arXiv subjects
Publications and source records attributed to H. Montani.
We study integrable systems on double Lie algebras in absence of Ad-invariant bilinear form by passing to the semidirect product with the $τ$-representation. We show that in this stage a natural Ad-invariant bilinear form does exist, allowing for a straightforward application of the AKS theory, and giving rise to Manin triple structure, thus bringing the problem to the realm of Lie bialgebras and Poisson-Lie groups.
We study the restriction to a family of second class constrained submanifolds in the cotangent bundle of a double Lie group, equipped with a 2-cocycle extended symplectic form, building the corresponding Dirac brackets. It is shown that, for a 2-cocycle vanishing on each isotropic subspaces of the associated Manin triple, the Dirac bracket contains no traces of the cocycle. We also investigate the restriction of the left translation action of the double Lie group on its cotangent bundle, where it fails in to be a symmetry a canonical transformation. However, the hamiltonian symmetry is restored on some special submanifolds. The main application is on loop groups, showing that a WZNW-type model on the double Lie group with a quadratic Hamilton function in the momentum maps associated with the left translation action on the cotangent bundle with the canonical symplectic form, restricts to a collective system on some special submanifolds. There, the lagrangian version coincides with so called Poisson-Lie $σ$-model.
We study integrable systems on the semidirect product of a Lie group and its Lie algebra as the representation space of the adjoint action. Regarding the tangent bundle of a Lie group as phase space endowed with this semidirect product Lie group structure, we construct a class of symplectic submanifolds equipped with a Dirac bracket on which integrable systems (in the Adler-Kostant-Symes sense) are naturally built through collective dynamics. In doing so, we address other issues as factorization, Poisson-Lie structures and dressing actions. We show that the procedure becomes recursive for some particular Hamilton functions, giving rise to a tower of nested integrable systems.
We construct fermionic Lagrangians with anisotropic scaling z=2, the natural counterpart of the usual z=2 Lifshitz field theories for scalar fields. We analyze the issue of chiral symmetry, construct the Noether axial currents and discuss the chiral anomaly giving explicit results for two-dimensional case. We also exploit the connection between detailed balance and the dynamics of Lifshitz theories to find different z=2 fermionic Lagrangians and construct their supersymmetric extensions.
In this work we study some symplectic submanifolds in the cotangent bundle of a factorizable Lie group defined by second class constraints. By applying the Dirac method, we study many issues of these spaces as fundamental Dirac brackets, symmetries, and collective dynamics. This last item allows to study integrability as inherited from a system on the whole cotangent bundle, leading in a natural way to the AKS theory for integrable systems.
We study the deep connection between integrable models and Poisson-Lie T-duality working on a finite dimensional example constructed on SL(2,C) and its Iwasawa factors SU(2) and B. We shown the way in which Adler-Kostant-Symes theory and collective dynamics combine to solve the equivalent systems from solving the factorization problem of an exponential curve in SL(2,C). It is shown that the Toda system embraces the dynamics of the systems on SU(2) and B.
We describe a general framework for studying duality between different phase spaces which share the same symmetry group $\mathrm{H}$. Solutions corresponding to collective dynamics become dual in the sense that they are generated by the same curve in $\mathrm{H}$. Explicit examples of phase spaces which are dual with respect to a common non trivial coadjoint orbit $\mathcal{O}_{c,0}(\mathbfα,1) \subset\mathfrak{h}^{\ast}$ are constructed on the cotangent bundles of the factors of a double Lie group $\mathrm{H}=\mathrm{N}\Join\mathrm{N}^{\ast}$. In the case $\mathrm{H}=LD$, the loop group of a Drinfeld double Lie group $D$, a hamiltonian description of Poisson-Lie T-duality for non trivial monodromies and its relation with non trivial coadjoint orbits is obtained.
We carry out a Hamiltonian analysis of Poisson-Lie T-duality based on the loop geometry of the underlying phases spaces of the dual sigma and WZW models. Duality is fully characterized by the existence of equivariant momentum maps on the phase spaces such that the reduced phase space of the WZW model and a pure central extension coadjoint orbit work as a bridge linking both the sigma models. These momentum maps are associated to Hamiltonian actions of the loop group of the Drinfeld double on both spaces and the duality transformations are explicitly constructed in terms of these actions. Compatible dynamics arise in a general collective form and the resulting Hamiltonian description encodes all known aspects of this duality and its generalizations.
