Distribution of low-dimensional Malcev over finite fields into isomorphism and isotopism classes

Oscar´ Falc´on1, Ra´ulFalc´on2, Juan N´u˜nez1 [email protected]

1Department of of Geometry and Topology. 2Department of Applied I. University of Seville.

Rota. July 6, 2015.

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields CONTENTS

1 Preliminaries. 2 Algebraic sets related to Mn,p. 3 Distribution into isotopism and isomorphism classes.

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Preliminaries

Malcev algebras. Isotopisms of algebras. Algebraic geometry.

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Preliminaries

Malcev algebras. Isotopisms of algebras. Algebraic geometry.

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Malcev algebras

A Malcev (Malcev, 1955) is an anticommutative algebra m such that

((uv)w)u + ((vw)u)u + ((wu)u)v = (uv)(uw), (1)

Anatoly Ivanovich Maltsev for all u, v, w ∈ m. 1909-1967

(1) is equivalent to the Malcev identity

M(u, v, w) = J(u, v, w)u − J(u, v, uw) = 0,

where J is the Jacobian

J(u, v, w) = (uv)w + (vw)u + (wu)v.

If J(u, v, w) = 0 for all u, v, w ∈ m, then this is a .

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Malcev algebras

A Malcev algebra (Malcev, 1955) is an anticommutative algebra m such that

((uv)w)u + ((vw)u)u + ((wu)u)v = (uv)(uw), (1)

Anatoly Ivanovich Maltsev for all u, v, w ∈ m. 1909-1967

Malcev algebras appear in a natural way in Quantum Mechanics. They constitute the nonassociative algebraic structure defined by velocities and coordinates of an electron moving in the field of a constant magnetic charge distribution (G¨unaydin and Zumino, 1985). The commutators of the velocities yield −→ −→ J(v1, v2, v3) = −∇ · B , −→ where B denotes the magnetic field.

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Malcev algebras

A Malcev algebra (Malcev, 1955) is an anticommutative algebra m such that

((uv)w)u + ((vw)u)u + ((wu)u)v = (uv)(uw), (1)

Anatoly Ivanovich Maltsev for all u, v, w ∈ m. 1909-1967

Associator:[u, v, w] = (uv)w − u(vw). Commutator:[u, v] = uv − vu. :[u, u, v] = [v, u, u] = 0

Lemma If a is an alternative algebra, then the algebra a− defined from the commutator product in a is a Malcev algebra.

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Malcev algebras

A Malcev algebra (Malcev, 1955) is an anticommutative algebra m such that

((uv)w)u + ((vw)u)u + ((wu)u)v = (uv)(uw), (1)

Anatoly Ivanovich Maltsev for all u, v, w ∈ m. 1909-1967

Lemma Every Malcev algebra is binary-Lie: any two elements generate a Lie subalgebra. As a consequence, every 2-dimensional Malcev algebra is a Lie algebra. every 2-dimensional non-Abelian Malcev algebra is isomorphic to the Malcev algebra of basis {e1, e2} defined by the product

e1e2 = e1

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Malcev algebras

A Malcev algebra (Malcev, 1955) is an anticommutative algebra m such that

((uv)w)u + ((vw)u)u + ((wu)u)v = (uv)(uw), (1)

Anatoly Ivanovich Maltsev for all u, v, w ∈ m. 1909-1967 The centralizer of S ⊆ m is

Cenm(S) = {u ∈ m | uv = 0, for all v ∈ S}.

The center of m is the ideal Z(m) = Cenm(m). If dim Z(m) = n, then m is called Abelian.

Lemma Let n ≥ 2. Every n-dimensional anticommutative algebra m such that dim Z(m) ≥ n − 2 is a Malcev algebra. In particular, every 2-dimensional anticommutative algebra is a Malcev algebra.

