(so_n) of the Lie algebra so_n which does not coincide with the Drinfeld--Jimbo quantum algebra U_q(so_n). In particular, it is shown that there exists an isomorphism from U'_q(so_n) to U_q(sl_n) and that finite dimensional irreducible representations of U'_q(so_n) separate elements of this algebra. Irreducible representations of the algebras U'_q(so_n) for q a root of unity q^p=1 are given. The main class of these representations act on p^N-dimensional linear space (where N is a number of p…
We analyse the Witten-Woronowicz's type deformations of the Lie superalgebra osp(2,2) and obtain a deformation parametrized by three independent parameters. For some of these algebras, finite dimensional representations are formulated in terms of a finite difference operator, providing operators that are relevant for the classification of quasi exactly solvable, finite difference equations. Similar representations are also pointed out for the Lie superalgebra osp(1,2).
We provide a very brief review of the description of colored invariants for the Hopf link in terms of characters, which need to be taken at a peculiar deformation of the topological locus, depending on one of the two representations associated with the two components of the link. Most important, we extend the description of this locus to conjugate and, generically, to composite representations and also define the "adjoint" Schur functions emerging in the dual description.
Representations and Classification of the compact quantum groups $U_q(2)$ for complex deformation parameters
本文研究了复变形参数下紧量子群$U_q(2)$的表示与分类，通过小$q$-雅各比多项式描述表示矩阵系数，并建立了彼得-沃伊尔分解和普朗查尔公式。
Beurling-Fourier algebras of $ q $-deformations of compact semisimple Lie groups and complexification
本文研究紧半单李群 $q$-形变的 Beurling-Fourier 代数，证明其 Drinfeld double 函数代数的不可约表示可由特定中心权族对应的加权 Fourier 代数不可约表示穷尽。
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