Adapting the idea of twisted tensor products to the category of finitely generated algebras, we define on its opposite, the category QLS of quantum linear spaces, a family of objects hom(B,A)^{op}, one for each pair A^{op},B^{op} there, with analogous properties to its internal Hom ones, but representing spaces of transformations whose coordinate rings hom(B,A) and the ones of their respective domains B^{op} do not commute among themselves. The mentioned non commutativity is controlled by a collection of twisting maps τ_{A,B}. We show that the (bi)algebras end(A)=hom(A,A), under certain circumstances, are 2-cocycle twistings of the quantum semigroups end(A) in the untwisted case. This fact generalizes the twist equivalence (at a semigroup level) between, for instance, the quantum groups GL_{q}(n) and their multiparametric versions GL_{q,ϕ}(n).
Given a family of monodromy matrices {T_u; u=0,1,...,K-1} corresponding to integrable anisotropic vertex models of A_{(n_u)-1}-type, we build up a related mixed vertex model by means of glueing the lattices on which they are defined, in such a way that integrability property is preserved. Algebraically, the glueing process is implemented through one dimensional representations of rectangular matrix algebras A(R_p,R_q), namely, the `glueing matrices' zeta_u. Here R_n indicates the Yang-Baxter operator associated to the standard Hopf algebra deformation of the simple Lie algebra A_{n-1}. We show there exists a pseudovacuum subspace with respect to which algebraic Bethe ansatz can be applied. For each pseudovacuum vector we have a set of nested Bethe ansatz equations identical to the ones corresponding to an A_{m-1} quasi-periodic model, with m equal to the minimal range of involved glueing matrices.
We consider a twisted version of quantum groups corepresentations. This generalization amounts to include in the theory the case where quantum space coordinates and its endomorphism matrix entries belong to a non-commutative quadratic algebra.
We develop a Lagrangian approach for constructing a symplectic structure for singular systems. It gives a simple and unified framework for understanding the origin of the pathologies that appear in the Dirac-Bergmann formalism, and offers a more general approach for a symplectic formalism, even when there is no Hamiltonian in a canonical sense. We can thus overcome the usual limitations of the canonical quantization, and perform an algebraically consistent quantization for a more general set of Lagrangian systems.
We study a suitable q-deformed version of the Moriya's superexchange interaction theory by means of its underlying quantum group structure. We show that the one-dimensional chain case is associated with the non-standard quantum group $GL_{pq}(2) $, evidencing the integrability structure of the system. This biparametric deformation of $GL(2,{\bf C}) $ arise as a twisting of $GL_{q}(2) $ and it match exactly the local rotation appearing in the Shekhtman's work \cite{Sh}. This allow us to express the frustration condition in terms of this twisting, also showing that effect of the Moriya's vector amounts to a twisting of the boundary condition.
We built up a explicit realization of (0+1)-dimensional q-deformed superspace coordinates as operators on standard superspace. A q-generalization of supersymmetric transformations is obtained, enabling us to introduce scalar superfields and a q-supersymmetric action. We consider a functional integral based on this action. Integration is implemented, at the level of the coordinates and at the level of the fields, as traces over the corresponding representation spaces. Evaluation of these traces lead us to standard functional integrals. The generation of a mass term for the fermion field leads, at this level, to an explicitely broken version of supersymmetric quantum mechanics.
We carry out a generalization of quantum group co-representations in order to encode in this structure those cases where non-commutativity between endomorphism matrix entries and quantum space coordinates happens.
We construct the chiral Wess-Zumino term as a solution for the Batalin-Vilkovisky master equation for anomalous two-dimensional gauge theories, working in an extended field-antifield space, where the gauge group elements are introduced as additional degrees of freedom. We analyze the Abelian and the non-Abelian cases, calculating in both cases the BRST generator in order to show the physical equivalence between this chiral solution for the master equation and the usual (non-chiral) one.