Non-Associative Structures and Their Applications in Differential Equations

Abstract

This article establishes a connection between nonlinear DEs and linear PDEs on the one hand, and non-associative algebra structures on the other. Such a connection simplifies the formulation of many results of DEs and the methods of their solution. The main link between these theories is the nonlinear spectral theory developed for algebra and homogeneous differential equations. A nonlinear spectral method is used to prove the existence of an algebraic first integral, interpretations of various phase zones, and the separatrices construction for ODEs. In algebra, the same methods exploit subalgebra construction and explain fusion rules. In conclusion, perturbation methods may also be interpreted for near-Jordan algebra construction.

1. Introduction

Foreword

The author’s desire to write this kind of article has existed for a long time. In this work, we want to build bridges between mathematicians working in different fields such as algebra, geometry, analysis, and differential equations.

Perhaps few mathematicians have paid attention to the fact that any linear homogeneous system of partial differential equations and all polynomial systems of ordinary differential equations are, in fact, the equations in (non-associative) algebra.

Traditionally, one defines an n-dimensional (non-associative) m-ary algebra $\mathbb{A}$ over field $\mathbb{K}$ of characteristic char $\left(\mathbb{K}\right)=0$ as a vector space ${\mathbb{K}}^n$ endowed with m-linear operation:

Unary	Binary	Ternary
$\ast x={\sum }_{ij}{a}_i^k{x}^i{e}_k$	$x\circ y={\sum }_{ijk}{a}_{ij}^k{x}^i{y}^j{e}_k$	$(x,y,z)={\sum }_{ijkl}{a}_{ijk}^l{x}^i{y}^j{z}^k{e}_l$

Unary and binary algebras will be denoted as $\mathbb{A}=({\mathbb{K}}^n,\ast )$, $\mathbb{A}=({\mathbb{K}}^n,\circ )$.

1.1. Meeting Point of the Non-Associative Algebraic Structures and DEs

Consider the following three types of DEs and their algebraic formulations:

$$\begin{array}{ccc}\hfill ccc\frac{d}{dt}{x}^i\left(t\right)=\sum _j{a}_j^i{x}^j\left(t\right),i=1\cdots n& (\mathrm{linear} \mathrm{ODE})& \dot{x}=\ast x\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill ccc\frac{d}{dt}{x}^i\left(t\right)=\sum _{jk}{a}_{jk}^i{x}^j\left(t\right)x^k\left(t\right),i=1\cdots n& (\mathrm{Riccati} \mathrm{ODE})& \dot{x}=x\circ x\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill ccc\sum _{jk}{a}_k^{ij}\frac{\partial }{{\partial }_{x^j}}{u}^k\left(x\right)=0,i=1\cdots n& (\mathrm{Dirac} \mathrm{PDE})& D\circ u=0\hfill \end{array}$$

(3)

Here, $D=\{e_1{\partial }_{x^1}+e_2{\partial }_{x^2}+\dots +e_n{\partial }_{x^n}\}$ is a Dirac operator in the finite-dimensional algebra $\mathbb{A}=({\mathbb{R}}^n,\circ )$ or $\mathbb{A}=({\mathbb{C}}^n,\circ )$ with basis $e_1,e_2,\dots ,e_n$.

All subsequent content in the article is aimed at answering the following:

Question: What benefits do we gain from establishing a correspondence between algebra and DEs?

First, the association with algebra simplifies the formulation of many results in DEs and the methods for their solution.

The study of any little-known topic must begin with a clear set of examples, and so we begin with:

Let the Riccati Equation (5) occur in an algebra $\mathbb{A}$. We refer to the following well-known facts formulated in purely algebraic notation:

Theorem 1

(Lyapunov function [1]). Suppose that there exists a symmetric, positive definite, bilinear form $b:{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ satisfies $b(x,x^2)=0$ for all $x\in \mathbb{A}$; then, the origin is stable.

Theorem 2

(Boundedness [2]). Suppose $\mathbb{A}$ is a rank three algebra [3]. Then, there exists a bounded solution to (5) if $\mathbb{A}$ has a complete complex structure (the existence of a complete complex structure equivalent to the non-trivial solubility of two equations $x^2\circ x^2=-x^2$ and $y\circ y^2=-y$ in the algebra $\mathbb{A}$).

Theorem 3

(Periodic solutions [4]). If $\mathbb{A}$ is power associative, then (5) has no periodic solutions.

Theorem 4

(Decomposition [5]). If $\mathbb{A}$ is semisimple, then (5) decouples into a system of equations in the simple algebras.

It should be noted that in all the above theorems, the results are given in purely algebraic terms, essentially simplifying their formulation. The above-stated list of links between algebra and DEs is not exhaustive. See further results in Section 2.

