415 research outputs found
Quantum Lie algebras associated to and
Quantum Lie algebras \qlie{g} are non-associative algebras which are
embedded into the quantized enveloping algebras of Drinfeld and Jimbo
in the same way as ordinary Lie algebras are embedded into their enveloping
algebras. The quantum Lie product on \qlie{g} is induced by the quantum
adjoint action of . We construct the quantum Lie algebras associated to
and . We determine the structure constants and the
quantum root systems, which are now functions of the quantum parameter .
They exhibit an interesting duality symmetry under .Comment: Latex 9 page
The structure of quantum Lie algebras for the classical series B_l, C_l and D_l
The structure constants of quantum Lie algebras depend on a quantum
deformation parameter q and they reduce to the classical structure constants of
a Lie algebra at . We explain the relationship between the structure
constants of quantum Lie algebras and quantum Clebsch-Gordan coefficients for
adjoint x adjoint ---> adjoint. We present a practical method for the
determination of these quantum Clebsch-Gordan coefficients and are thus able to
give explicit expressions for the structure constants of the quantum Lie
algebras associated to the classical Lie algebras B_l, C_l and D_l.
In the quantum case also the structure constants of the Cartan subalgebra are
non-zero and we observe that they are determined in terms of the simple quantum
roots. We introduce an invariant Killing form on the quantum Lie algebras and
find that it takes values which are simple q-deformations of the classical
ones.Comment: 25 pages, amslatex, eepic. Final version for publication in J. Phys.
A. Minor misprints in eqs. 5.11 and 5.12 correcte
Solutions to the Quantum Yang-Baxter Equation with Extra Non-Additive Parameters
We present a systematic technique to construct solutions to the Yang-Baxter
equation which depend not only on a spectral parameter but in addition on
further continuous parameters. These extra parameters enter the Yang-Baxter
equation in a similar way to the spectral parameter but in a non-additive form.
We exploit the fact that quantum non-compact algebras such as
and type-I quantum superalgebras such as and are
known to admit non-trivial one-parameter families of infinite-dimensional and
finite dimensional irreps, respectively, even for generic . We develop a
technique for constructing the corresponding spectral-dependent R-matrices. As
examples we work out the the -matrices for the three quantum algebras
mentioned above in certain representations.Comment: 13 page
Quantum affine Toda solitons
We review some of the progress in affine Toda field theories in recent years,
explain why known dualities cannot easily be extended, and make some
suggestions for what should be sought instead.Comment: 16pp, LaTeX. Minor revision
Representations of the Generalized Lie Algebra sl(2)_q
We construct finite-dimensional irreducible representations of two quantum
algebras related to the generalized Lie algebra \ssll (2)_q introduced by
Lyubashenko and the second named author. We consider separately the cases of
generic and at roots of unity. Some of the representations have no
classical analog even for generic . Some of the representations have no
analog to the finite-dimensional representations of the quantised enveloping
algebra , while in those that do there are different matrix
elements.Comment: 14 pages, plain-TEX file using input files harvmac.tex, amssym.de
On Quantum Lie Algebras and Quantum Root Systems
As a natural generalization of ordinary Lie algebras we introduce the concept
of quantum Lie algebras . We define these in terms of certain
adjoint submodules of quantized enveloping algebras endowed with a
quantum Lie bracket given by the quantum adjoint action. The structure
constants of these algebras depend on the quantum deformation parameter and
they go over into the usual Lie algebras when .
The notions of q-conjugation and q-linearity are introduced. q-linear
analogues of the classical antipode and Cartan involution are defined and a
generalised Killing form, q-linear in the first entry and linear in the second,
is obtained. These structures allow the derivation of symmetries between the
structure constants of quantum Lie algebras.
The explicitly worked out examples of and illustrate the
results.Comment: 22 pages, latex, version to appear in J. Phys. A. see
http://www.mth.kcl.ac.uk/~delius/q-lie.html for calculations and further
informatio
Boundary breathers in the sinh-Gordon model
We present an investigation of the boundary breather states of the
sinh-Gordon model restricted to a half-line. The classical boundary breathers
are presented for a two parameter family of integrable boundary conditions.
Restricting to the case of boundary conditions which preserve the \phi -->
-\phi symmetry of the bulk theory, the energy spectrum of the boundary states
is computed in two ways: firstly, by using the bootstrap technique and
subsequently, by using a WKB approximation. Requiring that the two descriptions
of the spectrum agree with each other allows a determination of the
relationship between the boundary parameter, the bulk coupling constant, and
the parameter appearing in the reflection factor derived by Ghoshal to describe
the scattering of the sinh-Gordon particle from the boundary.Comment: 16 pages amslate
From Quantum Universal Enveloping Algebras to Quantum Algebras
The ``local'' structure of a quantum group G_q is currently considered to be
an infinite-dimensional object: the corresponding quantum universal enveloping
algebra U_q(g), which is a Hopf algebra deformation of the universal enveloping
algebra of a n-dimensional Lie algebra g=Lie(G). However, we show how, by
starting from the generators of the underlying Lie bialgebra (g,\delta), the
analyticity in the deformation parameter(s) allows us to determine in a unique
way a set of n ``almost primitive'' basic objects in U_q(g), that could be
properly called the ``quantum algebra generators''. So, the analytical
prolongation (g_q,\Delta) of the Lie bialgebra (g,\delta) is proposed as the
appropriate local structure of G_q. Besides, as in this way (g,\delta) and
U_q(g) are shown to be in one-to-one correspondence, the classification of
quantum groups is reduced to the classification of Lie bialgebras. The su_q(2)
and su_q(3) cases are explicitly elaborated.Comment: 16 pages, 0 figures, LaTeX fil
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