On solutions of the Yang-Mills equations in the algebra of h-forms

We study the Yang-Mills equations in the algebra of h-forms, which is developed in the works of N. G. Marchuk and the author. The algebra of h-forms is a special geometrization of the Clifford algebra and is a generalization of the Atiyah-K¨ahler algebra. We discuss an invariant subspace of the constant Yang-Mills operator in the algebra of h-forms and present particular classes of solutions of the Yang-Mills equations.


Introduction
The algebra of h-forms is developed by N. G. Marchuk [1,2]. The algebra of h-forms is a generalization of the Atiyah-Kähler algebra [3,4,5,6] and the Clifford algebra. We use the algebra of h-forms in the works [7,8,9,10] related to the spin connection of general form and the Yang-Mills equations.
In this paper, we discuss an invariant subspace of the constant Yang-Mills operator in the algebra of h-forms and present particular classes of solutions of the Yang-Mills equations.

Yang-Mills equations in pseudo-Euclidean space
Let us consider n-dimensional pseudo-Euclidean space R p,q , p + q = n ≥ 1, with Cartesian coordinates x µ , µ = 1, . . . , n. The metric tensor of R p,q is given by the diagonal matrix with its first p entries equal to 1 and the last q entries equal to −1 on the diagonal. We can raise or lower indices of components of tensor fields with the aid of the metric tensor. For example, F µν = η µα η νβ F αβ . We denote partial derivatives by ∂ µ = ∂ ∂x µ . Let G be a semisimple Lie group and g be the real Lie algebra of the Lie group G. Multiplication of elements of g is given by the Lie bracket [U, V ] = −[V, U ]. Consider the Yang-Mills equations where A µ : R p,q → g is the potential of the Yang-Mills field, F µν : R p,q → g is the strength of the Yang-Mills field, and J ν : R p,q → g is the (non-Abelian) current. The equation (2) can be considered as a definition of the strength F µν . We can substitute F µν from (2) into (3) and obtain One suggests that A µ (and F µν ) are unknown and J ν is known. The current (3) satisfies the (non-Abelian) conservation law The equations (2) -(5) are gauge invariant w.r.t. the transformations where S = S(x) : R p,q → G.

The algebra of h-forms
Let us consider the real Clifford algebra (or geometric algebra) C p,q , p + q = n [11,12,13], with the generators e a , a = 1, . . . , n, which satisfy e a e b + e b e a = 2η ab e, where η = (η ab ) = (η ab ) is the diagonal matrix (1) and e is the identity element. The basis elements of C p,q are enumerated by ordered multi-indexes of length from 0 to n: e a 1 ...a k = e a 1 · · · e a k , a 1 < · · · < a k , k = 0, 1, . . . , n.
We have The projection of an arbitrary element U ∈ C p,q onto the subspace C 0 p,q is denoted by U 0 . Let us consider a vector field with values in the Clifford algebra h µ = h µ (x): R p,q → C p,q h µ (x) = y µ (x)e + y µ a (x)e a + a<b y µ ab (x)e ab + · · · + y µ 1...n (x)e 1...n , which satisfy the same conditions as generators of Clifford algebra (7) in any point of pseudo-Euclidean space: In the case of odd n, the condition h 1 (x)h 2 (x) · · · h n (x) 0 = 0 is also required (see the details in [8]). The expression h µ is called a Clifford field vector. The expression where are skewsymmetric tensor fields of rank j and ∧ is the wedge or exterior product [2], is called an h-form. The set of such h-forms is called an algebra of h-forms C [h] p,q .
In the Atiyah-Kähler algebra, we have differentials dx µ instead of Clifford field vectors h µ . The subspaces of grades k are denoted by We have

