Todor Popov

• PhD
• Associate Professor at Bulgarian Academy of Sciences & American University in Bulgaria

The Landau problem and harmonic oscillator in the plane share a Hilbert space that carries the structure of Dirac’s remarkable so(2,3) representation. We show that the orthosymplectic algebra osp(1|4) is the spectrum generating algebra for the Landau problem and, hence, for the 2D isotropic harmonic oscillator. The 2D harmonic oscillator is in dual...

The color neutrality of hadrons is interpreted as an expression of conformal symmetry of strong interaction, the latter being signaled through the detected “walking” at low transferred momenta, \(\lim _{Q^2\rightarrow 0}\alpha _s(Q^2)/\pi \rightarrow 1 \), of the strong coupling toward a fixed value (\(\alpha _s \) “freezing”). The fact is that con...

The color neutrality of hadrons is interpreted as an expression of conformal symmetry of strong interaction, the latter being signaled through the detected "walking" at low transferred momenta, $\lim_{Q^2\to 0}\alpha_s(Q^2)/\pi\to 1 $, of the strong coupling toward a fixed value ($\alpha_s $ "freezing" ). The fact is that conformal symmetry admits...

We show that the dynamical group of an electron in a constant magnetic feld is the group of symplectomorphisms Sp(4, R). It is generated by the spinorial realization of the conformal algebra so(2,3) considered in Dirac’s seminal paper ”A Remarkable Representation of the 3 + 2 de Sitter Group”. The symplectic group Sp (4,R) is the double covering of...

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 Kähler polarization of the plane has been used for the geometric quantization. Under the assumption of quasi-periodicity of the wavefunction, the Zak’s magnetic translations in the...

A bstract The topology of closed manifolds forces interacting charges to appear in pairs. We take advantage of this property in the setting of the conformal boundary of AdS 5 spacetime, topologically equivalent to the closed manifold S ¹ × S ³ , by considering the coupling of two massless opposite charges on it. Taking the interaction potential as...

The topology of closed manifolds forces interacting charges to appear in pairs. We take advantage of this property in the setting of the conformal boundary of $\mathrm{AdS}_5$ spacetime, topologically equivalent to the closed manifold $S^1\times S^3$, by considering the coupling of two massless opposite charges on it. Taking the interaction potenti...

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 Brava...

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 relation...

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 relation...

It has been realized long ago that the 15 dimensional conformal group, extending the Poincaré group with dilations and conformal inversions is a symmetry of the Maxwell equations. Conformal action yields a special mass-zero representation preserving the causal structure of Minkowski spacetime. On the other hand in the seventies in the works of Baru...

We show that the causal automorphisms of the Minkowski space-time and the dynamical group of the hydrogen atom stem both from the Tits–Kantor–Koecher conformal construction for a Jordan algebra.

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 closel...

We review the Robinson–Schensted–Knuth correspondence in the light of the quantum Schur–Weyl duality. The quantum plactic algebra is defined to be a Schur functor mapping a tower of left modules of Hecke algebras into a tower of \({U_q{\mathfrak {gl}}}\)-modules. The functions on the quantum group carry a \({U_q{\mathfrak {gl}}}\)-bimodule structur...

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 a...

We consider the free 2-step nilpotent Lie algebra and its cohomology ring. The homotopy transfer induces a homotopy commutative algebra on its cohomology ring which we describe. We show that this cohomology is generated in degree 1 as C-infinity-algebra only by the induced binary and ternary operations.

We consider the free 2-nilpotent graded Lie algebra $\mathfrak{g}$ generated in degree one by a finite dimensional vector space V. We recall the beautiful result that the cohomology $H^ \cdot \left( {\mathfrak{g},\mathbb{K}} \right)$ of $\mathfrak{g}$ with trivial coefficients carries a GL(V)-representation having only the Schur modules V wit...

Quantum Lie algebras related to multi-parametric Drinfeld-Jimbo $R$-matrices of type $GL(m|n)$ are classified.

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.

The homotopy transfer theorem due to Tornike Kadeishvili induces the structure of a homotopy commutative algebra, or C ∞ -algebra, on the cohomology of the free 2-nilpotent Lie algebra. The latter C ∞ -algebra is shown to be generated in degree one by the binary and the ternary operations.

We explore the Fock spaces of the parafermionic algebra closed by the creation and annihilation operators 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 hel...

We review the Poirier-Reutenauer Hopf structure on Standard Young Tableaux and show that it is a distinguished member of a family of Hopf structures. The family in question is related to deformed parastatistics.

We describe a quantum Lie algebra based on the Cremmer-Gervais R-matrix. The algebra arises upon a restriction of an infinite-dimensional quantum Lie algebra.

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...

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...

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 class...

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 mon...

We replace the ice Ansatz on matrix solutions of the Yang-Baxter equation by a weaker condition which we call "rime". Rime solutions include the standard Drinfeld-Jimbo R-matrix. Solutions of the Yang--Baxter equation within the rime Ansatz which are maximally different from the standard one we call "strict rime". A strict rime non-unitary solution...

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...

We consider the algebras spanned by the creation parafermionic and parabosonic operators which give rise to generalized parastatistics Fock spaces. The basis of such a generalized Fock space can be labelled by Young tableaux which are combinatorial objects. By means of quantum deformations a nice combinatorial structure of the algebra of the placti...

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 commutat...

Manin associated to a quadratic algebra (quantum space) the quantum matrix group of its automorphisms. This Note aims to demonstrate that Manin's construction can be extended for quantum spaces which are non-quadratic homogeneous algebras. The Artin–Schelter classification of regular algebras of global dimension three contains two types of algebra:...

As shown in [3] the concepts developed for quadratic algebras such as dual algebra, Koszul complexes and Koszul algebra [15, 13] have counterparts for homogeneous algebras of any order N(N ≥ 2). Here we apply these generalized notions on two particular types of cubic algebras. The first one is the parafermionic(parabosonic) algebra generated (only)...

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 (paraboson...

We present some new results on the rational solutions of the Knizhnik-Zamolodchikov (KZ) equation for the four-point conformal blocks of isospin I primary fields in the SU (2) k Wess-Zumino-Novikov-Witten (WZNW) model. The rational solutions corresponding to integrable representations of the affine algebra (2) k have been classified in [#!MST!#,...