Covariant (hh')-Deformed Bosonic and Fermionic Algebras as Contraction Limits of q-Deformed Ones

Abstract

$G{L}_h(n)\times G{L}_{h^{\prime }}(m)$-covariant (hh')-bosonic (or (hh')-fermionic) algebras ${\mathcal{A}}_{h{h}^{\prime }\pm }(n,m)$ are built in terms of the corresponding R_h and $R_{h^{\prime }}$-matrices by contracting the $G{L}_q(n)\times G{L}_{q^{\pm 1}}(m)$-covariant q-bosonic (or q-fermionic) algebras ${\mathcal{A}}_{q\pm }^{(\alpha )}(n,m)$, $\alpha =1,2$. When using a basis of ${\mathcal{A}}_{q\pm }^{(\alpha )}(n,m)$ wherein the annihilation operators are contragredient to the creation ones, this contraction procedure can be carried out for any n, m values. When employing instead a basis wherein the annihilation operators, as the creation ones, are irreducible tensor operators with respect to the dual quantum algebra $U_q(gl(n))\otimes U_{q^{\pm 1}}(gl(m))$, a contraction limit only exists for $n,m\in \{1,2,4,6,...\}$. For n=2, m=1, and n=m=2, the resulting relations can be expressed in terms of coupled (anti)commutators (as in the classical case), by using $U_h(sl(2))$ (instead of sl(2)) Clebsch-Gordan coefficients. Some U_h(sl(2)) rank-1/2 irreducible tensor operators, recently constructed by Aizawa, are shown to provide a realization of ${\mathcal{A}}_{h\pm }(2,1)$.

LaTeX, uses amssym.sty, 24 pages, no figure, to be published in Int. J. Theor. Phys