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Todor Popov

Publications and source records attributed to Todor Popov.

15 recordsLinked to original sources
arXiv2026

Landau levels via Jordan superalgebras

The goal of this note is to show that Jordan algebras and superalgebras provide an elegant and concise language for formulating quantum mechanical problems with inherent (super)conformal symmetry. The superconformal symmetries of the quantum MICZ-Kepler model and its dual oscillator realization in ${\mathbb R}^2$ are reviewed through the lens of the Tits-Kantor-Koecher correspondence: Kaplansky ${\mathfrak J} {\mathbb R}^{1|2}$ and Exceptional ${\mathfrak J} F^{6|4}$ Jordan superalgebras provide a natural framework for reconstructing (variously extended) superconformal algebras hidden in the Landau levels of an electron in an external magnetic field.

hep-th↗
arXiv2025

A remarkable dynamical symmetry of the Landau problem

We show that the dynamical group of an electron in a constant magnetic field is the group of symplectomorphisms $Sp(4,\mathbb{R})$. It is generated by the spinorial realization of the conformal algebra $\mathfrak{so}(2,3)$ considered in Dirac's seminal paper "A Remarkable Representation of the 3 + 2 de Sitter Group". The symplectic group $Sp(4,\mathbb{R})$ is the double covering of the conformal group $SO(2,3)$ of 2+1 dimensional Minkowski spacetime which is in turn the dynamical group of a hydrogen atom in 2 space dimensions. The Newton-Hooke duality between the 2D hydrogen atom and the Landau problem is explained via the Tits-Kantor-Koecher construction of the conformal symmetries of the Jordan algebra of real symmetric $2 \times 2$ matrices. The connection between the Landau problem and the 3D hydrogen atom is elucidated by the reduction of a Dirac spinor to a Majorana one in the Kustaanheimo-Stiefel spinorial regularization.

math-ph↗
arXiv2021

Bloch waves and Non-commutative Tori of Magnetic Translations

We review the Landau problem of an electron in a constant uniform magnetic field. The magnetic translations are the invariant transformations of the free Hamiltonian. A K\"ahler polarization of the plane has been used for the geometric quantization. Under the assumption of quasi-periodicity of the wavefunction the magnetic translations in the Bravais lattice generate a non-commutative quantum torus. We concentrate on the case when the magnetic flux density is a rational number. The Bloch wavefunctions form a finite-dimensional module of the noncommutative torus of magnetic translations as well as of its commutant which is the non-commutative torus of magnetic translation in the dual Bravais lattice. The bi-module structure of the Bloch waves is shown to be the connecting link between two Morita equivalent non-commutative tori.

math-ph↗
arXiv2019

Quantum Diagonal Algebra and Pseudo-Plactic Algebra

The subalgebra of diagonal elements of a quantum matrix group has been conjectured by Daniel Krob and Jean-Yves Thibon to be isomorphic to a cubic algebra, coined the quantum pseudo-plactic algebra. We present a functorial approach to the conjecture through the quantum Schur-Weyl duality between the quantum group and the Hecke algebra. The relations of the quantum diagonal subalgebra are found to be the image of the braid relations of the underlying Hecke algebra by an appropriate Schur functor which gives a straightforward proof of the conjecture.

math.QA↗
arXiv2017

Pre-Plactic Algebra and Snakes

We study a factor Hopf algebra $\mathfrak{PP}$ of the Malvenuto-Reutenauer convolution algebra of functions on symmetric groups ${\mathfrak{S}}=\oplus_{n\geq 0} \mathbb C[{\mathfrak{S}}_n] $ that we coined pre-plactic algebra. The pre-plactic algebra admits the Poirier-Reutenauer algebra based on Standard Young Tableaux as a factor and it is closely related to the quantum pseudo-plactic algebra introduced by Krob and Thibon in the non-commutative character theory of quantum group comodules. The connection between the quantum pseudo-plactic algebra and the pre-plactic algebra is similar to the connection between the Lascoux-Schützenberger plactic algebra and the Poirier-Reutenauer algebra. We show that the dimensions of the pre-plactic algebra are given by the numbers of alternating permutations (coined snakes after V.I. Arnold). Pre-plactic algebra is instrumental in calculating the Hilbert-Poincaré series of the quantum pseudo-plactic algebra.

math.QA↗
arXiv2014

Parafermionic algebras, their modules and cohomologies

We explore the Fock spaces of the parafermionic algebra introduced by H.S. Green. Each parafermionic Fock space allows for a free minimal resolution by graded modules of the graded 2-step nilpotent subalgebra of the parafermionic creation operators. Such a free resolution is constructed with the help of a classical Kostant's theorem computing Lie algebra cohomologies of the nilpotent subalgebra with values in the parafermionic Fock space. The Euler-Poincaré characteristics of the parafermionic Fock space free resolution yields some interesting identities between Schur polynomials. Finally we briefly comment on parabosonic and general parastatistics Fock spaces.