Secondly, the association of DEs with algebra allows us to draw parallels and compare the results in both areas separately.

We trace back this interconnectedness in Section 3, where we choose the spectral theory developed in the works of Peirce [6,7] for algebras and that developed by Kovalevskaya [8] for the homogeneous differential equations.

Third, the connection of algebra with ODEs allows us to use results that are unknown in algebra but widely known in DEs.

Section 4 is devoted to some examples of this kind of interaction, where it is shown how the theory of bifurcation and resolution of singularities of vector fields can be used to construct algebras close in their properties to a specific one (near-associative or near-Jordan algebras), as well as how it may clarify some identities in algebras.

1.2. Historical Remarks

To the best of our knowledge, there is no systematic exposition of the role of differential equation theories in algebras, although many attempts have been made to define fundamental notions such as derivatives and integrals in algebras. Here, we list two appropriate sources relevant to our discussion.

In 1883, G. Scheffers published a pioneering article on the possibility of constructing a theory of analytical functions in algebras. His article [9] was devoted to the generalization of the theory of analytic functions of a complex variable to the case of functions defined on quaternions. Significant interest in their article arose only in the 1890s. We can say that their work contributed to the development of modern theories of functions in algebra.

It was not until 1960 that an understanding of the connection between the solution of ordinary differential equations with algebraic structures emerged. L. Markus, in [10], was the first to propose consideration of commutative non-associative algebras as essential tools for studying diffeomorphisms of quadratic vector fields.

2. The Scope of the Riccati and Dirac Equation in Algebra

Theorem 5

(Kinyon and Sagle [11], p. 79, Proposition 2.1). Let

$$\frac{d^{m}z}{d{t}^m}=P\left(z,\frac{dz}{dt},\dots ,\frac{d^{m-1}z}{d{t}^{m-1}}\right),z\in {\mathbb{R}}^n$$

(4)

be a system of polynomial ODEs in ${\mathbb{R}}^n$; i.e., $P(z,z_1,\dots ,z_{m-1})$ is a polynomial in the $z_i$’s. Then, the solution to (4) may be obtained either from the solution of a linear or quadratic (Riccati) system

$$\dot{x}=x^2$$

(5)

occurred in a suitable algebra $\mathbb{A}=({\mathbb{R}}^N,\circ )$.

Example 1.

Given equation $\dot{x}=Q\left(x\right)$, $x=\{x_1,x_2\}$ with the non-degenerate cubic $Q\left(x\right)$, $degQ\mathsf{\le }3$.

Using new variables $U=\{x_1,x_2,x_1^2,x_1{x}_2,x_2^2\}$, the same equation may be rewritten as a Riccati equation $\dot{U}=U^2$ in some algebra $\mathbb{A}=({\mathbb{R}}^5,\circ \}$.

Theorem 6

(The proof is trivial). In any system of first-order PDEs with constant coefficients [12], it may be considered that the Dirac equation $D\circ u=0$ has occurred in a suitable algebra $\mathbb{A}$.

Example 2.

One can rewrite the Lame equation

$$\lambda \mathrm{grad}u+\mu \mathrm{curl}\overline{v}=0,\mathrm{div}\overline{v}=0.$$

equivalently as the Dirac equation $D\circ U\left(x\right)=0$, for $U=\{u,\overline{v}\}$ in an algebra $\mathbb{A}=({\mathbb{R}}^4,\circ )$ with the following multiplication:

$$e_0\circ e_i=0,e_i\circ e_0=\lambda e_i,i\ne 0,e_i\circ e_j=\mu {\epsilon }_{ij}^k{e}_k,i,j\in \{1,2,3\}.$$

Here ${\epsilon }_{ij}^k$ is a fundamental skew-symmetric tensor, $D={\partial }_t{e}_0+{\partial }_x{e}_1+{\partial }_y{e}_2+{\partial }_z{e}_3$. $e_0$ is a left annihilator in $\mathbb{A}$, $e_i$, $i=1,2,3$ are the right eigenvectors of an operator $L_{e_0}$ as multiplication on $e_0$.

2.1. The Main Paradigm

Two differential equations are equivalent if one can be transformed into another by a certain change of variables. In particular, this change of variables transforms solutions of one equation into solutions of another.

Suppose the differential Equation (DE) occurs in an algebra $\mathbb{A}$. The statement below shows that passing from a differential equation to the corresponding algebra is not just a formal trick!

Theorem 7.

Two quadratic Riccati systems (5) are diffeomorphic (linearly equivalent) if and only if the algebras associated with them are isomorphic. Two Dirac systems (3) are equivalent [13,14] if and only if the algebras associated with them are isotopic.

