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B. Abdesselam

Publications and source records attributed to B. Abdesselam.

29 records · Page 2Linked to original sources
arXiv1996

Centre and Representations of U_q(sl(2|1)) at Roots of Unity

Quantum groups at roots of unity have the property that their centre is enlarged. Polynomial equations relate the standard deformed Casimir operators and the new central elements. These relations are important from a physical point of view since they correspond to relations among quantum expectation values of observables that have to be satisfied on all physical states. In this paper, we establish these relations in the case of the quantum Lie superalgebra U_q(sl(2|1)). In the course of the argument, we find and use a set of representations such that any relation satisfied on all the representations of the set is true in U_q(sl(2|1)). This set is a subset of the set of all the finite dimensional irreducible representations of U_q(sl(2|1)), that we classify and describe explicitly.

q-alg↗
arXiv1996

Irreducible Representations of Jordanian Quantum Algebra ${\cal U}_h(sl(2))$ Via a Nonlinear Map

The generators of the Jordanian quantum algebra ${\cal U}_h(sl(2))$ are expressed as nonlinear invertible functions of the classical $sl(2)$ generators. This permits immediate explicit construction of the finite dimensional irreducible representations of the algebra ${\cal U}_h(sl(2))$. Using this construction, new finite dimensional solutions of the Yang-Baxter equation may be obtained.

q-alg↗
arXiv1996

On ${\cal U}_h(sl(2))$, ${\cal U}_h(e(3))$ and their Representations

By solving a set of recursion relations for the matrix elements of the ${\cal U}_h(sl(2))$ generators, the finite dimensional highest weight representations of the algebra were obtained as factor representations. Taking a nonlinear combination of the generators of the two copies of the ${\cal U}_h(sl(2))$ algebra, we obtained ${\cal U}_h(so(4))$ algebra. The latter, on contraction, yields ${\cal U}_h(e(3))$ algebra. A nonlinear map of ${\cal U}_h(e(3))$ algebra on its classical analogue $e(3)$ was obtained. The inverse mapping was found to be singular. It signifies a physically interesting situation, where in the momentum basis, a restricted domain of the eigenvalues of the classical operators is mapped on the whole real domain of the eigenvalues of the deformed operators.

q-alg↗
arXiv1996

On a Deformation of $sl(2)$ with Paragrassmannian Variables

We propose a new structure ${\cal U}^{r}_{\displaystyle{q}}(sl(2)) $. This is realized by multiplying $δ$ ($q=e^δ$, $δ\in \CC$) by $θ$, where $θ$ is a real nilpotent -paragrassmannian- variable of order $r$ ($θ^{r+1}=0$) that we call the order of deformation, the limit $r\rightarrow \infty$ giving back the standard ${\cal U}_{\displaystyle {q}}(sl(2))$. In particular we show that, for $r=1$, there exists a new ${\cal R}$-matrix associated with $sl(2)$. We also proof that the restriction of the values of the parameters of deformation give nonlinear algebras as particular cases.

q-alg↗
arXiv1996

On Nonlinear Angular Momentum Theories, Their Representations and Associated Hopf Structures

Nonlinear $sl(2)$ algebras subtending generalized angular momentum theories are studied in terms of undeformed generators and bases. We construct their unitary irreducible representations in such a general context. The linear $sl(2)$-case as well as its $q$-deformation are easily recovered as specific examples. Two other physically interesting applications corresponding to the so-called Higgs and quadratic algebras are also considered. We show that these two nonlinear algebras can be equipped with a Hopf structure.

q-alg↗
arXiv1995

On the Fundumental Invariant of the Hecke Algebra $H_{n}(q)$

The fundumental invariant of the Hecke algebra $H_{n}(q)$ is the $q$-deformed class-sum of transpositions of the symmetric group $S_{n}$. Irreducible representations of $H_{n}(q)$, for generic $q$, are shown to be completely characterized by the corresponding eigenvalues of $C_{n}$ alone. For $S_{n}$ more and more invariants are necessary as $n$ inereases. It is pointed out that the $q$-deformed classical quadratic Casimir of $SU(N)$ plays an analogous role. It is indicated why and how this should be a general phenomenon associated with $q$-deformation of classical algebras. Apart from this remarkable conceptual aspect $C_{n}$ can provide powerful and elegant techniques for computations. This is illustrated by using the sequence $C_{2}$, $C_{3}, \cdots,\; C_{n}$ to compute the characters of $H_{n}(q)$.