math-ph↗
arXiv2012

Young tableaux and homotopy commutative algebras

A homotopy commutative algebra, or $C_{\infty}$-algebra, is defined via the Tornike Kadeishvili homotopy transfer theorem on the vector space generated by the set of Young tableaux with self-conjugated Young diagrams. We prove that this $C_{\infty}$-algebra is generated in degree 1 by the binary and the ternary operations.

math.QA↗
arXiv2008

Parastatistics Algebra, Young Tableaux and the Super Plactic Monoid

The parastatistics algebra is a superalgebra with (even) parafermi and (odd) parabose creation and annihilation operators. The states in the parastatistics Fock-like space are shown to be in one-to-one correspondence with the Super Semistandard Young Tableaux (SSYT) subject to further constraints. The deformation of the parastatistics algebra gives rise to a monoidal structure on the SSYT which is a super-counterpart of the plactic monoid.

math-ph↗
arXiv2007

On rime Ansatz

The ice Ansatz on matrix solutions of the Yang-Baxter equation is weakened to a condition which we call rime. Generic rime solutions of the Yang-Baxter equation are described. We prove that the rime non-unitary (respectively, unitary) R-matrix is equivalent to the Cremmer-Gervais (respectively, boundary Cremmer-Gervais) solution. Generic rime classical r-matices satisfy the (non-)homogeneous associative classical Yang-Baxter equation.

math.QA↗
arXiv2007

Parastatistics Algebra and Super Semistandard Young Tableaux

We consider the parastatistics algebra with both parabosonic and parafermionic operators and show that the states in the universal parastatistics Fock space are in bijection with the Super Semistandard Young Tableaux (SSYT). Using deformation of the parastatistics algebra we get a monoid structure on SSYT which is a super version of the plactic monoid.

math-ph↗
arXiv2006

Automorphisms of Regular Algebras

Manin associated to a quadratic algebra (quantum space) the quantum matrix group of its automorphisms. This Talk aims to demonstrate that Manin's construction can be extended for quantum spaces which are non-quadratic homogeneous algebras. Here given a regular Artin-Schelter algebra of dimension 3 we construct the quantum group of its symmetries, i.e., the Hopf algebra of its automorphisms. For quadratic Artin-Schelter algebras these quantum groups are contained in the the classification of the GL(3) quantum matrix groups due to Ewen and Ogievetsky. For cubic Artin-Schelter algebras we obtain new quantum groups which are automorphisms of cubic quantum spaces.

math.QA↗
arXiv2006

Hopf Structure and Green Ansatz of Deformed Parastatistics Algebras

Deformed parabose and parafermi algebras are revised and endowed with Hopf structure in a natural way. The noncocommutative coproduct allows for construction of parastatistics Fock-like representations, built out of the simplest deformed bose and fermi representations. The construction gives rise to quadratic algebras of deformed anomalous commutation relations which define the generalized Green ansatz.

math-ph↗
arXiv2002

Homogeneous algebras, statistics and combinatorics

After some generalities on homogeneous algebras, we give a formula connecting the Poincaré series of a homogeneous algebra with the homology of the corresponding Koszul complex generalizing thereby a standard result for quadratic algebras. We then investigate two particular types of cubic algebras: The first one called the parafermionic (parabosonic) algebra is the algebra generated by the creation operators of the universal fermionic (bosonic) parastatics with $D$ degrees of freedom while the second is the plactic algebra that is the algebra of the plactic monoid with entries in $\{1,2,..., D\}$. In the case D=2 we describe the relations with the cubic Artin-Schelter algebras. It is pointed out that the natural action of GL(2) on the parafermionic algebra for D=2 extends as an action of the quantum group $GL_{p,q}(2)$ on the generic cubic Artin-Schelter regular algebra of type $S_1$; $p$ and $q$ being related to the Artin-Schelter parameters. It is claimed that this has a counterpart for any integer $D\geq 2$.

math.QA↗
arXiv2001

On the rational solutions of the su(2)_k Knizhnik-Zamolodchikov equation

We present some new results on the rational solutions of the Knizhnik-Zamolodchikov equation for the four-point conformal blocks of isospin I primary fields in the SU(2)_k Wess-Zumino-Novikov-Witten model. The rational solutions corresponding to integrable representations of the affine algebra su(2)_k have been classified by Michel, Stanev and Todorov; provided that the conformal dimension is an integer, they are in one-to-one correspondence with the local extensions of the chiral algebra. Here we give another description of these solutions as specific braid-invariant combinations of the so called regular basis and display a new series of rational solutions for isospins I = k+1 corresponding to non-integrable representations of the affine algebra.

hep-th↗