Searcharxiv⌕ Search

arXiv subjects

Salih Celik

Publications and source records attributed to Salih Celik.

At least 19 recordsLinked to original sources
arXiv2021

Right-covariant differential calculus on ${\cal F}({\mathbb C}_q^{2|1})$

We define a new ${\mathbb Z}_2$-graded quantum (2+1)-space and show that the extended ${\mathbb Z}_2$-graded algebra of polynomials on this ${\mathbb Z}_2$-graded quantum space, denoted by ${\cal F}({\mathbb C}_q^{2\vert1})$, is a ${\mathbb Z}_2$-graded Hopf algebra. We construct a right-covariant differential calculus on ${\cal F}({\mathbb C}_q^{2\vert1})$ and define a ${\mathbb Z}_2$-graded quantum Weyl algebra and mention a few algebraic properties of this algebra. Finally, we explicitly construct the dual ${\mathbb Z}_2$-graded Hopf algebra of ${\cal F}({\mathbb C}_q^{2\vert1})$.

math.QA↗
arXiv2018

Quantum de Rham complex on ${\mathbb A}_{h}^{1|2}$

Introducing $h$- and $h'$-deformations of ${\mathbb Z}_2$-graded (1+2)- and (2+1)-spaces, denoted by ${\mathbb A}_h^{1|2}$ and ${\mathbb A}_{h'}^{2|1}$, a two-parameter first order differential calculus, de Rham complex, on ${\mathbb A}_{h}^{1|2}$ explicitly constructed.

math.QA↗
arXiv2018

On $q$- and $h$-deformations of 3d-superspace

In this paper, we introduce non-standard deformations of (1+2)- and (2+1)-superspaces via a contraction using standard deformations of them. This deformed superspaces denoted by ${\mathbb A}_h^{1|2}$ and ${\mathbb A}_{h'}^{2|1}$, respectively. We find a two-parameter $R$-matrix satisfying quantum Yang-Baxter equation and thus obtain a {\it new} two-parameter non-standard deformation of the supergroup ${\rm GL}(1|2)$. Finally, we get a new superalgebra derived from the super-Hopf algebra of functions on the quantum superspace ${\mathbb A}_{p,q}^{1|2}$.

math.QA↗
arXiv2016

Covariant Differential Calculi on $SP_q^{1|2}$

A unitary orthosymplectic quantum supergroup is introduced. Two covariant differential calculi on the quantum superspace $SP_q^{1|2}$ are presented. The $h$-deformed symplectic superspaces via a contraction of the $q$-deformed symplectic superspaces are obtained. A new $h$-deformation of the Heisenberg superalgebra is given.

math.QA↗
arXiv2016

Some topics on the $Z_3$-graded exterior algebras

A $Z_3$-graded Hopf algebra structure of exterior algebra with two generators is introduced. Two covariant differential calculus on the $Z_3$-graded exterior algebra are presented. Using the generators and their partial derivatives a Grassmann-Heisenberg algebra is constructed. An R-matrix which satisfies graded Yang-Baxter equations is obtained. A $Z_3$-graded universal enveloping algebra $U_q(\widetilde{gl}(2))$ is constructed with the quadratic elements of the Grassmann-Weyl algebra.

math.QA↗
arXiv2016

A new $Z_3$-graded quantum group

We introduce a $Z_3$-graded version of exterior (Grassmann) algebra with two generators and using this object we obtain a new $Z_3$-graded quantum group denoted by $O(\widetilde{GL}_q(2))$. We also discuss some properties of ${ O}(\widetilde{GL}_q(2))$.

math.QA↗
arXiv2015

Bicovariant differential calculus on the superspace ${\mathbb R}_q(1|2)$

Super Hopf algebra structure on the function algebra on the extended quantum superspace has been defined. It is given a bicovariant differential calculus on the superspace. The corresponding (quantum) Lie superalgebra of vector fields and its Hopf algebra structure are obtained. The dual Hopf algebra is explicitly constructed. A new quantum supergroup that is the symmetry group of the differential calculus is found.

math.QA↗
arXiv2015

A differential calculus on Z$_3$-graded quantum superspace ${\mathbb R}_q(2|1)$

We introduce a Z$_3$-graded quantum $(2+1)$-superspace and define Z$_3$-graded Hopf algebra structure on algebra of functions on the Z$_3$-graded quantum superspace. We construct a differential calculus on the Z$_3$-graded quantum superspace, and obtain the corresponding Z$_3$-graded Lie superalgebra. We also find a new Z$_3$-graded quantum supergroup which is a symmetry group of this calculus.

math.QA↗
arXiv2014

A Differential Calculus on the $(h,j)$-Deformed Z$_3$-Graded Superplane

In this work, Z$_3$-graded quantum $(h,j)$-superplane is introduced with a help of proper singular $g$ matrix and a Z$_3$-graded calculus is constructed over this new $h$-superplane. A new Z$_3$-graded $(h,j)$-deformed quantum (super)group is constructed via the obtained calculus.

math.QA↗
arXiv2006

Cartan calculi on the quantum superplane

Cartan calculi on the extended quantum superplane are given. To this end, the noncommutative differential calculus on the extended quantum superplane is extended by introducing inner derivations and Lie derivatives.

math.QA↗
arXiv2002

Differential calculus on the h-superplane

A non-commutative differential calculus on the $h$-superplane is presented via a contraction of the $q$-superplane. An R-matrix which satisfies both ungraded and graded Yang-Baxter equations is obtained and a new deformation of the $(1+1)$ dimensional classical phase space (the super-Heisenberg algebra) is introduced.

math.QA↗
arXiv2002

Z$_3$-graded differential geometry of quantum plane

In this work, the Z$_3$-graded differential geometry of the quantum plane is constructed. The corresponding quantum Lie algebra and its Hopf algebra structure are obtained. The dual algebra, i.e. universal enveloping algebra of the quantum plane is explicitly constructed and an isomorphism between the quantum Lie algebra and the dual algebra is given.

math.QA↗
arXiv2001

The properties of the quantum supergroup $GL_{p,q}(1|1)$

In this paper properties of the quantum supermatrices in the quantum supergroup $GL_{p,q}(1|1)$ are discussed. It is shown that any element of $GL_{p,q}(1|1)$ can be expressed as the exponential of a matrix of non-commuting elements, like the group $GL_q(1|1)$. An explicit construction of this exponential representation is presented.

math.QA↗
arXiv2001

Two-parameter nonstandard deformation of 2x2 matrices

We introduce a two-parameter deformation of 2x2 matrices without imposing any condition on the matrices and give the universal R-matrix of the nonstandard quantum group which satisfies the quantum Yang-Baxter relation. Although in the standard two-parameter deformation the quantum determinant is not central, in the nonstandard case it is central. We note that the quantum group thus obtained is related to the quantum supergroup $GL_{p,q}(1|1)$ by a transformation.

math.QA↗