相关论文: A Holstein-Primakoff and a Dyson realization for t…
The Holstein-Primakoff and the Dyson realizations of the Lie superalgebra $gl(n/m)$ are generalized to the class of the quantum superalgebras $U_q[gl(n/m)]$ for any $n$ and $m$. It is shown how the elements of $U_q[gl(n/m)]$ can be…
The known Holstein-Primakoff and Dyson realizations for the Lie algebras $gl(n+1),\;n=1,2,\ldots$ in terms of Bose operators are generalized to the class of the Lie superalgebras $gl(m/n+1)$ for any $n$ and $m$. Formally the expressions are…
An analogue of the Holstein-Primakoff and of the Dyson realization for the Lie superalgebra $sl(1/n)$ is written down. The expressions are formally the same as for the Lie algebra $sl(n+1)$, however in the latter the Bose operators have to…
We give explicit expression of recurrency formulae of canonical realization for quantum enveloping algebras $U_{q}(sl(n+1,C))$. In these formulas the generators of the algebra $U_{q}(sl(n+1,C))$ are expressed by means of n-canonical q-boson…
Fock space representations of the Lie superalgebra $sl(n+1|m)$ and of its quantum analogue $U_q[sl(n+1|m)]$ are written down. The results are based on a description of these superalgebras via creation and annihilation operators. The…
We show that our construction of realizations for Lie algebras and quantum algebras can be generalized to quantum superalgebras, too. We study an example of quantum superalgebra $U_q(gl(2/1))$ and give the boson-fermion realization with…
We recall the relation between the Lie superalgebra $osp(1/2n)$ and para-Bose operators. The quantum superalgebra $U_q[osp(1/2n)]$, defined as usual in terms of its Chevalley generators, is shown to be isomorphic to an associative algebra…
Inspired by a result in [Ga], we locate two $ k[q,q^{-1}] $-integer forms of $ F_q[SL(n+1)] $, along with a presentation by generators and relations, and prove that for $ q=1 $ they specialize to $ U({\mathfrak{h}}) $, where $…
The Lie algebra $so(2n+1)$ and the Lie superalgebra $osp(1/2n)$ are quantized in terms of $3n$ generators, called preoscillator generators. Apart from $n$ "Cartan" elements the preoscillator generators are deformed para-Fermi operators in…
We give the Heisenberg realization for the quantum algebra $U_q(sl_n)$, which is written by the $q$-difference operator on the flag manifold. We construct it from the action of $U_q(sl_n)$ on the $q$-symmetric algebra $A_q(Mat_n)$ by the…
As an alternative to Chevalley generators, we introduce Jacobson generators for the quantum superalgebra $U_q[sl(n+1|m)]$. The expressions of all Cartan-Weyl elements of $U_q[sl(n+1|m)]$ in terms of these Jacobson generators become very…
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…
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…
A representation of the quantum superalgebra Uq(sl(M+1|N+1)) is constructed based on the q-differential operators acting on the coherent states parameterized by coordinates. These coordinates correspond to the local ones of the flag…
In the paper, we further realize the higher rank quantized universal enveloping algebra $U_q(sl_{n+1})$ as certain quantum differential operators in $\mathcal W_q(2n)$ defined over the quantum divided power algebra $\mathcal{A}_q(n)$ of…
The observation that $n$ pairs of para-Bose (pB) operators generate the universal enveloping algebra of the orthosymplectic Lie superalgebra $osp(1/2n)$ is used in order to define deformed pB operators. It is shown that these operators are…
In this paper, the module-algebra structures of $U_q(sl(m+1))$ on the quantum $n$-space $A_q(n)$ are studied. We characterize all module-algebra structures of $U_q(sl(m+1))$ on $A_q(2)$ and $A_q(3)$ when $m\geq 2$. The module-algebra…
The concept of a quantum algebra is made easy through the investigation of the prototype algebras $u_{qp}(2)$, $su_q(2)$ and $u_{qp}(1,1)$. The latter quantum algebras are introduced as deformations of the corresponding Lie algebras~; this…
We define for real $q$ a unital $*$-algebra $U_q(\mathfrak{sl}(2,\mathbb{R}))$ quantizing the universal enveloping $*$-algebra of $\mathfrak{sl}(2,\mathbb{R})$. The $*$-algebra $U_q(\mathfrak{sl}(2,\mathbb{R}))$ is realized as a…
We give a selfcontained introduction to the theory of quantum groups according to Drinfeld highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras,…