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Malcev algebras

A Malcev algebra (Malcev, 1955) is an anticommutative algebra m such that

((uv)w)u + ((vw)u)u + ((wu)u)v = (uv)(uw), (1)

Anatoly Ivanovich Maltsev for all u, v, w ∈ m. 1909-1967

The lower central series of a Malcev algebra m is defined as the series of ideals

C1(m) = m ⊇ C2(m) = [m, m] ⊇ ... ⊇ Ck (m) = [Ck−1(m), m] ⊇ ...

If there exists a natural m such that Cm(m) ≡ 0, then m is called nilpotent. If dim Ck (m) = n − k, for all k ∈ {2,..., n}, then m is called filiform.

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Malcev algebras

A Malcev algebra (Malcev, 1955) is an anticommutative algebra m such that

((uv)w)u + ((vw)u)u + ((wu)u)v = (uv)(uw), (1)

Anatoly Ivanovich Maltsev for all u, v, w ∈ m. 1909-1967

Let p be a prime number. In this talk, we focus on the sets Mn,p, Ln,p, Fn,p and An,p of n-dimensional Malcev algebras, Lie algebras, filiform Lie algebras and alternative algebras over the finite field Fp. If p 6= 2, then (1) is equivalent to the Sagle identity (Sagle, 1961) S(u, v, w, t) = (uv·w)t+(vw·t)u+(wt·u)v+(tu·v)w−uw·vt = 0, for all u, v, w, t ∈ m. This identity is Linear in each variable. Invariant under cyclic permutations of the variables. If p = 3, then simple Malcev algebra ⇒ Lie algebra.

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Preliminaries

Malcev algebras. Isotopisms of algebras. Algebraic geometry.

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Isotopisms of algebras

Two n-dimensional algebras a and a0 are isotopic (Albert, 1942) if there exist three regular linear transformations f , g, h: a → a0 such that

f (u)g(v) = h(uv), for all u, v ∈ a. Abraham Adrian Albert

1905-1972

The triple (f , g, h) is an isotopism between a and a0. To be isotopic is an equivalence relation among algebras. f = g = h ⇒ Isomorphism of algebras.

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Isotopisms of algebras

Literature on isotopisms of of algebras: Division: Albert (1942, 1944, 1947, 1952, 1961, 1961a), Benkart (1981, 1981a), Darpo (2007, 2012, 2012a), Deajim (2011), Dieterich (2005), Petersson (1971), Sandler (1962), Schwarz (2010). Lie: Albert (1942), Allison (2009, 2012), Bruck (1944), Jim´enez-Gestal(2008). Jordan: Loos (2006), McCrimmon (1971, 1973), Oehmke (1964), Petersson (1969, 1978, 1984), Ple (2010), Thakur (1999). Alternative: Babikov (1997), McCrimmon (1971), Petersson (2002). Absolute valued: Albert (1947), Cuenca (2010). Structural: Allison (1981). Real two-dimensional commutative: Balanov (2003).

What about isotopisms of Malcev algebras?

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Preliminaries

Malcev algebras. Isotopisms of algebras. Algebraic geometry.

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic geometry

Let Fp[x] be the ring of polynomials in x = {x1,..., xn} over the finite field Fp.

A term order < on the set of monomials of Fp[x] is a multiplicative well-ordering that has the constant monomial 1 as its smallest element.

The largest monomial of a polynomial in Fp[x] with respect to the term order < is its leading monomial. The ideal generated by the leading monomials of all the non-zero elements of an ideal is its initial ideal. Those monomials of polynomials in the ideal that are not leading monomials are called standard monomials. A Gr¨obnerbasis of an ideal I is any subset G of polynomials in I whose leading monomials generate the initial ideal. It is reduced if all its polynomials are monic and no monomial of a polynomial in G is generated by the leading monomials.

