arXiv subjects

Laszlo Feher

Publications and source records attributed to Laszlo Feher.

7 recordsLinked to original sources

arXiv2022

A note on quadratic Poisson brackets on ${\mathrm{gl}}(n,{\mathbb{R}})$ related to Toda lattices

It is well known that the compatible linear and quadratic Poisson brackets of the full symmetric and of the standard open Toda lattices are restrictions of linear and quadratic $r$-matrix Poisson brackets on the associative algebra ${\mathrm{gl}}(n,{\mathbb{R}})$. We here show that the quadratic bracket on ${\mathrm{gl}}(n,{\mathbb{R}})$, corresponding to the $r$-matrix defined by the splitting of ${\mathrm{gl}}(n,{\mathbb{R}})$ into the direct sum of the upper triangular and orthogonal Lie subalgebras, descends by Poisson reduction from a quadratic Poisson structure on the cotangent bundle $T^*{\mathrm{GL}}(n,{\mathbb{R}})$. This complements the interpretation of the linear $r$-matrix bracket as a reduction of the canonical Poisson bracket of the cotangent bundle.

math-ph↗

arXiv1997

KdV type systems and W-algebras in the Drinfeld-Sokolov approach

The generalized Drinfeld-Sokolov construction of KdV systems is reviewed in the case of an arbitrary affine Lie algebra paying particular attention to Hamiltonian aspects and $\W$-algebras. Some extensions of known results as well as a new interpretation of the construction are also presented.

hep-th↗

arXiv1997

Extended matrix Gelfand-Dickey hierarchies: reduction to classical Lie algebras

The Drinfeld-Sokolov reduction method has been used to associate with $gl_n$ extensions of the matrix r-KdV system. Reductions of these systems to the fixed point sets of involutive Poisson maps, implementing reduction of $gl_n$ to classical Lie algebras of type $B, C, D$, are here presented. Modifications corresponding, in the first place to factorisation of the Lax operator, and then to Wakimoto realisations of the current algebra components of the factorisation, are also described.

solv-int↗

arXiv1996

Wakimoto realizations of current algebras: an explicit construction

A generalized Wakimoto realization of $\widehat{\cal G}_K$ can be associated with each parabolic subalgebra ${\cal P}=({\cal G}_0 +{\cal G}_+)$ of a simple Lie algebra ${\cal G}$ according to an earlier proposal by Feigin and Frenkel. In this paper the proposal is made explicit by developing the construction of Wakimoto realizations from a simple but unconventional viewpoint. An explicit formula is derived for the Wakimoto current first at the Poisson bracket level by Hamiltonian symmetry reduction of the WZNW model. The quantization is then performed by normal ordering the classical formula and determining the required quantum correction for it to generate $\widehat{\cal G}_K$ by means of commutators. The affine-Sugawara stress-energy tensor is verified to have the expected quadratic form in the constituents, which are symplectic bosons belonging to ${\cal G}_+$ and a current belonging to ${\cal G}_0$. The quantization requires a choice of special polynomial coordinates on the big cell of the flag manifold $P\backslash G$. The effect of this choice is investigated in detail by constructing quantum coordinate transformations. Finally, the explicit form of the screening charges for each generalized Wakimoto realization is determined, and some applications are briefly discussed.

hep-th↗

arXiv1996

Regularization of Toda lattices by Hamiltonian reduction

The Toda lattice defined by the Hamiltonian $H={1\over 2} \sum_{i=1}^n p_i^2 + \sum_{i=1}^{n-1} ν_i e^{q_i-q_{i+1}}$ with $ν_i\in \{ \pm 1\}$, which exhibits singular (blowing up) solutions if some of the $ν_i=-1$, can be viewed as the reduced system following from a symmetry reduction of a subsystem of the free particle moving on the group $G=SL(n,\Real )$. The subsystem is $T^*G_e$, where $G_e=N_+ A N_-$ consists of the determinant one matrices with positive principal minors, and the reduction is based on the maximal nilpotent group $N_+ \times N_-$. Using the Bruhat decomposition we show that the full reduced system obtained from $T^*G$, which is perfectly regular, contains $2^{n-1}$ Toda lattices. More precisely, if $n$ is odd the reduced system contains all the possible Toda lattices having different signs for the $ν_i$. If $n$ is even, there exist two non-isomorphic reduced systems with different constituent Toda lattices. The Toda lattices occupy non-intersecting open submanifolds in the reduced phase space, wherein they are regularized by being glued together. We find a model of the reduced phase space as a hypersurface in ${\Real}^{2n-1}$. If $ν_i=1$ for all $i$, we prove for $n=2,3,4$ that the Toda phase space associated with $T^*G_e$ is a connected component of this hypersurface. The generalization of the construction for the other simple Lie groups is also presented.

hep-th↗

arXiv1996

An explicit construction of Wakimoto realizations of current algebras

It is known from a work of Feigin and Frenkel that a Wakimoto type, generalized free field realization of the current algebra $\widehat{\cal G}_k$ can be associated with each parabolic subalgebra ${\cal P}=({\cal G}_0+{\cal G}_+)$ of the Lie algebra ${\cal G}$, where in the standard case ${\cal G}_0$ is the Cartan and ${\cal P}$ is the Borel subalgebra. In this letter we obtain an explicit formula for the Wakimoto realization in the general case. Using Hamiltonian reduction of the WZNW model, we first derive a Poisson bracket realization of the ${\cal G}$-valued current in terms of symplectic bosons belonging to ${\cal G}_+$ and a current belonging to ${\cal G}_0$. We then quantize the formula by determining the correct normal ordering. We also show that the affine-Sugawara stress-energy tensor takes the expected quadratic form in the constituents.

hep-th↗

arXiv1995

Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction

The $p\times p$ matrix version of the $r$-KdV hierarchy has been recently treated as the reduced system arising in a Drinfeld-Sokolov type Hamiltonian symmetry reduction applied to a Poisson submanifold in the dual of the Lie algebra $\widehat{gl}_{pr}\otimes {\Complex}[λ, λ^{-1}]$. Here a series of extensions of this matrix Gelfand-Dickey system is derived by means of a generalized Drinfeld-Sokolov reduction defined for the Lie algebra $\widehat{gl}_{pr+s}\otimes {\Complex}[λ,λ^{-1}]$ using the natural embedding $gl_{pr}\subset gl_{pr+s}$ for $s$ any positive integer. The hierarchies obtained admit a description in terms of a $p\times p$ matrix pseudo-differential operator comprising an $r$-KdV type positive part and a non-trivial negative part. This system has been investigated previously in the $p=1$ case as a constrained KP system. In this paper the previous results are considerably extended and a systematic study is presented on the basis of the Drinfeld-Sokolov approach that has the advantage that it leads to local Poisson brackets and makes clear the conformal ($\cal W$-algebra) structures related to the KdV type hierarchies. Discrete reductions and modified versions of the extended $r$-KdV hierarchies are also discussed.

hep-th↗