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We propose the notion of -differential graded algebra, which generalizes the notions of -differential graded algebra and q-differential graded algebra. We construct two examples of -differential graded algebra, where the first one is constructed by means of the generalized Clifford algebra with two generators (reduced quantum plane), where we use a -twisted graded q-commutator. In order to construct the second example, we introduce the notion of -pre-cosimplicial algebra.
Skew-derivations or -derivations are generalized derivations obtained by twisting the Leibniz rule by means of an algebra map. They have been considered by physicists to study quantum groups and to obtain q-deformations of algebra of vector fields like Virasoro algebra and also Heisenberg algebras (oscillator algebras); see [1,2,3,4,5]. The main example is given by the Jackson derivative and leads for example to q-deformation of , Witt algebra, and Virasoro algebra. A natural generalization consists of -derivations involving two twist maps . It turns out that when using -derivations, the commutator bracket no longer satisfies the Jacobi condition. This was a starting point of studying hom-type algebras, where the usual identities are twisted by homomorphisms [7,8]. Later development in the field of research of hom-Lie algebras led to the introduction of the notion of -differential graded algebra , which generalizes the notion of differential graded algebra.
It is well known that the concept of differential graded algebra is based on the equation , where d is a differential of differential graded algebra. In order to generalize the concept of differential graded algebra, one can consider instead of a more general equation , where N is an integer greater than or equal to two. This generalization was proposed and studied in . We would like to mention that equation leads to generalized cohomologies, which can be applied in the quantum Wess–Zumino–Novikov–Witten (WZNW) model for the realization of the space of physical states . Later, an algebraic structure, based on the equation , was developed in , where the authors proposed the notion of q-differential graded algebra, where q is a primitive root of unity. It is worth mentioning that particularly in the case of (primitive square root of unity, ), the definition of q-differential graded algebra gives the definition of differential graded algebra. Later, it was shown that q-differential graded algebras can be applied in the field theories  and in noncommutative geometry to develop a generalization of the notion of connection .
In this paper, we propose and study the notion of -differential algebra, where q is the primitive root of unity and two degree zero endomorphisms of a graded algebra. This notion generalizes the notion of -differential graded algebra, introduced in , as well as the notion of q-differential graded algebra, which was studied in [15,16,17]. We construct several examples of -differential graded algebras by means of the generalized Clifford algebra with two generators (also called a reduced quantum plane) and by means of pre-cosimplicial algebra.
2. First Order -Differential Calculus with Right Partial Derivatives
Let be an associative unital algebra over , where is either the field of real or complex numbers . Let be two algebra endomorphisms of .
A first order -differential calculus over an algebra is the triple , where is an -bimodule and is a linear mapping, which satisfies the -Leibniz rule:
In particular, if are the identity transformations of algebra , i.e., , then the notion of first order -differential calculus amounts to the notion of first order differential calculus over an associative unital algebra.
Now, we assume that is a first order -differential calculus, where is a free finite right -module of rank r with a basis , i.e., any element u of can be uniquely written as:
Then, a structure of the -bimodule of is uniquely determined by the commutation relations:
where the linear mappings satisfy:
It is useful to compose the order square matrix such that a linear mapping is its entry at the intersection of the column and row. Then, (3) can be written in matrix form as follows . Hence, the linear mapping , where is the algebra of order matrices over an algebra , is the algebra homomorphism. Define the right partial derivatives (in a basis ) by the formula:
If is a first order -differential calculus over an algebra and is a free finite right -module with a basis , whose -bimodule structure is determined by the commutation relations (2), then the right partial derivatives, defined in (4), satisfy:
According to the definition of right partial derivatives, we can write:
On the other hand, making use of the -Leibniz rule, we get:
A first order -differential calculus , where is a free right -module of rank r, whose -bimodule structure is determined by the commutation rule (2) and the right derivatives are defined by (4), will be referred to as a first order -differential calculus with right partial derivatives. If is a first order -differential calculus with right partial derivatives, an algebra is generated by variables , and are the basis for a free right -module , then this first order -differential calculus will be referred to as a coordinate first order -differential calculus with right partial derivatives.
3. -Differential Graded Algebra
Let be a graded associative unital algebra and be degree zero endomorphisms of . The degree of a homogeneous element u will be denoted by . In what follows, q will be a primitive root of unity, where For instance, we can take . We give the following definition:
is said to be a -differential graded algebra if is endowed with a degree one linear mapping such that it satisfies the following conditions:
d commutes with endomorphisms , i.e.,
d satisfies the -Leibniz rule:
for any element .