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic geometry

Let I be an ideal in Fp[x]. The algebraic set defined by I is the set

n V(I )= {a ∈ Fp : f (a) = 0 for all f ∈ I }. I is zero-dimensional if V(I ) is finite. In particular,

|V(I )| ≤ dimFp Fp[x]/I . I is radical if

m {f ∈ I ⇒ q ∈ I }, for all f ∈ Fp[x] and m ∈ N.

Theorem If I is zero-dimensional and radical, then

|V(I )| = dimFp Fp[x]/I and coincides with the number of standard monomials of I .

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic geometry

The computation of the reduced Gr¨obnerbasis is fundamental in the calculus of |V(I )|.

Theorem (Lakshman and Lazard, 1991) The complexity of computing the reduced Gr¨obnerbasis of a zero-dimensional ideal is d O(n), where d is the maximal degree of the polynomials of the ideal. n is the number of variables.

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic sets related to Mn,p.

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic sets related to Mn,p

Every Malcev algebra in Mn,p of basis {e1,..., en} is characterized by its k structure constants cij ∈ Fp such that

n X k ei ej = cij ek , for all i, j ≤ n. k=1 Particularly, k cii = 0, for all i, k ≤ n. k k cji = −cij , for all i, j, k ≤ n such that i < j.

Let Fp[c] be the polynomial ring over the finite field Fp in the set of k variables c = {cij : i, j, k ≤ n}.

Let an,p be the n-dimensional algebra over Fp[c] with basis {e1,..., en}, such that n X k ei ej = cij ek , for all i, j ≤ n. k=1

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic sets related to Mn,p

Let puijk be the polynomial in Fp[c] that constitutes the coefficient of ek in the Malcev identity M(u, ei , ej ) = 0, for all u ∈ an,p and i, j ≤ n.

Theorem

The set Mn,p is identified with the algebraic set defined by the zero-dimensional radical ideal in Fp[c]

M k k p k In,p = h cii : i, k ≤ n i + h cji − cij : i, j, k ≤ n, i < j i+

h puijk : u ∈ an,p, i, j, k ≤ n i. M Besides, |Mn,p| = dimFp (Fp[c]/In,p).

3 Complexity: max{3, p}O(n ). 2 2 n (n−1) n 3 Maximum number of polynomials: n + 2 + p n .

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic sets related to Mn,p

If p 6= 2, let qijklm be the polynomial in Fp[c] that constitutes the coefficient of em in the Sagle identity S(ei , ej , ek , el ) = 0, for all i, j, k, l ≤ n.

Theorem

If p 6= 2, the set Mn,p is identified with the algebraic set defined by the zero-dimensional radical ideal in Fp[c]

S k k p k In,p = h cii : i, k ≤ n i + h cji − cij : i, j, k ≤ n, i < j i+

h qijklm : i ≤ j, k, l, m ≤ n} i. S Besides, |Mn,p| = dimFp (Fp[c]/In,p).

3 Complexity: pO(n ). 2 2 4 2 n (n−1)  n (n−1)  Maximum number of polynomials: n + 2 + 2 .

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic sets related to Mn,p

Let rijkl be the polynomial in Fp[c] that constitutes the coefficient of el in the Jacobi identity J(ei , ej , ek ) = 0, for all i, j, k ≤ n.

Theorem

The set Ln,p is identified with the algebraic set defined by the zero-dimensional radical ideal in Fp[c]

J k k p k In,p = h cii : i, k ≤ n i + h cji − cij : i, j, k ≤ n, i < j i+

h rijkl : i, j, k, l ≤ n, i < j < k} i. J Besides, |Ln,p| = dimFp (Fp[c]/In,p).

3 Complexity: pO(n ). 2 2 3 2 n (n−1)  n (n−1)  Maximum number of polynomials: n + 2 + 2 .

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic sets related to Mn,p

0 Let sijk and sijk be the polynomials in Fp[c] that constitute the coefficients of ek in the Alternative identities [ei , ei , ej ] = 0 and [ej , ei , ei ] = 0, for all i, j ≤ n.