Proof. 

Global diffeomorphism of a homogeneous polynomial quadratic ordinary differential equation is always linear; therefore, any associated algebras are isomorphic for diffeomorphic homogeneous quadratic systems [15].

Concerning the Dirac partial differential Equation (see [16,17]), linear transformations act differently on differentials and independent variables. Therefore, the only thing expected in this case is the isotopy of corresponding algebras. □

The above proposition suggests that studying the dynamics of quadratic systems via the properties of underlying algebras would be beneficial.

2.2. Symmetric Powers

As usual, non-associativity causes immediate algebraic difficulties. Symmetrization may help in some cases (see [18]).

Symmetric powers $x^{\left[m\right]}$ are defined in any non-associative algebra $\mathbb{A}$ by the recurrent formula:

$$a^{\left[1\right]}:=a,a^{\left[2\right]}=a^2,a^{[m+1]}=\frac{1}{m}\sum _{k=1}^m{a}^{\left[k\right]}\circ a^{[m-k+1]}$$

(6)

In power-associative algebra, the term symmetric power joins closely together with the conventional power.

Surprisingly, with the symmetric powers in hand, one can construct both solutions in non-associative algebra $\mathbb{A}$ to Riccati equation, in addition to the polynomial solution of the Dirac equation in the unital associative algebra [7].

2.3. Solution to the Riccati Equation

If $\mathbb{A}$ is the unital division, the solution to the initial value problem (IVP) for the Riccati equation

$$\dot{x\left(t\right)}=x{\left(t\right)}^2,t>0,x\left(0\right)=a,$$

(7)

may be written as $x\left(t\right)={(a^{-1}-t1)}^{-1}$.

A standard power series technique for building an appropriate formal solution of the Riccati equation for each algebra $\mathbb{A}$ yields:

Theorem 8

([19]). The formal solution of the initial value problem (7) for the Ricatti equation in the binary algebra $\mathbb{A}$ may be written using the symmetric powers

$$x\left(t\right)=a+a^{2}t+\dots +a^{[n+1]}{t}^n+\dots$$

(8)

As per the Peano fundamental theorem, the solution to the IVP for (5) coincides with its formal solution. Furthermore, if a structure in algebra to be metricized is defined, then it is possible to estimate the radius of convergence of such a series.

2.4. Polynomial Solution to the Dirac Equation

Theorem 9

([19]). Assume that $e_0,e_1,\dots ,e_n$ forms an orthogonal basis in unital associative algebra $\mathbb{A}=({\mathbb{R}}^{n+1},\circ )$ (for example, in Clifford algebra), and let $e_0$ be the two-sided unit element in $\mathbb{A}$.

Denote by $z_i=x_0{e}_i-x_i{e}_0$, $i=1,\dots ,n$. By construction, $D\circ z_i=0$. Then, any polynomial solution to the Dirac equation in $\mathbb{A}$ may be represented by the superposition of the following homogenous monomials:

$$z^{[m_1,m_2,\dots ,m_k]}=\frac{1}{k!}\sum _{\pi (m_1,\dots ,m_k)}{z}_{m_1}\circ z_{m_2}\circ \dots \circ z_{m_k}$$

(9)

where the sum runs over all distinguishable permutations of $m_1,\dots ,m_k$.

3. Statement of the Nonlinear Spectral Problem

A well-founded spectral theory of nonlinear operators should inherit as many properties of linear spectral theory as possible. Such a theory should be satisfactorily compatible with the standard spectral theory of linear operators.

However, a reasonable generalization of a “nonlinear spectrum” is an entirely different concept. For example, classical eigenvalues are an integral part of the spectra of linear operators, by which one can judge almost all of their properties. However, classic eigenvalues of nonlinear operator linearization depend on where such a linearization was carried out. Therefore, such a concept requires rethinking.

One idea is to consider the local action of a nonlinear operator in small neighborhoods of a set of specially chosen points.

Thus, one comes to the following:

Question: What set of points is a good (well-defined and meaningful) choice for constructing the spectral theory of a nonlinear operator?

The main argument in favor of the choice should be that the linearization spectra at all these points uniquely determine the global behavior of the entire map.

Many significant results and examples are related to considering sets of fixed points of the mapping. The union of the linearization spectra of the nonlinear operator F at its fixed points will be called the cumulative spectrum (also called point spectrum) $\sigma \left(F\right)$ of a nonlinear operator F. Explicitly computing $\sigma \left(F\right)$ seems to be a difficult task of formidable complexity. In cases where F is a homogeneous quadratic map, $\sigma \left(F\right)$ coincides with a collection of idempotents together with the Peirce numbers at all of them.