In particular, if we choose in the above definition and , then we get the definition of differential graded algebra. If we fix , but N is an arbitrary integer greater than two, then the notion of -differential graded algebra gives the notion of q-differential graded algebra [12,13,14,15,16,17,18]. If we put (then ) and are different from , then the definition of -differential graded algebra gives the definition of -differential algebra . An root of unity q determines a graded structure of -differential graded algebra, and when it increases (then ), a graded structure of -differential graded algebra differs more and more from the classical -graded structure of differential graded algebra, because there appear more and more subspaces of different degrees, which can be labeled by integers .
Let be a -differential graded algebra. Obviously, is the subalgebra of . Next, it is easy to see that every subspace of homogeneous elements can be considered as an -bimodule, where the left (right) -module structure of is determined by the left (right) multiplication by degree zero elements, i.e.,
Hence, the triple is the first order -differential calculus, because is the -bimodule and satisfies in this case the -Leibniz rule:
4. Construction of -Differential Graded Algebra by Means of the Graded -Commutator
In this section, we show that given an associative unital graded algebra, one can construct a -differential graded algebra with the help of the graded q-commutator, where q is a primitive root of unity.
Let be a graded algebra over , be its unit element, and be two degree zero endomorphisms of .
Let be an element of degree one. Define the degree one linear mapping by the following formula:
q is a primitive root of unity,
, where λ is a non-zero complex number,
then a graded algebra endowed with the degree one linear mapping d is the -differential graded algebra.
First we prove . For any homogeneous , we have:
Analogously, we can prove . Starting with the right-hand side of the -Leibniz rule (Definition 2), we get:
and the -Leibniz rule is proven. For the power of d, we have the following power expansion (see ):
where and . According to the assumption, q is a primitive root of unity, which implies for the quantum Newton binomial coefficients:
and the terms in the power expansion (7) labeled by vanish. Thus, there are only two non-trivial terms in (7):
If N is an odd positive integer, then:
If N is an even positive integer, then:
In order to construct a matrix example of -differential graded algebra, we apply Theorem 1 to the generalized Clifford algebra. We recall you that the generalized Clifford algebra is an associative unital algebra over generated by variables , which are subjected to the relations:
where q is a primitive root of unity and is the unit element of the generalized Clifford algebra. The generalized Clifford algebra will be denoted by , where are independent integers, n is the number of generators, and is an exponent at which the power of every generator equals the identity element .
We consider the generalized Clifford algebra with two generators . Then, from the relations (8), it follows:
The associative unital algebra generated by two variables , which are subjected to the relations (9), is also called the algebra of functions on a reduced quantum plane. This algebra has the matrix representation by order complex matrices. Indeed, we can identify the generators with the Weyl pair, i.e., two order matrices:
Since the Weyl pair satisfies the relations (9), it provides the matrix representation for the generalized Clifford algebra with two generators. It is worth mentioning that the matrices (10) generate the whole algebra of order complex matrices , and they are widely used in quantum information processing theory .
In order to define a structure of graded algebra on , we attribute degree zero to the unit element and the generator x, degree one to the generator , and extend this degree to any product of the generators by defining the degree of a product as the sum of degrees of its cofactors. Then, the whole algebra splits into the direct sum of subspaces of homogeneous elements, and a subspace of elements of degree k will be denoted by , where k runs over the residue classes modulo N, i.e., . Obviously, the subalgebra of elements of degree zero will be generated by the generator x, i.e., is the algebra of polynomials of x. We consider the generator x as an analog of the coordinate function of one-dimensional space. Thus, the algebra of “functions” is the algebra of polynomials of x. In order to emphasize that we consider the elements of as analogs of functions, we will denote the elements of by .
Let be two degree zero endomorphisms of the generalized Clifford algebra such that they commute, , , and is the invertible element of . According to Theorem 1, we define the differential by the formula:
where u is a homogeneous element of the generalized Clifford algebra and is its degree. Since all the assumptions of Theorem 1 are fulfilled, the algebra of order complex matrices endowed with the structure of -graded algebra, which is based on , and with the differential is the -differential graded algebra.
We conclude this section by considering the structure of the first order differential calculus of matrix -differential graded algebra . It is easy to show that is the coordinate first order -differential calculus with right derivative. Indeed, we have:
Because we assume that is an invertible element of , the differential of coordinate function x can serve as the basis for the right -module. The commutation relation, which determines the -bimodule structure of , has the form:
where is the automorphism of the algebra. Thus, according to (4), the differential induces the right derivative:
where f is a polynomial of x. According to Proposition 1, this derivative satisfies:
Since are linear mappings, the left-hand side of (12) can be written as:
is the inverse of . From this, it follows:
The derivative (14) is called the -twisted Jackson q-derivative . Thus, the differential , defined in (11), determines the coordinate first order -differential calculus with the -twisted Jackson type q-derivative.
