On the ${\\mathbb Z}_3$-Graded Structures

Authors : Salih Celik, Sultan Çelik

Pages : 31-40

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Publication Date : 2023-12-30

Article Type : Research Paper

Abstract :After introducing some ${\\mathbb Z}_3$-graded structures, we first give the definition of a ${\\mathbb Z}_3$-graded quantum space and show that the algebra of functions on it, denoted by ${\\cal O}(\\widetilde{\\mathbb C}_q^{1|1|1})$, has a ${\\mathbb Z}_3$-graded Hopf algebra structure. Later, we obtain a new ${\\mathbb Z}_3$-graded quantum group, denoted by $\\widetilde{\\rm GL}_q(1|1)$, and show that the algebra of functions on this group is a ${\\mathbb Z}_3$-graded Hopf algebra. Finally, we construct two non-commutative differential calculi on the algebra ${\\cal O}(\\widetilde{\\mathbb C}_q^{1|1})$ which are left covariant with respect to the ${\\mathbb Z}_3$-graded Hopf algebra ${\\cal O}(\\widetilde{\\rm GL}_q(1|1))$.
Keywords : Z3 graded vector space, Z3 graded algebra, Z3 graded Hopf algebra, Z3 graded quantum group, Z3 graded differential calculus