It is exciting and unexpected that the spectral theory of homogeneous quadratic operators was developed in parallel and independently in the works of B. Peirce. (see [7], p. 36) for algebras and Kovalevskaya [8] concerning the Riccati equation.

3.1. Elements of the Peirce Spectral Theory in Algebras $\mathbb{A}=({\mathbb{R}}^n,\circ )$

If $\mathbb{A}$ is a unary algebra, denote by $Eig\left(\mathbb{A}\right)\subset \mathbb{C}$ the set of its non-zero eigenvectors ($\lambda$-idempotents) as a solution to the equation $\ast x=\lambda x$ with (possible complex) $\lambda$.

Given a binary algebra $\mathbb{A}$, the solution to the equation $x^2=\lambda x$ makes sense only for $\lambda =0,1$, (on real ternary algebras, only for $\lambda =0,\pm 1$).

Denote by $\mathcal{P}\left(\mathbb{A}\right)$ (resp. $\mathcal{N}\left(\mathbb{A}\right)$) the set of non-zero idempotents (resp. nilpotents) in $\mathbb{A}$ (in real ternary algebra, one can distinguish the set of 3-nilpotents, tripotents, and negative tripotents).

Denote by $L_x$ the operator in $\mathbb{A}$ as a left multiplication by x: $L_{x}y=x\circ y.$

Now define the characteristic polynomial by

$$p_x\left(t\right)=det(L_x-tI),t\in K.$$

Let $\sigma \left(x\right)$ denote the set of (in the general complex) roots of the characteristic equation $p_x\left(t\right)=0$ counting multiplicity. By the made assumption, $\sigma \left(x\right)$ is well defined and is said to be the Peirce spectrum of x.

Suppose that $c\in \mathcal{P}\left(\mathbb{A}\right)$ is a nonzero idempotent. Then, $t=1$ is an obvious eigenvalue of $L_c$ (corresponding to c), thus $1\in \sigma \left(c\right)$. Distinct elements of the Peirce spectrum $\sigma \left(c\right)$ are called Peirce numbers.

An idempotent c is called semisimple if A is decomposable as the sum of the corresponding Peirce subspaces:

$$A=\underset{i}{⨁}{A}_c\left({\lambda }_i\right),$$

where ${\lambda }_i$ represents the Peirce numbers of c.

3.2. Stability near the Ray Solution to the Riccati ODEs

Concerning the Riccati equation in an algebra $\mathbb{A}$, the idempotent $c\in \mathbb{A}$ is also called “Darboux point”, and Peirce numbers are known as “Kovalevskaja exponents”.

Consider the initial value problem for the Riccati equation:

$$\dot{x_c}\left(t\right)=x_c{\left(t\right)}^2,x_c\left(0\right)=c,\mathrm{where}c\mathrm{is}\mathrm{any}\mathrm{idempotent}\mathrm{in}\mathbb{A}.$$

(10)

In this case, its solution $x_c\left(t\right)=\frac{c}{1-t}$ is one dimensional and called the ray solution. The direction along the idempotents may be observed as accepting a line in the phase portrait of the Riccati equation. One can see this phenomenon in Figure 1 below.

Peirce numbers serve as essential tools for dynamics of the Riccati flow studying:

Let $\mathbb{A}$ be a commutative non-associative algebra over a field K of $\mathrm{char}\left(K\right)=0$ unless otherwise stated explicitly. Any semisimple idempotent $0\ne c=c^2\in A$ induces the corresponding Peirce decomposition:

$$A=\underset{\lambda \in \sigma \left(c\right)}{⨁}{A}_c\left(\lambda \right),$$

where $cx=xc=\lambda x$ for any $x\in A_c\left(\lambda \right)$ and $\sigma \left(c\right)$ is the Peirce spectrum of c. The Peirce spectrum $\sigma \left(A\right)=\{{\lambda }_1,\dots ,{\lambda }_s\}$A represents the set of all possible distinct eigenvalues ${\lambda }_i$ in $\sigma \left(c\right)$ when c triggers all A idempotents. A fusion (or multiplication) rule is a multiplication rule on eigenspaces of chosen idempotent. It may be written in the following form:

$$A_c\left({\lambda }_i\right)A_c\left({\lambda }_j\right)\subset \underset{k\in \mathcal{F}(i,j)}{⨁}{A}_{{\lambda }_k},\mathcal{F}(i,j)\subset \{1,2\dots ,s\}$$

Definition 1.

Given a Riccati ODE, let l be the ray along an idempotent. Then, define a ray solution type in the Peirce numbers range. Namely:

Definition 2.

