Idempotent geometry in generic algebras

Using the syzygy method, established in our earlier paper, we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.


Introduction
The aim of this paper is to further develop the geometrical approach to generic commutative (maybe nonassociative) algebras initiated in [6] and based on the topological index theory and singularity theory. More precisely, we are interested in the following question: How the geometry of idempotents in a generic algebra is determined by its algebraic structure, and vice versa. It is classically known that idempotents play a distinguished role in associative (matrix, Clifford) and nonassociative (Jordan, octonions) algebra structures [14], [10]; see also very recent results for the so-called axial algebras [5] and nonassociative algebras of cubic minimal cones [12], [15].
A key ingredient of our approach in [6] is the Euler-Jacobi formula which gives an algebraic relation between the critical points of a polynomial map and their indices. Indeed, there is a natural bijection between fixed points of the squaring map ψ : x → x 2 in a nonassociative algebra A and its idempotents. By Bezóut's theorem, the complexification A C has either 2 dim A distinct idempotents (counting properly multiplicities and nilpotents which can be thought of as idempotents at infinity), or there are infinitely many idempotents. In the first case, the algebra A is called generic. Then the dichotomy can be essentially reformulated as follows: an algebra is generic if and only if 1 2 is not in the Peirce spectrum of the algebra. The Euler-Jacobi formula applied to the squaring map ψ yields several obstructions both on the spectrum of the idempotents and also the idempotent configurations, the socalled syzygies. In particular, information about the spectrum of sufficiently many idempotents completely determines the rest part of the spectrum and also prescribes possible idempotent constellations.
In this paper, we focus on the vector syzygies, i.e. the obstruction on the possible 'geometry' or configurations of idempotents. The main results are presented for two-dimensional real generic algebras where we are able to obtain a complete characterization. The reality assumption is needed, in particular, because we are interested in certain topological invariants. We also relate our results to potential applications and known facts from qualitative theory of quadratic ODEs.

Preliminaries
We begin by recalling some standard concepts and definitions of nonassociative algebra. By A we denote a finite dimensional commutative nonassociative algebra over a subfield K of the field of complex numbers. The algebra multiplication is denoted by juxtaposition. Thus, xy = yx for all x, y ∈ A but x(yz) and (xy)z are not necessarily equal. By a slight abuse of terminology, an algebra always mean a commutative nonassociative algebra.
If not explicitly stated otherwise, we shall assume that K = R, the field of real numbers. By A C we denote the complexification of A obtained in an obvious way by extending of the ground field such that dim A = dim C A C .
An operator of multiplication by y ∈ A is denoted by L y , i.e. L y x = yx. An element c is called idempotent if c 2 = c and 2-nilpotent if c 2 = 0. By Idm(A) = {0 = c ∈ A : c 2 = c} we denote the set of all nonzero idempotents of A and the complete set of idempotents will be denoted by By Nil 2 (A) we denote the set of 2-nilpotents, i.e. the elements x of A such that x 2 = 0.
Any semisimple idempotent 0 = c = c 2 ∈ A induces the corresponding Peirce decomposition: where cx = xc = λx for any x ∈ A c (λ) and σ(c) is the Peirce spectrum of c (i.e. the spectrum of the operator L c ). The Peirce spectrum σ(A) = {λ 1 , . . . , λ s } of the algebra A is the set of all possible distinct eigenvalues λ i in σ(c), when c runs all idempotents of A. Given a subset E ⊂ A we denote by Span(E) the subspace of A spanned by A. By Affin(E) we denote the affine span of E, i.e. the smallest affine subspace of A that contains E, regarded as a set of points in A. Equivalently, where e ∈ E is an arbitrary element and E − e = {x − e : x ∈ E}.

