Distribution of low-dimensional Malcev algebras over finite fields - - PowerPoint PPT Presentation

Distribution of low-dimensional Malcev algebras

• ver finite fields into isomorphism and

isotopism classes

´ Oscar Falc´

• n1, Ra´

ul Falc´

• n2, Juan N´

u˜ nez1 rafalgan@us.es

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

University of Seville.

• Rota. July 6, 2015.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Preliminaries

Malcev algebras. Isotopisms of algebras. Algebraic geometry.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Preliminaries

Malcev algebras. Isotopisms of algebras. Algebraic geometry.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Malcev algebras

Anatoly Ivanovich Maltsev 1909-1967

A Malcev algebra (Malcev, 1955) is an anticommutative algebra m such that ((uv)w)u + ((vw)u)u + ((wu)u)v = (uv)(uw), (1) for all u, v, w ∈ m. (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´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Malcev algebras

Anatoly Ivanovich Maltsev 1909-1967

A Malcev algebra (Malcev, 1955) is an anticommutative algebra m such that ((uv)w)u + ((vw)u)u + ((wu)u)v = (uv)(uw), (1) for all u, v, w ∈ m. 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´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Malcev algebras

Anatoly Ivanovich Maltsev 1909-1967

A Malcev algebra (Malcev, 1955) is an anticommutative algebra m such that ((uv)w)u + ((vw)u)u + ((wu)u)v = (uv)(uw), (1) for all u, v, w ∈ m. 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´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Malcev algebras

Anatoly Ivanovich Maltsev 1909-1967

A Malcev algebra (Malcev, 1955) is an anticommutative algebra m such that ((uv)w)u + ((vw)u)u + ((wu)u)v = (uv)(uw), (1) for all u, v, w ∈ m. 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´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Malcev algebras

Anatoly Ivanovich Maltsev 1909-1967

A Malcev algebra (Malcev, 1955) is an anticommutative algebra m such that ((uv)w)u + ((vw)u)u + ((wu)u)v = (uv)(uw), (1) for all u, v, w ∈ m. 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´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Malcev algebras

Anatoly Ivanovich Maltsev 1909-1967

A Malcev algebra (Malcev, 1955) is an anticommutative algebra m such that ((uv)w)u + ((vw)u)u + ((wu)u)v = (uv)(uw), (1) for all u, v, w ∈ m. 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´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Malcev algebras

Anatoly Ivanovich Maltsev 1909-1967

A Malcev algebra (Malcev, 1955) is an anticommutative algebra m such that ((uv)w)u + ((vw)u)u + ((wu)u)v = (uv)(uw), (1) for all u, v, w ∈ m. 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 = 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´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Preliminaries

Malcev algebras. Isotopisms of algebras. Algebraic geometry.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Isotopisms of algebras

Abraham Adrian Albert 1905-1972

Two n-dimensional algebras a and a′ are isotopic (Albert, 1942) if there exist three regular linear transformations f , g, h: a → a′ such that f (u)g(v) = h(uv), for all u, v ∈ a. The triple (f , g, h) is an isotopism between a and a′. To be isotopic is an equivalence relation among algebras. f = g = h ⇒ Isomorphism of algebras.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Preliminaries

Malcev algebras. Isotopisms of algebras. Algebraic geometry.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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¨

• bner basis 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´

• n, Ra´

ul Falc´

• n, 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 V(I) = {a ∈ Fn

p : 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 {f m ∈ 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´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Algebraic geometry

The computation of the reduced Gr¨

• bner basis is fundamental in the

calculus of |V(I)|. Theorem (Lakshman and Lazard, 1991) The complexity of computing the reduced Gr¨

• bner basis of a

zero-dimensional ideal is dO(n), where d is the maximal degree of the polynomials of the ideal. n is the number of variables.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Algebraic sets related to Mn,p.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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 structure constants ck

ij ∈ Fp such that

eiej =

n

• k=1

ck

ij ek,

for all i, j ≤ n. Particularly, ck

ii = 0, for all i, k ≤ n.

ck

ji = −ck ij , 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 variables c = {ck

ij : i, j, k ≤ n}.