Given a planar homogeneous ODE, define a sector as an available domain bounded by two subsequent ray solutions.

There exist precisely three non-exceptional types [1,16] of rays in the plane and six types of regular sectors $H,P,E$, presented in Figure 2 and Figure 3 below.

Type of l	Peirce numbers range	Adjustment trajectories to l
h	$\lambda <0$	at infinity
$p_1$	$\lambda >1/2$	at origin
$p_2$	$0<\lambda <1/2$	non
t	$\lambda$ is complex	with torsion

Figure 2. Types of the ray solution l: (a)—type h, (b)—type $p_1$, (c)—type $p_2$, (d)—type t ( in 3D only).

Figure 2. Types of the ray solution l: (a)—type h, (b)—type $p_1$, (c)—type $p_2$, (d)—type t ( in 3D only).

Type of ${\mathit{l}}_1$	Type of ${\mathit{l}}_2$	Sector Type
$p_1$	$p_2$	$P_1$
$p_1$	h	$P_2$
$p_1$	$p_1$	E
h	h	H
h	$p_2$	$H_1$
$p_2$	$p_2$	$H_2$

Figure 3. Non exceptional types of sectors.

Figure 3. Non exceptional types of sectors.

Definition 3.

A sector composed of two subsequent rays such that at least one is exceptional is called exceptional. (Exceptional rays are characterized by the presence of $\frac{1}{2}$ or 0 in the Peirce spectrum.)

It should be pointed out that all types of vector fields except a focus and a center can be broken down into sectors where their behavior is one of three types: parabolic, hyperbolic, or elliptic. Differential equations are classified precisely by the configuration and the type of sectors. It would be interesting to translate such a classification into algebraic language.

As an example of the application of the spectral theory, we present in the following subsection the conditions of the existence of the algebraic first integrals in the Riccati equation.

3.3. Algebraic First Integrals

Recall that the first integral of (5) is a function $\gamma :\mathbb{A}\to R$, which is constant on their trajectories, i.e., $\gamma \left(X\right(t\left)\right)=const$.

We denote by ${\mathbb{Z}}^{+}$ the set of non-negative integers. The following result is due to [20,21,22]:

Theorem 10

([21]). For the Riccati equation in $({\mathbb{R}}^n,\circ )$, let $c\in {\mathbb{R}}^n$ be an idempotent. Denote by ${\lambda }_1=1,{\lambda }_2,\cdots ,{\lambda }_n$ the Peirce numbers with respect to c. Define set

$$G=\{(k_1,\cdots ,k_n)\in {\left({\mathbb{Z}}^{+}\right)}^n:\sum _{i=1}^n{k}_i{\lambda }_i=0,\sum _{i=1}^n{k}_i>0\}.$$

If the Z-linear space generated by G has dimension r, then there exists $r\mathsf{\le }n$ functional independent analytic first integral ${\Phi }_1\left(x\right),\cdots ,{\Phi }_r\left(x\right)$.

More important is the property of the system to be completely (vollständig) integrable:

Theorem 11

([20]). If the system (5) is algebraically integrable, then all Kowalevskaya exponents are rational.

Recall that each element $(k_1,\cdots ,k_n)\in G$ is a resonant lattice of the Peirce values $\{{\lambda }_1,\cdots ,{\lambda }_n\}$. The theorem implies that the number of a functionally independent first integral is r.

Example 3.

Well known in cosmology, the Kasner DEs [23]

$$\dot{X}=-X^2+YZ,\dot{Y}=-Z^2+XY,\dot{Z}=-Y^2+XZ.$$

(11)

described the so-called Kasner’s metrics as being an exact solution to Einstein’s general relativity theory equations in a vacuum under certain assumptions. It may be considered to be a Riccati equation in Matsuo algebra $3C_{0,1}$. It is known that:

• The Kasner differential Equation (11) is completely integrable.
• $I=XY+YZ+ZX$ is the algebraic first integral of the Kasner differential Equation (11).
• All Peirce numbers of the Matsuo algebra $3C_{0,1}$ are rational and equal to $[1,1/2,-1/2]$.

The resonant lattice for the Kasner equation is:

$$k_1+\frac{1}{2}{k}_2+\left(-\frac{1}{2}\right)k_3=0$$

which is to be expected, as per Theorems 10 and 11.

3.4. Syzygies

The main result in [24] describes syzygies (obstructions) on the idempotent set of a finite-dimensional generic commutative non-associative algebra A. The generic mean, that there are exactly $2^n-1$ non-zero separate simple idempotents in $\mathbb{A}$.

Theorem 12.

