Narrow Results

In this paper, we investigate the property of schematicness introduced by Van Oystaeyen and Willaert [F. Van Oystaeyen and L. Willaert, Grothendieck topology, coherent sheaves and Serre’s theorem for schematic algebras, J. Pure Appl. Algebra104(1–3) (1995) 109–122] in the setting of skew Ore polynomials of higher order generated by homogenous quadratic relations defined by Golovashkin and Maksimov [A. V. Golovashkin and V. M. Maksimov, Skew Ore polynomials of higher orders generated by homogeneous quadratic relations, Russian Math. Surveys53(2) (1998) 384–386]. We establish sufficient conditions to guarantee whether some of these algebras are schematic or not.

The Dirac equation with scalar and vector potentials of equal magnitude is considered. For the two-dimensional harmonic oscillator superintegrable potential, the superintegrable potentials of E₈ (case (3b)), S₄ and S₂, the Schrödinger-like equations are studied. The quadratic algebras of these quasi-Hamiltonians are derived. By using the realization of the quadratic algebras in a deformed oscillator algebra, the structure function and the energy eigenvalues are obtained.

The spin and pseudospin symmetries in the Dirac Hamiltonian are investigated in the presence of the Hartmann and the Higgs oscillator superintegrable potentials. The Pauli-Dirac representation is used in the Dirac equation with scalar and vector potentials of equal magnitude. Then, the Dirac equation is reduced to a Schrödinger-like equation. The symmetry algebras of the Schrödinger-like equation corresponding to the superintegrable potentials are represented. Also, the associated irreducible representations are shown by means of the quadratic algebras. Finally, the relativistic energy spectra of the Hartmann and the Higgs oscillator superintegrable potentials are calculated.

We consider algebras over a field K with a presentation K<x₁,…,x_n:R>, where R consists of square-free relations of the form x_ix_j=x_kx_l with every monomial x_ix_j, i≠j, appearing in one of the relations. The description of all four generated algebras of this type that satisfy a certain non-degeneracy condition is given. The structure of one of these algebras is described in detail. In particular, we prove that the Gelfand–Kirillov dimension is one while the algebra is noetherian PI and semiprime in case when the field K has characteristic zero. All minimal prime ideals of the algebra are described. It is also shown that the underlying monoid is a semilattice of cancellative semigroups and its structure is described. For any positive integer m, we construct non-degenerate algebras of the considered type on 4^m generators that have Gelfand–Kirillov dimension one and are semiprime noetherian PI algebras.

Let $𝔽$ denote a field with $\mathrm{\text{char}}𝔽\ne 2$. The Racah algebra $\Re$ is the unital associative $𝔽$-algebra defined by generators and relations in the following way. The generators are $A$, $B$, $C$, $D$. The relations assert that

The set $\{{\mathrm{\Omega }}_A,{\mathrm{\Omega }}_B,{\mathrm{\Omega }}_C\}$ is invariant under a faithful $D_6$-action on $\Re$. Moreover, we show that any Casimir element $\mathrm{\Omega }$ is algebraically independent over $ℭ$; if $\mathrm{\text{char}}𝔽=0$, then the center of $\Re$ is $ℭ[\mathrm{\Omega }]$.

The two-dimensional Dirac equation with spin and pseudo-spin symmetries is investigated in the presence of the maximally superintegrable potentials. The integrals of motion and the quadratic algebras of the superintegrable quantum $E{3}^{\prime }$, anisotropic oscillator and the Holt potentials are studied. The corresponding Casimir operators and the structure functions of the mentioned superintegrable systems are found. Also, we obtain the relativistic energy spectra of the corresponding superintegrable systems. Finally, the relativistic energy eigenvalues of the generalized Yang–Coulomb monopole (YCM) superintegrable system (a $SU(2)$ non-Abelian monopole) are calculated by the energy spectrum of the eight-dimensional oscillator which is dual to the former system by Hurwitz transformation.

Let S be a scheme such that 2 is not a zero divisor. In this paper, we address the following question: given a quadratic algebra over S, how can we parametrize its Picard group in terms of quadratic forms? In 2011, Wood established a set-theoretical bijection between isomorphism classes of primary binary quadratic forms over S and isomorphism classes of pairs $({𝒞},{ℳ})$ where ${𝒞}$ is a quadratic algebra over S and ${ℳ}$ is an invertible ${𝒞}$-module. Unexpectedly, examples suggest that a refinement of Wood’s bijection is needed in order to parametrize Picard groups. This is why we start by classifying quadratic algebras over S; this is achieved by using two invariants, the discriminant and the parity. Extending the notion of orientation of quadratic algebras to the non-free case is another key step, eventually leading us to the desired parametrization. All along the paper, we illustrate various notions and obstructions with a wide range of examples.