Theorem

The set An,p is identified with the algebraic set defined by the zero-dimensional radical ideal in Fp[c]

A k k p k In,p = h cii : i, k ≤ n i + h cji − cij : i, j, k ≤ n, i < j i+

0 h sijk , sijk : i, j, k ≤ n} i. A Besides, |An,p| = dimFp (Fp[c]/In,p).

3 Complexity: pO(n ). 2 2 n (n−1) 3 Maximum number of polynomials: n + 2 + 2n .

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic sets related to Mn,p

These results have been implemented in a library malcev.lib in the open computer algebra system for polynomial computations Singular. Available in http://www.personal.us.es/raufalgan/LS/malcev.lib

proc MalcevAlg(int n, int p, list C, int opt1, int opt2) "USAGE: MalcevAlg(n, p, C, opt1, opt2); int n, int p, list C, int opt1, int opt2 PURPOSE: Study the set of n-dimensional Malcev algebras over the finite Fp with structure constants C. RETURN: There are several options: opt1: Option 1: Use the Malcev identity. Option 2: Use the Sagle identity (p must be distinct of 2). Option 3: Use the Jacobi identity (for Lie algebras). Option 4: Use the Alternative identity (for Alternative algebras). opt2: Option 1: The number of algebras. Option 2: The list of algebras. "

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic sets related to Mn,p

int i,j,k,s,t,u,v,w,d; list L,L’,L’’,V; ring R=p,c(1..n)(1..n)(1..n),dp; ideal I,J,K,GB; if (opt1==2 and p==2){ "Warning! Sagle identity can only be used for characteristic distinct of 2."; return(); } for (i=1; i<=size(C); i++){ K=K+(c(C[i][1])(C[i][2])(C[i][3])-C[i][4]); } for (i=1; i<=n; i++){ L[i]=0; } for (i=1; i<=n; i++){ L’’[i]=L; L’’[i][i]=1; for (k=1; k<=n; k++){ J=J+c(i)(i)(k); for (j=1; j<=n; j++){ if (i

L=list(); if (opt1==1){ for (i=1; i<=p; i++){ V[1]=i-1; L[i]=V; } s=2; while (s<=n){ t=0; L’=list(); for (i=1; i<=size(L); i++){ for (j=0; j

} else{ if (opt1==2){ for (u=1; u<=size(L’’); u++){ for (v=u; v<=size(L’’); v++){ for (w=u; w<=size(L’’); w++){ for (j=u; j<=size(L’’); j++){ L’=SagleId(L’’[u],L’’[v],L’’[w],L’’[j]) for (i=1; i<=size(L’); i++){ I=I+reduce(L’[i],std(K)); } } } } } } else{ if (opt1==3){ for (u=1; u<=size(L’’); u++){ for (v=u+1; v<=size(L’’); v++){ for (w=v+1; w<=size(L’’); w++){ L’=JacobiId(L’’[u],L’’[v],L’’[w]); for (i=1; i<=size(L’); i++){ I=I+reduce(L’[i],std(K)); } } } } } Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic sets related to Mn,p

else{ for (u=1; u<=size(L’’); u++){ for (v=u+1; v<=size(L’’); v++){ L’=AlternativeId(L’’[u],L’’[v]); for (i=1; i<=size(L’); i++){ I=I+reduce(L’[i],std(K)); }}}}}} L=elimlinearpart(interred(I+J+K)); if (L[1]!=1 and L[1]!=0){ def R2 = tolessvars(L[1],"dp"); setring R2; ideal GB=slimgb(IMAG); d=vdim(GB); if (opt2==1){ return(d); } list S=ffsolve(GB); matrix M[1][size(S[1])]; for (i=1; i<=size(S[1]); i++){ M[1,size(S[1])-i+1]=jet(S[1][i],1)-jet(S[1][i],0); } "The " + string(d) + " Malcev algebras are those with structure constants"; string(M); for (i=1; i<=size(S); i++){ for (j=1; j<=size(S[i]); j++){ M[1,size(S[i])-j+1]=-jet(S[i][j],0); } string(M); } } Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic sets related to Mn,p We have implemented the procedure in a system with an Intel Core i7-2600, with a 3.4 GHz processor and 16 GB of RAM.