Let A be a generic commutative non-associative algebra over $\mathbb{C}$, $dimA=n$. Then

$$\sum _{c\in \mathcal{P}\left(\mathbb{A}\right)}\frac{p_c\left(t\right)}{p_c\left(\frac{1}{2}\right)}\equiv 2^n,\mathrm{for}\mathrm{all}t\in \mathbb{R}.$$

(12)

where $p_c\left(t\right)$ denotes the characteristic polynomial of the multiplication matrix on the idempotent $c\in \mathcal{P}\left(\mathbb{A}\right)$.

We emphasize that the absence of nilpotents and values $\frac{1}{2}$ in the Peirce spectrum of idempotents guarantees the genericity of the algebra.

Example 4.

Suppose the Peirce spectrum of the finite-dimensional algebra $\mathbb{A}$ is composed of two values: $Spect\left(\mathbb{A}\right)=\{1,\lambda \}$. Then, either an algebra $\mathbb{A}$ is not generic, or all its idempotents are not primitive. Recall that an idempotent c is primitive if any 1-eigenvector of a left multiplication by idempotent operator $L_c$ is 1-dimensional. (Theorem 12 does not allow algebra with the prescribed properties to occur).

Theorem 12 can be exploited to derive test cases of algebras. In particular, to build different axial algebras, a class of commutative non-associative algebras introduced by Hall, Rehren, and Shpectorov [25]. Furthermore, one can use Theorem 12 to obtain qualitative results for ODEs.

Example 5.

Berlinskii result for Riccati equation on the plane. Recall that Berlinski’s Theorem [26] classifies the critical points of quadratic systems depending on their distribution in the plane. It turns out that not all configurations are possible. For instance, if a quadratic system has four critical points at the vertices of a convex quadrilateral, then a pair of opposite critical points are saddles whereas the other two are anti-saddles.

3.5. Inverse Spectral Problem

Definition 4.

k vectors $(v_1,v_2,\dots ,v_k)$ in ${\mathbb{R}}^n$ are called quadratically independent if their Kronecker’s squares $(v_1\otimes v_1,v_2\otimes v_2,\dots ,v_k\otimes v_k)$ are linearly independent.

The above-stated definition’s importance clears up with the following fact [27]:

Theorem 13.

For a given set of $\frac{1}{2}n(n+1)$ quadratically independent vectors, one can uniquely and constructively define the commutative real n-dimensional algebra $\mathbb{A}$, such that all these vectors are either idempotents or nilpotents.

Example 6.

Recall that for any vector $v=(x,y)\in {\mathbb{R}}^2$, its Kronecker square is a vector $v^{\otimes 2}=(x^2,xy,y^2)\in {\mathbb{R}}^3$. Three vectors $e_1=(1,0),e_2=(1,1),e_3=(1,-1)$ are evidently linearly dependent in ${\mathbb{R}}^2$ but quadratically independent (their Kronecker’s squares, are (1,0,0), (1,1,1), (1,−1,1)).

The only algebra $\mathbb{A}=({\mathbb{R}}^2,\circ )$ with multiplication $x\circ y=\{x^1{y}^1,\frac{1}{2}(x^1{y}^2+x^2{y}^1)\}$ has these three vectors as idempotents.

Given algebra $\mathbb{A}$. Theorem 13 establishes the following conditions for the existence of a subalgebra and receipt of its construction:

Theorem 14.

Any m-dimensional subalgebra of an algebra $\mathbb{A}$ should contain the set of at least $\frac{1}{2}m(m+1)$ quadratically independent idempotents and/or nilpotents counting with multiplicities.

4. On the Deformation of Singularities and Asymptotic Laws by Bifurcation

Determining the multiplicities of an idempotent usually based on the local algebra notion.

The work [28] fetches a universal method of algebraic factorization. It exploits small topological deformations for quadratic maps. It can be especially useful for establishing the nature of multiple idempotents in the Jordan algebra.

We deal with the following simple criteria for multiple roots identification (see Lemma 4.1 in [27]).

4.1. Multiple Roots and Divisibility of Polynomials

A simple algebraic criterion for a singular point to be multiple is frequently used in what follows:

Lemma 1

(Factorization Lemma [28]). Let $U\subset {\mathbb{R}}^n$ be a small ball around a and let $P=(p_1,\cdots ,p_n):U\to {\mathbb{R}}^n$ be an analytic map with $P\left(a\right)=0$. Then, a is an isolated multiple root of P if and only if there exist coordinates $x=(x_1,\cdots x_n)$ in ${\mathbb{R}}^n$ and natural $m\ge 2$ such that

$$p_k\left(x\right)={(x_1-a_1)}^m{q}_{k1}\left(x_1\right)+\sum _{i=2}^n(x_i-a_i)q_{ki}\left(x\right)(k=1,\cdots ,n),$$

(13)

where $Q\left(x\right)={\left\{q_{ki}\left(x\right)\right\}}_{k,i=1}^n$ is a matrix with analytic entries such that the column vector $q_1\left(x_1\right):={\{q_{11}\left(x_1\right),\dots ,q_{n1}\left(x_1\right)\}}^t$ depends only on $x_1$ and is not equal to zero identically. Representation (13) is unique to the order of the Jordan blocks of linearization of P.

