Deformations of enveloping algebra of Lie superalgebrasl(m,n)

Theq-deformations of the universal enveloping algebra ofsl(m, n) are considered, a Poincaré-Birkhoff-Witt type theorem is proved for these deformations, and the extra relations which are needed to definesl(m, n) as a contragredient algebra in addition to the Serre-type relations are identified with proof.     

Quantized universal enveloping superalgebra of type Q and a super-extension of the Hecke algebra

The quantized universal enveloping algebra U _q(q(n)) of the strange Lie superalgebra q(n) and a super-analogue HC _q (N) of the Hecke algebra H _q (N) are constructed. These objects are in a duality similar to the known duality between U _q (gl(n)) and H _q (N).     

Polynomial deformations of enveloping algebra U(sl(2,2102))

We define the polynomial deformation, U _t ^p, of enveloping algebra U(sl(2,)) as an analogue of quantum group and construct a nice set D _p ^even of parameters on which we described irreducible representations and classify the irreducible ones in terms of the highest weights. Also, we construct the Casimir elements and using them we see that every finite dimensional U _t ^p-module is semisimple.     

The two-parameter deformation of the supergroup GL(1|1), its differential calculus and its Lie algebra

We construct a right-invariant differential calculus on the quantum supergroup GL _q,s (1|1) and we show that its quantum Lie algebra with comultiplication is isomorphic to that which we obtain using the Reshetikhin-Takhtajan-Faddeev approach.     

The enveloping algebra of a covariant system

In an earlier work, Doplicher, Kastler and Robinson have examined a mathematical structure consisting of a pair (A, G), whereA is aC*-algebra andG is a locally compact automorphism group ofA. We call such a structure a covariant system. The enveloping von Neumann algebraA(A, G) of (A, G) is defined as a *-algebra of operator valued functions (called options) on the space of covariant representations of (A, G). The system (A, G) is canonically embedded in, and in fact generates, the von Neumann algebraA(A, G). Further we show there is a natural one-to-one correspondence between the normal *-representations ofA(A, G) and the proper covariant representations of (A, G). The relation ofA(A, G) to the covarainceC*-algebraC*(A, G) is also examined.     

Induced representations of the two-parameter Jordanian quantum algebra

We construct induced infinite-dimensional representations of the two-parameter quantum algebraUg,h(gl(2)) which is in duality with the deformationGLg,h(2). The representations depend on two representation parameters, but split into one-parameter representations of a one-generator central subalgebra and the three-generator Jordanian quantum subalgebraU (sl(2)), =g + h. The representations of the latter can be mapped to representations in one complex variable, which give anew deformation of the standard one-parameter vector-field realization ofsl(2). These infinite-dimensional representations are reducible for some values of the representation parameters, and then we obtain canonically the finite-dimensional representations ofU (sl(2)). Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. Permanent address of V.K.D.     

The Clebsch-Gordan coefficients ofsl(2,R) in the parabolic basis

We realize the Lie algebra sl(2,R) in terms of second-order differential operators defined on a dense common domain of square-integrable functions on a two-chart space, where the self-adjoint extension(s) (families) lead to all (and only) self-adjoint irreducible representations of the algebra, single- as well as multi-valued over the group. This allows for a rather straight-forward evaluation of the Clebsch-Gordan coefficients of sl(2,R) in the parabolic subalgebra basis.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.I would like to thank the hospitality of Prof. J. A. de Azcárraga, at the Instituto de Fí'sica Teórica, Universidad de Valencia, where this review was written.     

Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra

Let be a complex simple Lie algebra. We show that whent is not a root of 1 all finite dimensional representations of the quantum analogU _t are completely reducible, and we classify the irreducible ones in terms of highest weights. In particular, they can be seen as deformations of the representations of the (classical)U .     

Rank of quantized universal enveloping algebras and modular functions

We compute an intrinsic rank invariant for quasitriangular Hopf algebras in the case of general quantum groupsU _q (g). As a function ofq the rank has remarkable number theoretic properties connected with modular covariance and Galois theory. A number of examples are treated in detail, including rank (U _q (su(3))) and rank (U _q (e ₈)). We briefly indicate a physical interpretation as relating Chern-Simons theory with the theory of a quantum particle confined to an alcove ofg.     

Evolution of a two-parameter chaotic dynamics from universal attractors

We observed endlessly repeating sequences of the period-doubling universal route to chaos followed by ordered states in the dynamics of a two-parameter 1-D system. The coalescence of these sequences are reported together with measurements of a strange attractor which evolves from this overlap.     

A three-parameter Hopf deformation of the algebra of Feynman-like diagrams*

We construct a three-parameter deformation of the Hopf algebra LDIAG. This is the algebra that appears in an expansion in terms of Feynman-like diagrams of the product formula in a simplified version of quantum field theory. This new algebra is a true Hopf deformation which reduces to LDIAG for some parameter values and to the algebra of matrix quasi-symmetric functions (MQSym) for others, and thus relates LDIAG to other Hopf algebras of contemporary physics. Moreover, there is an onto linear mapping preserving products from our algebra to the algebra of Euler–Zagier sums.     

A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions

A quantum deformation of the Virasoro algebra is defined. The Kac determinants at arbitrary levels are conjectured. We construct a bosonic realization of the quantum deformed Virasoro algebra. Singular vectors are expressed by the Macdonald symmetric functions. This is proved by constructing screening currents acting on the bosonic Fock space.     

The algebra and geometry ofSU(3) matrices

We give an elementary treatment of the defining representation and Lie algebra of the three-dimensional unitary unimodular groupSU(3). The geometrical properties of the Lie algebra, which is an eight dimensional real linear vector space, are developed in anSU(3) covariant manner. Thef andd symbols ofSU(3) lead to two ways of ‘multiplying’ two vectors to produce a third, and several useful geometric and algebraic identities are derived. The axis-angle parametrization ofSU(3) is developed as a generalization of that forSU(2), and the specifically new features are brought out. Application to the dynamics of three-level systems is outlined.     

Big deformation in ~(17)C

Reaction and interaction cross sections of ¹⁷C on a carbon target have been re-analyzed using the modified Glauber model. The analysis with a deformed Woods-Saxon density/potential suggests a big deformation structure for ¹⁷C. The existence of a tail in the density distribution supports the possibility of it being a one-neutron halo structure. Under a deformed core plus a single-particle assumption, analysis shows a dominant d-wave of the valence neutron in ¹⁷C.     

A three-parameter deformation of the Weyl-Heisenberg algebra: differential calculus and invariance

We define a three-parameter deformation of the Weyl-Heisenberg algebra that generalizes the q-oscillator algebra. By a purely algebraical procedure, we set up on this quantum space two differential calculi that are shown to be invariant on the same quantum group, extended to a ten-generator Hopf-star-algebra. We prove that when the values of the parameters are related, the two differential calculi reduce to one that is invariant under two quantum groups.     

An isomorphism between the Para-Fermi algebra and the algebra O(n,n+1)

It is shown that the relations which define the para-Fermi creation and annihilation operatorsb _i ,a _i (i = 1,...,n) can be considered as commutation relations of the algebra 0(n, n+1). It turns out that every representation of 0(n, n+1) determines a representation of the para-Fermi algebra and vice versa.