5. Construction of -Differential Graded Algebra by Means of -Pre- Cosimplicial Algebra
In this section, we introduce the notion of -pre-cosimplicial algebra and show that a -differential algebra can be constructed with the help of a -pre-cosimplicial algebra.
First, we recall the notion of a pre-cosimplicial vector space . A pre-cosimplicial vector space is a positive graded vector space together with coface homomorphisms (linear mappings of vector spaces) , where i runs from zero to , such that:
Thus, every pair of vector spaces is equipped with the coface homomorphisms , where . For example, in the case of vector spaces , there are three coface homomorphisms , which satisfy:
The following definition generalizes the notion of a pre-cosimplicial algebra, which can be found in .
Let be two degree zero endomorphisms of a pre-cosimplicial vector space such that they commute with coface homomorphisms, i.e.,:
A pre-cosimplicial vector space is said to be a -pre-cosimplicial algebra if:
is a graded algebra,
are degree zero endomorphisms of a graded algebra ,
for any homogeneous elements and any integer , we have:
In particular, if we take , then the above definition reduces to the definition of a pre-cosimplicial algebra.
Let be a -pre-cosimplicial algebra. Define the degree one linear mapping by:
where q is a primitive root of unity. Then, a -pre-cosimplicial algebra endowed with the degree one linear mapping d is the -differential graded algebra.
According to Definition 3, degree zero endomorphisms of a positive graded vector space commute with coface homomorphisms , and this immediately implies that commute with d. It is proven in  that for any pre-cosimplicial vector space , the degree one linear mapping d, defined in (17), satisfies , i.e., according to the terminology adopted in , d is N-differential. Since the right-hand side of the formula for d does not depend on degree zero endomorphisms , the same result holds in the case of -pre-cosimplicial algebra. Hence we only need to prove that d satisfies the -Leibniz rule.
Let be a homogeneous element of degree k and . Our aim is to prove:
Making use of Formulas (16) and (17), we can write the left-hand side of the -Leibniz rule as follows:
The right-hand side of the same formula can be written in the form:
where the crossed out terms cancel each other because of the second relation in (16). Comparing (19) with (20), we see that their left-hand sides are equal, and this ends the proof. □
Let be an associative unital algebra, whose unit element will be denoted by , and be two endomorphisms of . The tensor algebra is the graded algebra, where a subspace of elements of degree n is the tensor product , i.e., , and the algebra multiplication , where and are homogeneous elements of degree n and m, respectively, is defined by:
We extend endomorphisms to the tensor algebra by:
Obviously, are degree zero endomorphisms of graded algebra .
For any , define the linear mappings , where , by:
Then, is the -pre-cosimplicial algebra, and are its coface homomorphisms. If we endow the -pre-cosimplicial algebra with the N-differential d, defined in (17), then becomes the -differential graded algebra.
Let be two homogeneous elements of tensor algebra . Then, their product:
is the element of the subspace , and consequently, we have coface homomorphisms from the subspace to the subspace . For the coface homomorphisms , we have to prove the formula:
According to the definition of coface homomorphisms, we can write the left-hand side as follows:
The right-hand side can be written as follows:
Since is an endomorphism of algebra, we have , and Formula (23) is proven. For coface homomorphisms , we have to prove:
The proof of this formula is similar to the proof of (23). Finally, we have to prove the second relation in (16), i.e.,
The left-hand side of this relation can be written as:
and the right-hand side can be written as:
and we see that they are equal. □
Investigation, V.A., O.L. and A.M.; Writing-Original Draft Preparation, V.A.
The first two authors gratefully acknowledge that this work was financially supported by the institutional funding IUT20-57 of the Estonian Ministry of Education and Research. The first author also gratefully acknowledges the financial support of the Département de Mathématiques, Université de Haute-Alsace, within the framework of the visiting professor program.
We thank the reviewer of our paper for valuable suggestions, particularly for drawing our attention to the term Weyl pair, which is used for matrices generating the matrix representation of the generalized Clifford algebra with two generators. The first author also expresses his gratitude for the invitation to visit the Département de Mathématiques, Université de Haute-Alsace, for hospitality during his stay, and for a very creative atmosphere while preparing this paper.
Conflicts of Interest
The authors declare no conflict of interest.
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