General facts on the geometry of idempotents
Recall [6] that an nonassociative algebra over K is called generic if its complexification A C contains exactly 2 n distinct idempotents. Then the following characterization holds: Proposition 3.1 (Theorem 3.2 in [6]). If A is a commutative generic algebra then 1 2 ∈ σ(A). In the converse direction: if 1 2 ∈ σ(A) and A does not contain 2-nilpotents then A is generic.
equivalently, c k+1 lies in the affine subspace Affin(c 1 , . . . , c k ). Conversely, if c k+1 lies in the affine subspace Affin(c 1 , . . . , c k ) then (2) holds. Then α i can be thought of as a sort of barycentric coordinates (even if c i are linearly dependent).
Proof. Let us suppose by contradiction that there exist 5 distinct idempotents c i , 1 ≤ i ≤ 5 in Π. If some of c i = 0 then Π is a 2-dimensional subspace. Let, for example, c 5 = 0. Since any two nonzero idempotents non-collinear, c 1 , c 2 is a basis in Π, hence c 3 = αc 1 + βc 2 , where αβ = 0 because c 3 is distinct from c 1 , c 2 . This yields i.e. Π = Span(c 1 , c 2 ) is a subalgebra. Since Π contains 5 distinct idempotents (counting 0) then the subalgebra Π is non-generic, therefore A is non-generic too, a contradiction. Thus, Π 0. Let {c i , c j , c k } be any triple with distinct indices 1 ≤ i, j, k ≤ 5. Since A is generic, c i , c j , c k cannot lie on the same line. Also, by the above argument, the affine subspace Affin(c i , c j , c k ) does not contain the origin, thus {c i , c j , c k } are linearly independent. For dimension reasons, Π is contained in the 3dimensional subspace V ijk := Span(c i , c j , c k ) of A. In particular, we can write (where the latter condition follows from the fact that no pair of the idempotents can lie on the same line). Applying Lemma 3.3, we find 1 Since Then it follows from (5) that Expanding the latter identity yields hence, . This shows that (λ−1)c 1 +u is an eigenvector of L c1 with eigenvalue 1 2 , i.e. 1 2 ∈ σ(A). This proves by Proposition 3.1 that A is not generic, a contradiction.
Note however that there are (necessarily non-generic) algebras with 5 idempotents lying in a two-dimensional affine subspace Π ⊂ A. A typical situation is when these 5 idempotents lie on a quadric in Π, see two examples below.
Example 3.6. Consider an algebra A with the ground vector space R 3 equipped by the following multiplication rule 1 : Then the set of a nonzero idempotents is the circle and the set of a nonzero 2-nilpotents is In particular, there exist complex (non-real) 2-nilpotents in the complexification A C . Furthermore, it is straightforward to verify that the spectrum of any c ∈ Idm(A) is constant: σ(c) = {1, 1 2 , − 1 2 }. Example 3.7. Let b(x, y) be a bilinear form on R n , n ≥ 1. Consider an algebra A on R × R n with the following multiplication rule: The algebra is obviously commutative. It is well-known that the algebra A is Jordan, see [3]. It is called a spin-factor [10] or a Jordan algebra of Clifford type. Note that e = (1, 0) is the algebra unit. The set of nonzero idempotents is given by Conjecture. If Π is a k-dimensional affine subspace of a generic algebra then at most 2 k distinct idempotents can be contained in Π.

Two-dimensional real generic algebras
Though the two-dimensional nonassociative algebras is a very well-studied object, see for example H. Petersson's paper [13], our results below have a different nature. In this section we shall assume that A is a real generic algebra and dim R A = 2. Let c 0 = 0 and c 1 , c 2 , c 3 denote four distinct idempotents of A.
Then since dim A = 2 and 1 ∈ σ(c i ) for 1 ≤ i ≤ 3, the second eigenvalue λ i is also real. Let us associate to each c i the value of the characteristic polynomial of L ci evaluated at 1 2 : where σ(c i ) = {1, λ i }, 1 ≤ i ≤ 3. Since A is generic, λ i = 1 2 , hence χ ci ( 1 2 ) = 0. The sign of χ ci ( 1 2 ) has a clear topological interpretation. Let us consider the vector field Any zero c of ψ A is an idempotent of A and vice versa. If A is generic then all zeros of ψ A are isolated. Recall that the index of ψ A at an isolated zero c is the degree of the (normalized) mapping ψ A (x) nearby c [11]. Let c ∈ Idm(A) and let z ∈ A then therefore, the index ind c ψ A = sgn χ c ( 1 2 ) By the Poincaré-Hopf theorem, the total index of the vector field is equal to the Euler characteristic of A, i.e. vanishes. In particular, this yields the index at infinity Note that the latter sum is a topological invariant. This in particular means that ind ∞ ψ A is invariant under small deformations of a generic algebra.
Our main goal is to study possible configurations of the idempotent quadruple (c 0 , c 2 , c 2 , c 3 ) from topological point of view. To this end, note that there is a syzygy for the nontrivial part of the algebra spectrum established explicitly in [6] (see also [17]): Another way to present (11) in a generic algebra is see (40) in [6]. Furthermore, the idempotents also are balanced by the following 'mean value' type identity: By virtue of (12) let us rewrite the latter identity as where The vector of coefficients a := (a 1 , a 2 , a 3 ) ∈ R can be thought as the barycentric coordinates of c 0 = 0 in the affine coordinate system (c 1 , c 2 , c 3 ) in A.
Note that the case (−, −, −) is impossible by virtue of the second relation in (14). Therefore there are only three possible unordered sign configurations of (a 1 , a 2 , a 3 ): This yields the following three unordered configurations of sign: The sign of χ ci ( 1 2 ) is an important parameter here. Indeed, as it was remarked above, sgn χ ci ( 1 2 ) is exactly the index of the vector field ψ A (x) = x 2 − x, thus carrying information on the topological 'charge' of the corresponding idempotent c i . Note that since χ 0 (t) = t 2 , we have sgn χ c0 ( 1 2 ) = +1 ⇒ sgn a 0 = −1, i.e. the sign of the zero idempotent is always negative (the corresponding charge a 0 has the opposite sign, i.e. positive). The three possible configurations of all four idempotents with their charges are shown in Fig. 2. The three types can be described in terms of the Peirce spectrum as follows. To this end, note that it follows from (11) that where i, j, k is a permutation of 1, 2, 3. Therefore if λ i , λ j > 1 2 then λ k < 1 2 . In that case, a i , a j < 0 and a k > 0, therefore we have type (ii) configuration. Similarly, if λ i , λ j < 1 2 then λ k > 1 2 , hence a i , a j > 0 and a k < 0, therefore we have type (iii) configuration.
Let us show that all three types of generic algebras are realizable. It is natural to ask if the condition (11) is sufficient to have an algebra with the Peirce spectrum (λ 1 , λ 2 , λ 3 ). The following proposition shows that this is the case.
Proof. For the proof of the first part we refer to Theorem 4.7 in our recent paper [6]. Thus, let us suppose that λ 1 , λ 2 , λ 3 ∈ K satisfy (11). We consider a two-dimensional vector space V over K. Let c 1 , c 2 be a basis of V . Turn V into an algebra A = (V, •) by setting Then c 1 , c 2 are obviously idempotents in A, and it is straightforward to see that σ(c i ) = {1, λ i }, i = 1, 2. Next, let Then using the definition of the multiplicative table on A we obtain Hence c 3 is a nonzero idempotent distinct of c 1 , c 2 . Furthermore, using (16) we obtain σ(c 3 ) = {1, λ 3 }, hence A is a generic algebra with the desired Peirce spectrum.
Taking into account the latter theorem, we see that all three types are realizable. We give some explicit examples of each type.
We emphasize that any deformation (i.e. a continuous path in the variety of all nonassociative commutative algebras) of any generic algebra A switching its type must pass through non-generic algebras. ......

