714 research outputs found

### hbar-(Yangian) Deformation of Miura Map and Virasoro Algebra

An hbar-deformed Virasoro Poisson algebra is obtained using the Wakimoto
realization of the Sugawara operator for the Yangian double DY_\hbar(sl_2)_c at
the critical level c=-2.Comment: LaTeX file, 43kb, No Figures. Serious misprints corrected, one more
reference to E. Frenkel adde

### Reconstruction of universal Drinfeld twists from representations

Universal Drinfeld twists are inner automorphisms which relate the coproduct
of a quantum enveloping algebra to the coproduct of the undeformed enveloping
algebra. Even though they govern the deformation theory of classical symmetries
and have appeared in numerous applications, no twist for a semi-simple quantum
enveloping algebra has ever been computed. It is argued that universal twists
can be reconstructed from their well known representations. A method to
reconstruct an arbitrary element of the enveloping algebra from its irreducible
representations is developed. For the twist this yields an algebra valued
generating function to all orders in the deformation parameter, expressed by a
combination of basic and ordinary hypergeometric functions. An explicit
expression for the universal twist of su(2) is given up to third order.Comment: 24 page

### On reflection algebras and twisted Yangians

It is known that integrable models associated to rational $R$ matrices give
rise to certain non-abelian symmetries known as Yangians. Analogously
`boundary' symmetries arise when general but still integrable boundary
conditions are implemented, as originally argued by Delius, Mackay and Short
from the field theory point of view, in the context of the principal chiral
model on the half line. In the present study we deal with a discrete quantum
mechanical system with boundaries, that is the $N$ site $gl(n)$ open quantum
spin chain. In particular, the open spin chain with two distinct types of
boundary conditions known as soliton preserving and soliton non-preserving is
considered. For both types of boundaries we present a unified framework for
deriving the corresponding boundary non-local charges directly at the quantum
level. The non-local charges are simply coproduct realizations of particular
boundary quantum algebras called `boundary' or twisted Yangians, depending on
the choice of boundary conditions. Finally, with the help of linear
intertwining relations between the solutions of the reflection equation and the
generators of the boundary or twisted Yangians we are able to exhibit the
symmetry of the open spin chain, namely we show that a number of the boundary
non-local charges are in fact conserved quantitiesComment: 16 pages LATEX, clarifications and generalizations added, typos
corrected. To appear in JM

### Quantum Affine Lie Algebras, Casimir Invariants and Diagonalization of the Braid Generator

Let $U_q(\hat{\cal G})$ be an infinite-dimensional quantum affine Lie
algebra. A family of central elements or Casimir invariants are constructed and
their eigenvalues computed in any integrable irreducible highest weight
representation. These eigenvalue formulae are shown to absolutely convergent
when the deformation parameter $q$ is such that $|q|>1$. It is proven that the
universal R-matrix $R$ of $U_q(\hat{\cal G})$ satisfies the celebrated
conjugation relation $R^\dagger=TR$ with $T$ the usual twist map. As
applications, the braid generator is shown to be diagonalizable on arbitrary
tensor product modules of integrable irreducible highest weight $U_q(\hat{\cal
G})$-modules and a spectral decomposition formula for the braid generator is
obtained which is the generalization of Reshetikhin's and Gould's forms to the
present affine case. Casimir invariants acting on a specified module are also
constructed and their eigenvalues, again absolutely convergent for $|q|>1$,
computed by means of the spectral decomposition formula.Comment: 22 pages (many changes are made

### Topological Quantum Liquids with Quaternion Non-Abelian Statistics

Noncollinear magnetic order is typically characterized by a "tetrad" ground
state manifold (GSM) of three perpendicular vectors or nematic-directors. We
study three types of tetrad orders in two spatial dimensions, whose GSMs are
SO(3) = S^3/Z_2, S^3/Z_4, and S^3/Q_8, respectively. Q_8 denotes the
non-Abelian quaternion group with eight elements. We demonstrate that after
quantum disordering these three types of tetrad orders, the systems enter fully
gapped liquid phases described by Z_2, Z_4, and non-Abelian quaternion gauge
field theories, respectively. The latter case realizes Kitaev's non-Abelian
toric code in terms of a rather simple spin-1 SU(2) quantum magnet. This
non-Abelian topological phase possesses a 22-fold ground state degeneracy on
the torus arising from the 22 representations of the Drinfeld double of Q_8.Comment: 5 pages, 3 figure