Computing time Used memory M S M S n p |Mn,p| In,p In,p In,p In,p 2 2 4 0 s - 0 MB - 3 9 0 s 0 s 0 MB 0 MB 5 25 0 s 0 s 0 MB 0 MB 7 49 1 s 0 s 0 MB 0 MB 11 121 1 s 0 s 0 MB 0 MB 13 169 2 s 0 s 0 MB 0 MB 17 289 4 s 0 s 0 MB 0 MB ...... 53 2809 32 s 0 s 0 MB 0 MB

i p i 2 Reduced Gr¨obnerbasis: G = { c12 − c12 : i ≤ 2} ⇒ |V(G)| = p . Remark: Every 2-dimensional anticommutative algebra is a Malcev algebra.

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic sets related to Mn,p

Computing time Used memory M S M S n p |Mn,p| In,p In,p In,p In,p 3 2 120 0 s - 0 MB - 3 1,431 1 s 0 s 0 MB 0 MB 5 31,125 11 s 2 s 9 MB 5 MB 7 234,955 128 s 55 s 73 MB 59 MB 11 3,541,791 14,616 s 10,091 s 3.5 GB 2.8 GB 13 ? Run out of memory

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic sets related to Mn,p

Computing time Used memory A A n p |An,p| In,p In,p 3 2 8 0 s 0 MB 3 27 0 s 0 MB 5 125 0 s 0 MB 7 343 0 s 0 MB 11 1,331 0 s 0 MB 13 2,197 0 s 0 MB 4 2 169 0 s 1 MB 3 1,665 3 s 1 MB 5 26,833 82 s 4 MB 7 170,929 853 s 1021 MB

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic sets related to Mn,p

The values of some variables can be imposed to determine a subset of Mn,p.

Computing time Used memory n p |M | I M I S I M I S n,p|e3∈Z n,p n,p n,p n,p 3 2 8 0 s - 0 MB - 3 27 1 s 0 s 0 MB 0 MB 5 125 5 s 0 s 0 MB 0 MB 7 343 50 s 0 s 0 MB 0 MB 11 1,331 83 s 0 s 0 MB 0 MB 13 2,197 186 s 0 s 0 MB 0 MB 17 4,913 274 s 0 s 0 MB 0 MB

i p i 3 Reduced Gr¨obnerbasis: { c12 − c12 : i ≤ 3} ⇒ |V(G)| = p .

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic sets related to Mn,p

The values of some variables can be imposed to determine a subset of Mn,p.

Computing time Used memory n p |M | I M I S I M I S n,p|e3,e4∈Z n,p n,p n,p n,p 4 2 16 0 s - 0 MB - 3 81 11 s 1 s 0 MB 0 MB 5 625 89 s 1 s 0 MB 0 MB 7 2,401 347 s 1 s 2 MB 0 MB 11 14,641 2,087 s 1 s 2 MB 0 MB 13 28,561 3,997 s 1 s 2 MB 0 MB

i p i 4 Reduced Gr¨obnerbasis: { c12 − c12 : i ≤ 4} ⇒ |V(G)| = p .

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Algebraic sets related to Mn,p

The values of some variables can be imposed to determine a subset of Mn,p.