Real generic algebras and Riccati type ODEs
We finish this section by indicating some useful correspondence between the three algebra types and three possible phase portraits of quadratic systems of ODEs with two independent variables. Recall that given a system of quadratic ordinary differential equations where F : R n → R n is a homogeneous degree two map, one can associate a commutative algebra A F on R n with multiplication (note that the right hand side is obviously bilinear in x, y). In the introduced notation, (17) becomes a Riccati type ODĖ on the algebra A F . This construction is well-known and there exists a nice correspondence between the standard algebraic concepts and ODE tools, we refer the interested reader to [9], [16], [4], [7], [8]. For example, c ∈ Idm(A F ) if and only if the one-dimensional subspace spanned by c is an integral curve of (17) (i.e. the separate variable solution X(t) = f (t)c is a solution to (17)).
In ODEs the idempotent is often called a Darboux point and the Peirce numbers are known as the Kovalevskaja exponents, [18]. There are many well-established quadratic ODEs, in particular, in mathematical biology or dynamics of the so-called second order chemical reactions (i.e. the reactions with a rate proportional to the concentration of the square of a single reactant or the product of the concentrations of two reactants) [1].
In what follows, we focus on the particular case dim A F = n = 2. Recall that Berlinskii's Theorem [2] classifies the global behaviour of a quadratic ODE system in the plane, in particular, it describes all possible configurations of critical points. More precisely, if a plane ODE has four singular points in a finite part of the phase plane, then only one of the following cases is possible: (B1) three singular points are vertices of a triangle containing the fourth point inside and this point is a saddle while the others are antisaddles (a center, a focus or a node). (B2) three singular points are vertices of a triangle containing the fourth point inside, and this point is a antisaddle while the others are saddles. (B3) these points are vertices of a convex quadrangular, where two opposite vertices are saddles (antisaddles) and two others are antisaddles (saddles); The reader easily recognize that these cases correspond to exactly to the algebras of type (i), (ii) and (iii). Let us explain this in more detail in the case of the quadratic systems. To this end, let us consider (19) in a two-dimensional real generic algebra A with idempotents Idm(A) = {c 0 = 0, c 1 , c 2 , c 3 }. Then the rays F = (x 2 − y 2 , −2xy) F = (x 2 − y 2 , 2 3 xy) sF = (x 2 , y 2 ) Table 1. The ODE ↔ real generic NA correspondence split A ∼ = R 2 into 6 sector domains with vertices at the origin. Clearly, that each ± (c i ) is an integral curve of (19), thus each of the sector 6 domains is invariant under flow. Then the three different types of plane quadratic ODEs corresponding to the three two-dimensional real generic algebra types are shown on Table 5.

Some final remarks and questions
The case of real generic algebras of dim A ≥ 3 is more subtle. It is interesting to know which configurations of idempotents are realizable and which geometrical concepts are naturally appear in this analysis. Recall that in the two-dimensional case considered in the previous section, critical (non-generic) situations occur exactly when three idempotents fall on one line. The situation in three dimensions is more complicated and involves quadric surfaces instead. We are still able to get a nice description here but this requires a more delicate work with the syzygies in the spirit of [6]. Full details will appear elsewhere. Note that any idempotent generates a one-dimensional subalgebra. Twodimensional subalgebras of generic algebras appear naturally in the configurations considered above. This motivates a very natural question: what can be said about the possible number of subalgebras of a given generic algebra? For example, the direct algebra product R × R × R has 6 two-dimensional subalgebras. Is there exists more than six subalgebras in dimension three? The latter example is a unital algebra. Is the unitality necessary for that?