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 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 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
(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 Lie 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
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. Alternative algebra:[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 field 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