The invariant subspace of the constant Yang-Mills operator
The algebra of h-forms C [h] p,q can be considered as a Lie algebra with respect to the commutator Particular classes of solutions of the Yang-Mills equations in the case of the Lie algebra C [h] p,q are considered in [1,8].
Let us consider the following system of equations Solutions of the system of equations (12) - (13) are also solutions of the Yang-Mills system of equations (4). All constant (which do not depend on x ∈ R p,q ) solutions of the system (4) are solutions of the system (12) - (13). In some sense, the system (12) -(13) models certain aspects of the system of the Yang-Mills equations (4), see the details in [14]. Let us consider the operator We call the operator Q the constant Yang-Mills operator because the system (12) can be also interpreted as the system for constant solutions of the Yang-Mills equations. However, the system (12) (or the system (12) -(13)) may also have nonconstant solutions. Let us consider the subspace of grade 1 of the algebra of h-forms and  [15].
We call the subspace C [h] 1 p,q an invariant subspace of the constant Yang-Mills operator Q. Let us consider the algebraic system of equations (12) in the invariant subspace The operator (14) takes the form In the case of the identity matrix Σ = (σ µ ν ) (i.e. σ µ α = δ µ α ), we get In the case of the diagonal matrices Σ = (σ µ ν ) and E = ( µ ν ) with the diagonal elements σ k , k = 1, . . . , n, and k , k = 1, . . . , n, we get the following system of equations with known k , k = 1, . . . , n, and unknown σ k , k = 1, . . . , n. From our point of view, the system (18) deserves attention. In the case n = 3, the equations (18) are the SU(2) Yang-Mills equations for constant solutions because the element e 123 lies in the center of the Clifford algebra C 3,0 and the elements e k e 123 , k = 1, 2, 3 constitute a basis of the subspace C 2 3,0 , which is a Lie algebra of the spin group Spin(3) ∼ = SU(2). We use this fact and the method of the hyperbolic SVD [16] to present all constant solutions of the SU(2) Yang-Mills equations with arbitrary current in [17,18].
In the next section, we study the system (18) in the case of an arbitrary natural number n.
The general solution to this system in the cases n = 2, 3 is given in [17]. Note that in the case n = 3, the system (19) has the following symmetry. If the system (19) has a solution (σ 1 , σ 2 , σ 3 ) with all nonzero σ k , k = 1, 2, 3, then the system (19) has also a solution of the form ( K . Now let us consider the system with all the same := 1 = · · · = n (this is condition for the Yang-Mills current) but in the case of an arbitrary natural number n: Lemma 5. 1 The system (20), n ≥ 4, has the following symmetry. If the system (20) has a solution with all the same σ := σ 1 = · · · = σ k : (σ, σ, . . . , σ), then it has also the following n solutions (σ(n − 2)  Note that in the case n = 3 the symmetry is trivial: the system has a unique solution (not four) because all solutions (22) coincide with (21) in this case. In the cases n ≥ 4, the symmetry is not trivial.
Theorem 5.1 In the cases n = 2, 3, the system (20) with = 0 has a unique solution of the form In the cases n ≥ 4, the system (20) with = 0 has n + 1 solutions: the solution (23) and n solutions of the form with circular permutation.
Proof. The proof is given in Appendix A.
Note that the results of this paper can be generalized to the case of unitary and pseudounitary groups in the formalism of the algebra of h-forms (see about the realization of different classical matrix Lie groups in Clifford algebras in [15,19,20,21,22]).
In this paper, we discussed mathematical structures. The relationship of the proposed mathematical constructions with objects of the real world (elementary particles) is beyond the scope of this study. The explicit formulas (21) -(25) can have physical consequences.

Acknowledgments
The author is grateful to Prof. N. G. Marchuk for fruitful discussions. The author is grateful to the anonymous reviewers for their careful reading of the paper and helpful comments on how to improve the presentation. This work is supported by the Russian Science Foundation (project 21-71-00043), https://rscf.ru/en/project/21-71-00043/.
Appendix A. The proof of Theorem 5.1 The case of = 0 is trivial, we have the solutions (25).
We obtain the following expression for σ 1 6 and the following system of n − 1 equations for x 2 , . . . , x n : Equating the first expression with the second expression, we get Proceeding in the same way with the rest of the equations, we obtain the system of equations Let the expressions in the first brackets of all equations (A.2) are equal to zero, i.e. x k = 1, k = 2, . . . , n. Using (A.1) and σ k = x 1 x k , k = 2, . . . , n, we get the solution (23).
Let the expressions in the first brackets of all equations (A.2), except one, are equal to zero and the expression in the second brackets of one of the equations (for example, for k = n) is equal to zero. Then x k = 1, k = 2, . . . , n − 1 and x n − x 2 2 − · · · − x 2 n−1 = 0, i.e. x n = n − 2. Using (A.1), we get the solutions of type (24) with circular permutation.
The theorem is proved.