On a graded q-differential algebra

In this work, we consider the algebra $M_{N}(C)$ of $N\times N$ matrices as a cyclic quantum plane. We also analyze the coaction of the quantum group ${\cal F}$ and the action of its dual quantum algebra ${\cal H}$ on it. Then, we study the decomposition of $M_{N}(C)$ in terms of the quantum algebra representations. Finally, we develop the differential algebra of the cyclic group $Z_{N}$ with $d^{N}=0$, and treat the particular case N=3.

In this paper we show that a semi-commutative Galois extension of associative unital algebra by means of an element, whose Nth power is equal to the identity element of an algebra, where N is an integer greater or equal to two, induces a structure of graded q-differential algebra, where q is a primitive Nth root of unity. A graded q-differential algebra with differential d, whose Nth power is equal to zero, can be viewed as a generalization of graded differential algebra. The...

We recall the definition of $q$-differential algebras and discuss some representative examples. In particular we construct the $q$-analog of the Hochschild coboundary. We then construct the universal $q$-differential envelope of a unital associative algebra and study its properties. The paper also contains general results on $d^N=0$.

In this article, we describe the construction of graded $q$-differential algebra with ternary differential satisfying the property $d^3=0$ and the $q$-Leibniz rule. Our starting point is coordinate first order differential calculus on some complex algebra and the corresponding bimodule of second order differentials.

We present some results concerning the generalized homologies associated with nilpotent endomorphisms $d$ such that $d^N=0$ for some integer $N\geq 2$. We then introduce the notion of graded $q$-differential algebra and describe some examples. In particular we construct the $q$-analog of the simplicial differential on forms, the $q$-analog of the Hochschild differential and the $q$-analog of the universal differential envelope of an associative unital algebra.

We show that for $q\not=-1$ the q-graded tensor product fails to preserve the q-differential structure of the product algebra and therefore there is no natural tensor product construction for q-differential algebras.

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.

In this work, we construct the algebra of differential forms with the cube of exterior differential equal to zero on one-dimensional space. We prove that this algebra is a graded q-differential algebra where q is a cubic root of unity. Since the square of differential is not equal to zero the algebra of differential forms is generated not only by the first order differential but also by the second order differential of a coordinate. We study the bimodule generated by this sec...

These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.

An introduction to quantum groups and non-commutative differential calculus (Lecture at the III Workshop on Differential Geometry, Granada, September 1994)