### Twisting cocycles in fundamental representation and triangular bicrossproduct Hopf algebras

We find the general solution to the twisting equation in the tensor bialgebra
$T({\bf R})$ of an associative unital ring ${\bf R}$ viewed as that of
fundamental representation for a universal enveloping Lie algebra and its
quantum deformations. We suggest a procedure of constructing twisting cocycles
belonging to a given quasitriangular subbialgebra ${\cal H}\subset T({\bf R})$.
This algorithm generalizes Reshetikhin's approach, which involves cocycles
fulfilling the Yang-Baxter equation. Within this framework we study a class of
quantized inhomogeneous Lie algebras related to associative rings in a certain
way, for which we build twisting cocycles and universal $R$-matrices. Our
approach is a generalization of the methods developed for the case of
commutative rings in our recent work including such well-known examples as
Jordanian quantization of the Borel subalgebra of $sl(2)$ and the null-plane
quantized Poincar\'e algebra by Ballesteros at al. We reveal the role of
special group cohomologies in this process and establish the bicrossproduct
structure of the examples studied.Comment: 20 pages, LaTe

### Twisting 2-cocycles for the construction of new non-standard quantum groups

We introduce a new class of 2-cocycles defined explicitly on the generators
of certain multiparameter standard quantum groups. These allow us, through the
process of twisting the familiar standard quantum groups, to generate new as
well as previously known examples of non-standard quantum groups. In particular
we are able to construct generalisations of both the Cremmer-Gervais
deformation of SL(3) and the so called esoteric quantum groups of Fronsdal and
Galindo in an explicit and straightforward manner.Comment: 21 pages, AMSLaTeX, expanded introduction and a few other minor
corrections, to appear in JM

### Toroidal and level 0 U'_q(\hat{sl_{n+1}}) actions on U_q(\hat{gl_{n+1}}) modules

(1) Utilizing a Braid group action on a completion of U_q(\hat{sl_{n+1}}), an
algebra homomorphism from the toroidal algebra U_q(sl_{n+1,tor}) (n\ge 2) with
fixed parameter to a completion of U_q(\hat{gl_{n+1}}) is obtained.
(2) The toroidal actions by Saito induces a level 0 U'_q(\hat{sl_{n+1}})
action on level 1 integrable highest weight modules of U_q(\hat{sl_{n+1}}).
Another level 0 U'_q(\hat{sl_{n+1}}) action is defined by Jimbo, et al., in the
case n=1. Using the fact that the intertwiners of U_q(\hat{sl_{n+1}}) modules
are intertwiners of toroidal modules for an appropriate comultiplication, the
relation between these two level 0 U'_q(\hat{sl_{n+1}}) actions is clarified.Comment: Latex, 20 page

### q-Deforming Maps for Lie Group Covariant Heisenberg Algebras

We briefly summarize our systematic construction procedure of q-deforming
maps for Lie group covariant Weyl or Clifford algebras.Comment: latex file, 4 pages. Contribution to the proceedings of the 5th
Wigner Symposium. Slight modification

### Canonical and Lie-algebraic twist deformations of $\kappa$-Poincare and contractions to $\kappa$-Galilei algebras

We propose canonical and Lie-algebraic twist deformations of
$\kappa$-deformed Poincare Hopf algebra which leads to the generalized
$\kappa$-Minkowski space-time relations. The corresponding deformed
$\kappa$-Poincare quantum groups are also calculated. Finally, we perform the
nonrelativistic contraction limit to the corresponding twisted Galilean
algebras and dual Galilean quantum groups.Comment: 16 pages, no figures, v3: few changes provided - version for journal,
v2: submitted incidentally, v4: the page numbers for all references in
preprint version are provide

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