q-alg↗
arXiv1995

Representations of U_q(sl(N)) at Roots of Unity

The Gelfand--Zetlin basis for representations of $U_q(sl(N))$ is improved to fit better the case when $q$ is a root of unity. The usual $q$-deformed representations, as well as the nilpotent, periodic (cyclic), semi-periodic (semi-cyclic) and also some atypical representations are now described with the same formalism.

q-alg↗
arXiv1995

The character table of the Hecke algebra $H_n(q)$ in terms of traces of products of Murphy operators

The traces of the Murphy operators of the Hecke algebra $H_n(q)$, and of products of sets of Murphy operators with non-consecutive indices, can be evaluated by a straightforward recursive procedure. These traces are shown to determine all the reduced traces in this algebra, which, in turn, determine all other traces. To illustrate the procedure we obtain the set of reduced traces for $H_7(q)$ - the lowest order Hecke algebra whose character table has not hitherto been reported. This is preceded by the presentation of an explicit algorithm for the reduction of the trace of an arbitrary element of the Hecke algebra into a linear combination of traces of elements consisting of appropriately defined disjoint cycles; and of a proof, presented in order to make the present article reasonably self-contained, that a reduced trace depends only on the set of lengths of the disjoint cycles that it consists of.

q-alg↗
arXiv1995

The fundamental invariant of the Hecke algebra $H_n(q)$ characterizes the representations of $H_n(q)$, $S_n$, $SU_q(N)$ and $SU(N)$

The irreducible representations (irreps) of the Hecke algebra $H_n(q)$ are shown to be completely characterized by the fundamental invariant of this algebra, $C_n$. This fundamental invariant is related to the quadratic Casimir operator, ${\cal{C}}_2$, of $SU_q(N)$, and reduces to the transposition class-sum, $[(2)]_n$, of $S_n$ when $q\rightarrow 1$. The projection operators constructed in terms of $C_n$ for the various irreps of $H_n(q)$ are well-behaved in the limit $q\rightarrow 1$, even when approaching degenerate eigenvalues of $[(2)]_n$. In the latter case, for which the irreps of $S_n$ are not fully characterized by the corresponding eigenvalue of the transposition class-sum, the limiting form of the projection operator constructed in terms of $C_n$ gives rise to factors that depend on higher class-sums of $S_n$, which effect the desired characterization. Expanding this limiting form of the projection operator into a linear combination of class-sums of $S_n$, the coefficients constitute the corresponding row in the character table of $S_n$. The properties of the fundamental invariant are used to formulate a simple and efficient recursive procedure for the evaluation of the traces of the Hecke algebra. The closely related quadratic Casimir operator of $SU_q(N)$ plays a similar role, providing a complete characterization of the irreps of $SU_q(N)$ and - by constructing appropriate projection operators and then taking the $q\rightarrow 1$ limit - those of $SU(N)$ as well, even when the quadratic Casimir operator of the latter does not suffice to specify its irreps.

q-alg↗
arXiv1994

Representations of $SO(5)_{q}$ and Non-Minimal $q$-Deformation

Representations of $SO(5)_{q}$ can be constructed on bases such that either the Chevalley triplet $(e_{1},\;f_{1},\;h_{1})$ or $(e_{2},\;f_{2},\;h_{2})$ has the standard $SU(2)_{q}$ matrix elements. The other triplet in each cases has a more complicated action. The $q$-deformation of such representations present striking differences. In one case a {\bf non-minimal} deformation is found to be essential. This is explained and illustrated below. Broader interests of a parallel use of the two bases are pointed out.

hep-th↗