Related papers: ISO_q(3) and ISO_q(2,1)
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…
On the basis of the Schwinger boson representation for the Elliott model developed in (III), the Holstein-Primakoff representation for the su(3)-algebra is presented. The basic idea is the same as that adopted in (II), and as an example,…
We derive orthogonality relations for discrete q-ultraspherical polynomials and their duals by means of operators of representations of the quantum algebra su_q(1,1). Spectra and eigenfunctions of these operators are found explicitly. These…
On the basis of the Schwinger boson representation for the Lipkin model developed in (I), the Holstein-Primakoff representation for the su(3)-algebra is presented. Including the symmetric case, the representation obtained in this paper…
A possible generalization of the method of orbits to SLq(2,R) is discussed.
We describe an embedding of the Poincare Lie algebra into an extension of the Lie field of SO(2,3) (the anti-de Sitter group). We also describe higher dimensional analogs of this embedding, which connect SO(p,q) groups to their associated…
We give the complete set of irreducible representations of U(SU(2))_q when q is a m-th root of unity. In particular we show that their dimensions are less or equal to m. Some of them are not highest weight representations.
We derive an explicit isomorphism between the Hilbert modular group and certain congruence subgroups on the one hand and particular subgroups of the special orthogonal group $SO(2, 2)$ on the other hand. The proof is based on an application…
We show that the quantum Heisenberg group $H_{q}(1)$ can be obtained by means of contraction from quantum $SU_q(2)$ group. Its dual Hopf algebra is the quantum Heisenberg algebra $U_{q}(h(1))$. We derive left and right regular…
In this paper we study some combinatorial properties of the shuffle algebra $S_{q_{1},q_{2}}$ with two complex parameters and, in particular, obtain the Hilbert series for $S_{q_{1},q_{2}}/I$, where $I$ is the ideal generated by $z-\lambda$…
In this paper, we prove that $C(SO_q(4)/SO_q(2))$ is isomorphic to the $C^*$-algebra of the tight groupoid $\mathcal{G}_{\mathrm{tight}}$ associated with the inverse semigroup generated by the standard generators of its classical limit…
Using crystal basis, in the space of symmetric irreducible representations, we explicitly write invertible deforming functionals between the q-deformed universal enveloping algebra and the Lie algebras sl(n) and sp(2n).For sl(2) we obtain…
We realize all irreducible unitary representations of the group $\mathrm{SO}_0(n+1,1)$ on explicit Hilbert spaces of vector-valued $L^2$-functions on $\mathbb{R}^n\setminus\{0\}$. The key ingredient in our construction is an explicit…
Let $\sigma$ be an isometric representation of $\mathbb{N}^d$ on a Hilbert space $\mathcal{H}$. We induce $\sigma$ to an isometric representation $V$ of $\mathbb{R}_{+}^{d}$ on another Hilbert space $\mathcal{K}$. We show that the map…
Let $P$ and $Q$ be finite posets and $R$ a commutative unital ring. In the case where $R$ is indecomposable, we prove that the $R$-linear isomorphisms between partial flag incidence algebras $I^3(P,R)$ and $I^3(Q,R)$ are exactly those…
In this paper we study of *-representations for polynomial algebras on quantum matrix spaces. We deal with two special cases of the polynomial algebras, namely the algebra of polynomials on quantum complex matrices $\mathrm{Mat_2}$ and on…
The generators $(J_{\pm}, J_0)$ of the algebra $U_q(sl(2))$ is our starting point. An invertible nonlinear map involving, apart from q, a second arbitrary complex parameter h, defines a triplet $({\hat X},{\hat Y},{\hat H})$. The latter set…
The linear representation $T_n^*(\mathcal{K})$ of a point set $\mathcal{K}$ in a hyperplane of $\mathrm{PG}(n+1,q)$ is a point-line geometry embedded in this projective space. In this paper, we will determine the isomorphisms between two…
We show that a Hilbert space bounded linear operator has an $m$-isometric lifting for some integer $m\ge 1$ if and only if the norms of its powers grow polynomially. In analogy with unitary dilations of contractions, we prove that such…
Let $K$ be the attractor of the following IFS $$\{f_1(x)=\lambda x, f_2(x)=\lambda x +c-\lambda,f_3(x)=\lambda x +1-\lambda\}, $$ where $f_1(I)\cap f_2(I)\neq \emptyset, (f_1(I)\cup f_2(I))\cap f_3(I)=\emptyset,$ and $I=[0,1]$ is the convex…