scholarly journals Two-Parameter Deformation of the Poincaré Algebra

1997 ◽  
Vol 12 (05) ◽  
pp. 891-901 ◽  
Author(s):  
A. Stern ◽  
I. Yakushin
Keyword(s):  
Quadratic Algebra ◽  
Commuting Operators ◽  
Poincaré Algebra ◽  
Two Parameter ◽  
Parameter Deformation

We examine a two-parameter (ℏ,λ) deformation of the Poincaré algebra which is covariant under the action of SL q(2,C). When λ → 0 it yields the Poincaré algebra, while in the ℏ → 0 limit we recover the classical quadratic algebra discussed previously in Refs. 1 and 2. The analogues of the Pauli–Lubanski vector w and Casimirs p2 and w2 are found and a set of mutually commuting operators is constructed.

scholarly journals TOWARDS CONSTRUCTING ONE-PARTICLE REPRESENTATIONS OF THE DEFORMED POINCARÉ ALGEBRA

1998 ◽  
Vol 13 (13) ◽  
pp. 2103-2121
Author(s):  
I. YAKUSHIN
Keyword(s):  
Energy Spectrum ◽  
Infinite Number ◽  
Massive Particle ◽  
Rest State ◽  
Commuting Operators ◽  
Poincaré Algebra ◽  
Complete Set ◽  
Two Parameter ◽  
Parameter Deformation

We give a method for obtaining states of massive particle representations of the two-parameter deformation of the Poincaré algebra proposed in Refs. 1–3. We discuss four procedures to generate eigenstates of a complete set of commuting operators starting from the rest state. One result of this work is the fact that upon deforming to the quantum Poincaré algebra, the rest state is split into an infinite number of states. Another result is that the energy spectrum of these states is discrete. Some curious residual degeneracy remains: there are states constructed by applying different operators to the rest state which nevertheless are indistinguishable by eigenvalues of all the observables in the algebra.

scholarly journals String theory at order α′2 and the generalized Bergshoeff-de Roo identification

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Stanislav Hronek ◽  
Linus Wulff
Keyword(s):  
Field Theory ◽  
String Theory ◽  
Strong Evidence ◽  
Degrees Of Freedom ◽  
Heterotic String ◽  
Lorentz Transformations ◽  
Covariant Formalism ◽  
Double Field Theory ◽  
Two Parameter ◽  
Parameter Deformation

Abstract It has been shown by Marques and Nunez that the first α′-correction to the bosonic and heterotic string can be captured in the O(D, D) covariant formalism of Double Field Theory via a certain two-parameter deformation of the double Lorentz transformations. This deformation in turn leads to an infinite tower of α′-corrections and it has been suggested that they can be captured by a generalization of the Bergshoeff-de Roo identification between Lorentz and gauge degrees of freedom in an extended DFT formalism. Here we provide strong evidence that this indeed gives the correct α′2-corrections to the bosonic and heterotic string by showing that it leads to a cubic Riemann term for the former but not for the latter, in agreement with the known structure of these corrections including the coefficient of Riemann cubed.

scholarly journals COMMENT ON THE DIFFERENTIAL CALCULUS ON THE QUANTUM EXTERIOR PLANE

1998 ◽  
Vol 13 (20) ◽  
pp. 1645-1651 ◽  
Author(s):  
SALIH ÇELIK ◽  
SULTAN A. ÇELIK ◽  
METIN ARIK
Keyword(s):  
Symmetry Group ◽  
Differential Calculus ◽  
Baxter Equation ◽  
Oscillator Algebra ◽  
Quantum Deformation ◽  
R Matrix ◽  
Two Parameter ◽  
Parameter Deformation

We give a two-parameter quantum deformation of the exterior plane and its differential calculus without the use of any R-matrix and relate it to the differential calculus with the R-matrix. We prove that there are two types of solutions of the Yang–Baxter equation whose symmetry group is GL p,q(2). We also give a two-parameter deformation of the fermionic oscillator algebra.

scholarly journals Interval positroid varieties and a deformation of the ring of symmetric functions

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Allen Knutson ◽  
Mathias Lederer
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
Dimensional Subspace ◽  
Schubert Varieties ◽  
International Audience ◽  
Structure Constants ◽  
Schubert Classes ◽  
K Theory ◽  
Two Parameter ◽  
Parameter Deformation

International audience Define the <b>interval rank</b> $r_[i,j] : Gr_k(\mathbb C^n) →\mathbb{N}$ of a k-plane V as the dimension of the orthogonal projection $π _[i,j](V)$ of V to the $(j-i+1)$-dimensional subspace that uses the coordinates $i,i+1,\ldots,j$. By measuring all these ranks, we define the <b>interval rank stratification</b> of the Grassmannian $Gr_k(\mathbb C^n)$. It is finer than the Schubert and Richardson stratifications, and coarser than the positroid stratification studied by Lusztig, Postnikov, and others, so we call the closures of these strata <b>interval positroid varieties</b>. We connect Vakil's "geometric Littlewood-Richardson rule", in which he computed the homology classes of Richardson varieties (Schubert varieties intersected with opposite Schubert varieties), to Erd&odblac;s-Ko-Rado shifting, and show that all of Vakil's varieties are interval positroid varieties. We build on his work in three ways: (1) we extend it to arbitrary interval positroid varieties, (2) we use it to compute in equivariant K-theory, not just homology, and (3) we simplify Vakil's (2+1)-dimensional "checker games" to 2-dimensional diagrams we call "IP pipe dreams". The ring Symm of symmetric functions and its basis of Schur functions is well-known to be very closely related to the ring $\bigoplus_a,b H_*(Gr_a(\mathbb{C}^{(a+b)})$ and its basis of Schubert classes. We extend the latter ring to equivariant K-theory (with respect to a circle action on each $\mathbb{C}^{(a+b)}$, and compute the structure constants of this two-parameter deformation of Symm using the interval positroid technology above.

scholarly journals On a (p,k)-analogue of the Gamma function and some associated Inequalities

2016 ◽  
Vol 2 (2) ◽  
pp. 79-90 ◽  
Author(s):  
K. Nantomah ◽  
E. Prempeh ◽  
S. B. Twum
Keyword(s):  
Gamma Function ◽  
Two Parameter ◽  
Parameter Deformation

Abstract In this paper, we introduce a new two-parameter deformation of the classical Gamma function, which we call a (p,k)-analogue of the Gamma function. We also provide some identities generalizing those satisfied by the classical Gamma function. Furthermore, we establish some inequalities involving this new function.

REPRESENTATIONS OF THE TWO PARAMETER DEFORMATION OF THE VIRASORO ALGEBRA

2003 ◽  
Vol 23 (3) ◽  
pp. 309-315 ◽  
Author(s):  
Zilong Zhang ◽  
Gengsheng Zhang ◽  
Yuting Jia
Keyword(s):  
Virasoro Algebra ◽  
Two Parameter ◽  
Parameter Deformation
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