Computing time Used memory n p |M | I M I S I M I S n,p|e4∈Z n,p n,p n,p n,p 4 2 736 2 s - 0 MB - 3 26,001 12 s 2 s 3 MB 3 MB 5 2,340,625 270 s 130 s 46 MB 39 MB 7 ? Run out of memory

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Distribution into isotopism and isomorphism clases

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Distribution into isotopism and isomorphism clases

Theorem a) The dimension of the center of a Malcev algebra is preserved by isotopisms. b) The n-dimensional Abelian Malcev algebra is not isotopic to any other Malcev algebra. c) dim Z(m) = n − 2 > 0 ⇒ m is isomorphic to one of the following two non-isomorphic but isotopic Malcev algebras

e1e2 = e1. e1e2 = e3.

d) Let dm(m) = max{dim Cenm(h): h is an m-dimensional ideal of m}. Given two isotopic n-dimensional Malcev algebras, m and m0, and a 0 natural m ≤ n, it is dm(m) = dm(m ).

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Distribution into isotopism and isomorphism clases

Let mu,v,w be the Malcev algebra of basis {e1, e2, e3} in M3,p such that

e1e2 = u, e1e3 = v and e2e3 = w.

Theorem

There exist four isotopism classes in M3,p, for all prime p ≤ 7

m0,0,0, me1,0,0, me3,e2,0, me3,e2,e1 .

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Distribution into isotopism and isomorphism clases

0 Let m and m be two isomorphic Malcev algebras in Mn,p of 0 0 respective bases {e1,..., en} and {e1,..., en}. k 0k respective sets of structure constants {cij } and {c ij } Every isomorphism f : m → m0 is uniquely related to the square matrix Mf = (fij ) such that

n X 0 f (ei ) = fij ej , for all i ≤ n. j=1

2 Let Fp[f] be the polynomial ring over the finite field Fp in the set of n variables f = {f11,..., fnn}.

0 Let m and m be the algebras over Fp[f] having the same structure constants as m and m0.

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Distribution into isotopism and isomorphism clases

Let pijm be the polynomial in Fp[f] that constitutes the coefficient of em in the identity n n X k 0 X m 0 cij fkmem = ckl fik fjl em. k,m=1 k,l,m=1 It is related to the definition of isomorphism

f (ei ej ) = f (ei )f (ej ).

Theorem The set of isomorphisms between m and m0 is identified with the algebraic set defined by the zero-dimensional radical ideal

p−1 Im,m0 = h pijm : i, j, m ≤ n i + h det(Mf ) − 1 i ⊂ Fp[f ].

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Distribution into isotopism and isomorphism clases

Theorem There exist

a) seven isomorphism classes in M3,2

m0,0,0, me1,0,0, me3,0,0, me2,e3,0, me3,e2,0, me3,e2+e3,0, me3,e2,e1 .

b) nine isomorphism classes in M3,3

m0,0,0, me1,0,0, me3,0,0, me2,e3,0, me2+e3,e3,0,

me3,e2,0, me3,e2+e3,0, me3,0,e2+e3 , me3,e2,e1 .

c) eleven isomorphism classes in M3,5

m0,0,0, me1,0,0, me3,0,0, me2,e3,0, me3,e2,0, me3,e2+e3,0, me3,0,e1+e3 ,

m2e3,e2,0, m2e3,e2+e3,0, m2e3,e2+2e3,0, me3,e2,e1 .

d) thirteen isomorphism classes in M3,7

m0,0,0, me1,0,0, me3,0,0, me2,e3,0, me2+e3,e3,0, me3,e2,0, me3,e2+e3,0,

me3,0,e1+e3 , me3,0,e1 , m3e3,e2+e3,0, m3e3,e2+2e3,0, m3e3,e2+3e3,0, me3,e2,e1 .

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Distribution into isotopism and isomorphism clases

Lemma

Given a six-dimensional filiform Lie algebra g over a field K, there exist three numbers a, b, c ∈ K and an adapted basis of g such that

 [e , e ] = e , for all i > 1,  1 i+1 i  ∼ 6 [e4, e5] = ae2, g = gabc ≡ [e4, e6] = be2 + ae3,  [e5, e6] = ce2 + be3 + ae4.

Theorem There exist five isotopism classes of six-dimensional filiform Lie algebras over any field:

6 6 6 6 6 g000, g001, g010, g100 and g110.