4.2. Perturbation Formula

The factorization lemma combined with the scalar interpolation polynomial suggests the following formula for a deformation of P near a:

$$P_{\epsilon }\left(x\right)=\prod _{j=0}^{m-1}(x_1-a_1-\epsilon j)q_1\left(x_1\right)+\sum _{i=2}^n(x_i-a_i)q_i\left(x\right),$$

(14)

where $\epsilon$ is a sufficiently small real number. Clearly,

$$P_{\epsilon }\left(a\right)=P_{\epsilon }(a+\epsilon e_1)=\dots =P_{\epsilon }(a+\epsilon (m-1)e_1)=0,$$

(15)

4.3. Perturbations and Geometric Graphs

Let $\mathcal{P}$ is a perturbation near a decomposing a into several singularities; then, the same $\mathcal{P}$ can be viewed as a deformation gluing these singularities into a. Moreover, $\mathcal{P}$ can indicate the pairs of obtained singularities that can coalesce (independently of other singularities) and those that cannot coalesce. For example, deformation (14) shows that $a_k=a+\epsilon k{e}_1$ can coalesce with $a_{k+1}$, whereas $a_k$ cannot coalesce with $a_{k+2}$ (provided $m\ge 3$). These observations can be given a more formal description.

Definition 5.

Let $\mathcal{P}:[0,{\epsilon }_o]\times U\to {\mathbb{R}}^n$ be a deformation near a (in general, non good). Take a small positive ε and let $x^1=x^1\left(\epsilon \right),\cdots ,x^{\mu }=x^{\mu }\left(\epsilon \right)$ be all the singularities of $P_{\epsilon }$ close to a. We say that $x^i$ and $x^j$ are incident if there exists a deformation $\mathcal{R}:[0,{\delta }_o]\times V\to {\mathbb{R}}^n$ near some b such that:

(i) 

$P_{\epsilon }=R_{{\delta }_o}$;

(ii) 

all singularities of $P_{\epsilon }$ close to a and different from $x^i$ and $x^j$ are the roots of $R_{\delta }$ for all $\delta \in [0,{\delta }_0]$;

(iii) 

$x^i$ and $x^j$ coalesce into b

(here $R_{\delta }:=\mathcal{R}(\delta ,·)$ and V is is a small ball near b.

Using the concept of incidence, one can assign to a good deformation $\mathcal{P}$ near a a geometric graph$G\left(\mathcal{P}\right)$ as follows: for small $\epsilon >0$, the vertices of $G\left(\mathcal{P}\right)$ coincide with the singularities of $P_{\epsilon }$; an edge connects two vertices if the corresponding singularities are incident. Two graphs corresponding to different small positive values of $\epsilon$ are isomorphic.

Furthermore, given a good deformation $\mathcal{P}$ near a decomposing a into simple singularities, the graph $G\left(\mathcal{P}\right)$ indicates possible scenarios of gluing these singularities into a. These scenarios can be identified with successive operations of edge contractions on $G\left(\mathcal{P}\right)$ widely used in graph theory.

The idea behind the concept of a geometric graph can be traced back to the pioneering work of V. I. Arnold [29], where a deep connection between the hierarchy of singularities of gradient maps based on their possible decomposition on the one hand, and decomposition of Dynkin diagrams on the other hand was established.

It is not clear if two good deformations of the same polynomial map near the exact singularity yield isomorphic geometric graphs, in general.

4.4. Geometric Graphs and A-Equivalence

Given an arbitrary good deformation $\mathcal{P}$ near a, constructing the geometric graph $G\left(\mathcal{P}\right)$ is a problem of formidable complexity. This problem can be simplified by considering equivalences preserving the graph. In this subsection, we focus on one of them—the so-called A equivalence.

Definition 6.

Let $U\subset {\mathbb{R}}^n$ be a small ball centered at a and let $P,X:U\to {\mathbb{R}}^n$ be two analytic maps. Suppose there exists a matrix $Q\left(x\right)={\left\{q_{ki}\left(x\right)\right\}}_{k,i=1}^n$ with analytic entries such that

$$P\left(x\right)=Q\left(x\right)X\left(x\right)\mathrm{and}detQ\left(a\right)\ne 0.$$

(16)

Then, P and X are said to be A-equivalent near a.