Theorem There exist six isomorphism classes of six-dimensional filiform Lie algebras over a field of characteristic two: 6 6 6 6 6 6 g000, g001, g010, g011, g100 and g110. If the base field has characteristic distinct of two, then there exist five isomorphism classes, which 6 ∼ 6 correspond to the previous ones, keeping in mind that now, g010 = g011.

Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Distribution into isotopism and isomorphism clases

Lemma Let g be a seven-dimensional filiform Lie algebra over a field K of characteristic distinct of two. Then, there exist four numbers a, b, c, d ∈ K such that  [e1, ei+1] = ei , for all i > 1,  [e , e ] = ae ,  4 7 2 ∼ 7 g = gabcd ≡ [e5, e6] = be2,  [e5, e7] = ce2 + (a + b)e3,  [e6, e7] = de2 + ce3 + (a + b)e4.

If the base field K has characteristic two, then it can be also of the form: [e , e ] = [e , e ] = [e , e ] = e , [e , e ] = [e , e ] = e ,  1 3 4 6 5 7 2  3 7 4 6 2   [e4, e7] = [e5, e6] = e3, [e1, e4] = [e5, e6] = e3,   7 [e , e ] = e , 7 [e , e ] = e , g ∼= g ≡ 1 5 4 or g ∼= h ≡ 5 7 4 a [e , e ] = e , a [e , e ] = e ,  1 6 5  6 7 5   [e1, e7] = e6, [e1, e7] = e6,   [e6, e7] = e3 + ae4, for some a ∈ {0, 1}. [e4, e7] = ae2, for some a ∈ {0, 1}.

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Distribution into isotopism and isomorphism clases

Theorem a) There exist 10 isotopism classes of seven-dimensional filiform Lie algebras over a field of characteristic two:

7 7 7 7 7 7 7 7 7 7 g0000, g0001, g0010, g0100, g1000, g1100, g1110, g0, g1 and h0.

b) There exist eight isotopism classes of seven-dimensional filiform Lie algebras over an algebraically closed field of characteristic distinct of two and also over the finite field Fp, where p is a prime distinct of two:

7 7 7 7 7 7 7 7 g0000, g0001, g0010, g0100, g1000, g1100, g1(−1)00 and g1(−1)10.

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields Distribution into isotopism and isomorphism clases

Theorem

a) There exist 15 isomorphism classes of seven-dimensional filiform Lie algebras over a field of characteristic two: 7 7 7 7 7 7 7 7 g0000, g0001, g0010, g0011, g0100, g0110, g1000, g1010, 7 7 7 7 7 7 7 g1011, g1100, g1110, g0, g1, h0 and h1. b) Any seven-dimensional filiform Lie algebra over an algebraically closed field of characteristic distinct of two is isomorphic to one of the following isomorphism classes: 7 7 7 7 7 7 7 7 g0000, g0001, g0010, g0011, g0100, g1001, g1b00, and g1(−1)10, where b ∈ K. c) Given a prime p 6= 2, there exist p + 8 isomorphism classes of seven-dimensional filiform Lie algebras over the finite field Fp: 7 7 7 7 7 7 7 7 7 g0000, g0001, g0010, g0011, g0100, g1001, g100q, g1b00 and g1(−1)10,

where b ∈ K and q is a non-perfect square of Fp.

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields References

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Oscar´ Falc´on,Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields THANK YOU!!!

Distribution of low-dimensional Malcev algebras over finite fields into isomorphism and isotopism classes

Oscar´ Falc´on1, Ra´ulFalc´on2, Juan N´u˜nez1 [email protected]

1Department of of Geometry and Topology. 2Department of Applied Mathematics I. University of Seville.

Rota. July 6, 2015.

Oscar´ Falc´on, Ra´ulFalc´on,Juan N´u˜nez Distribution of low-dimensional Malcev algebras over finite fields