If P and X are A-equivalent near a and $\mathcal{X}$ is a good deformation for X near a, then $\mathcal{P}=Q\mathcal{X}$ is a good deformation for P near a. The geometric graphs $G\left(\mathcal{P}\right)$ and $G\left(\mathcal{X}\right)$ are isomorphic.

4.5. Good Deformations Near $A_m$-Singularities

Theorem 15

([28]). Take (13) and (14) and substitute

$${\omega }_{\epsilon ,m}\left(x_1\right):=\prod _{j=0}^{m-1}(x_1-a_1-\epsilon j).$$

(17)

Then, for each $k=0,\dots ,m-1$, one has:

$$det\left(D_{P_{\epsilon }}(a+k\epsilon e_1)\right)=\frac{{(-1)}^{m-1-k}(m-1)!}{\left(\begin{array}{c}m-1\\ k\end{array}\right)}{\epsilon }^{m-1}det\left(Q(a+\epsilon k{e}_1)\right);$$

(18)

$$\frac{{\partial }^{m-1}}{\partial x_1^{m-1}}det\left(D_P\left(x\right)\right){|}_{x=a}=m!det\left(Q\left(a\right)\right).$$

(19)

Calculation determinants of the Jacobi matrix using Formula (19) in the vicinity of splitting, not singular points of the perturbed system, makes it possible to obtain qualitative results and asymptotic laws under small perturbations of the Riccati Equation (see [24,28]).

This circumstance helps to understand the geometry of the arrangement of idempotents in algebra and the topological structure of the formation and/or splitting of multiples of idempotents with small perturbations of the multiplication table. Some asymptotic rules for splitting double singularities of vector fields are formulated in the following subsection. There are also examples of small perturbations of the multiplication table in algebras in which the set of multiple idemponents is split into simple ones.

4.6. Double Roots

• Double roots always allow decomposition along a straight line;
• The Jacobian matrices at regular roots are asymptotically invariant;
• The Jacobian determinants at simple roots have asymptotically the same magnitude but are opposite in sign (in particular, $\mathrm{ind}(a,P)=0$);
• Traces in simple roots are asymptotically equal to the trace of the linearization in a;
• The union of simple singularities: explicit asymptotic formulas for the linearization invariants in simple roots effectively allow calculation of the invariants of the original double root and also determine its topological type.

In contrast to the case of a double root, it is impossible to guarantee that there exists a deformation of P near a that satisfies property P is A-equivalent to $X_m$, $m=(3,1,\cdots ,1)$.

More precisely, the formula (13) with $m=2$ still holds, but there is the following alternative: either (i) P is A-equivalent to $X_3$ near a, in which case a good deformation of P near a can be obtained using the formula (14) or $det\left(Q\right(a\left)\right)=0$, and then a is necessarily the double root of $P_{\epsilon }$, whereas $a_{\epsilon }:=a+\epsilon e_1$ is a simple root (in particular, $\mathrm{ind}(a,P)=\mathrm{ind}(a_{\epsilon },P_{\epsilon })=\pm 1$). In case (ii), there are no more roots for $P_{\epsilon }$ near a. Therefore, applying a formula similar to (14) $P_{\epsilon }$ near a one can construct a nice deformation of P near a.

Example 7.

Consider the commutative algebra $\mathbb{A}=({\mathbb{R}}^2,\circ )$ with a square $x^2$ of

$$x^2=\left[\begin{array}{c}{x}_1^2-x_2^2\\ x_1{x}_2\end{array}\right]=x_1^2\left[\begin{array}{c}1\\ 0\end{array}\right]+x_1{x}_2\left[\begin{array}{c}0\\ 1\end{array}\right]+x_2^2\left[\begin{array}{c}-1\\ 0\end{array}\right].$$

(20)

By inspection, $c=[1,0]$ is the only triple idempotent in $\mathbb{A}$. After a small perturbation, the square in ${\mathbb{A}}_{\epsilon }$ will be similar to

$$x^2=\left[\begin{array}{c}{x}_1^2-x_2^2\\ (1+\epsilon )x_1{x}_2\end{array}\right]$$

(21)

However, now in ${\mathbb{A}}_{\epsilon }$ there are three idempotents $c_1=[1,0]$, $c_{2,3}=[1/(1+\epsilon ),\pm \sqrt{\epsilon }/(1+\epsilon )]$.

5. Summary

In this article, we tried to show the close relationship between algebraic methods and content widely used in practice. We did not attempt to provide comprehensive coverage of this problem. Still, we gave the most striking examples from our point of view, illustrating the possibility to apply the methods of non-associative algebras.