Let an,p be the n-dimensional algebra over Fp[c] with basis {e1, . . . , en}, such that eiej =

n

• k=1

ck

ijek,

for all i, j ≤ n.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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] I M

n,p = ck ii : i, k ≤ n +

• ck

ji

p − ck

ij : i, j, k ≤ n, i < j +

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

n,p).

Complexity: max{3, p}O(n3). Maximum number of polynomials: n2 + n2(n−1)

2

+ pnn3.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Algebraic sets related to Mn,p

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

n,p = ck ii : i, k ≤ n +

• ck

ji

p − ck

ij : i, j, k ≤ n, i < j +

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

n,p).

Complexity: pO(n3). Maximum number of polynomials: n2 + n2(n−1)

2

+

• n2(n−1)

2

4 .

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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] I J

n,p = ck ii : i, k ≤ n +

• ck

ji

p − ck

ij : i, j, k ≤ n, i < j +

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

n,p).

Complexity: pO(n3). Maximum number of polynomials: n2 + n2(n−1)

2

+

• n2(n−1)

2

3 .

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Algebraic sets related to Mn,p

Let sijk and s′

ijk 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] I A

n,p = ck ii : i, k ≤ n +

• ck

ji

p − ck

ij : i, j, k ≤ n, i < j +

sijk, s′

ijk : i, j, k ≤ n} .

Besides, |An,p| = dimFp(Fp[c]/I A

n,p).

Complexity: pO(n3). Maximum number of polynomials: n2 + n2(n−1)

2

+ 2n3.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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:

• pt1:

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).

• pt2:

Option 1: The number of algebras. Option 2: The list of algebras. "

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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<j){ J=J+(c(i)(j)(k)^p-c(i)(j)(k)); } else{ J=J+(c(i)(j)(k)+c(j)(i)(k)); } } } }

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Algebraic sets related to Mn,p

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<p; j++){ t++; L’[t]=insert(L[i],j,s-1); } } L=L’; s++; } for (u=2; u<=size(L); u++){ for (v=1; v<=size(L’’); v++){ for (w=v; w<=size(L’’); w++){ L’=MalcevId(L[u],L’’[v],L’’[w]); for (i=1; i<=size(L’); i++){ I=I+reduce(L’[i],std(K)); } } } }

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Algebraic sets related to Mn,p

} 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´

• n, Ra´

ul Falc´

• n, 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´

• n, Ra´

ul Falc´

• n, 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 n p |Mn,p| I M

n,p

I S

n,p

I M

n,p

I S

n,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 Reduced Gr¨

• bner basis: G = {
• ci

12

p − ci

12 : i ≤ 2} ⇒ |V(G)| = p2.

Remark: Every 2-dimensional anticommutative algebra is a Malcev algebra.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Algebraic sets related to Mn,p

Computing time Used memory n p |Mn,p| I M

n,p

I S

n,p

I M

n,p

I S

n,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´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Algebraic sets related to Mn,p

Computing time Used memory n p |An,p| I A

n,p

I A

n,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´

• n, Ra´

ul Falc´

• n, 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 |Mn,p|e3∈Z| I M

n,p

I S

n,p

I M

n,p

I S

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 Reduced Gr¨

• bner basis: {
• ci

12

p − ci

12 : i ≤ 3} ⇒ |V(G)| = p3.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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 |Mn,p|e3,e4∈Z| I M

n,p

I S

n,p

I M

n,p

I S

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 Reduced Gr¨

• bner basis: {
• ci

12

p − ci

12 : i ≤ 4} ⇒ |V(G)| = p4.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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 |Mn,p|e4∈Z| I M

n,p

I S

n,p

I M

n,p

I S

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´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Distribution into isotopism and isomorphism clases

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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 m′, and a natural m ≤ n, it is dm(m) = dm(m′).

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

Distribution into isotopism and isomorphism clases

Let m and m′ be two isomorphic Malcev algebras in Mn,p of respective bases {e1, . . . , en} and {e′

1, . . . , e′ n}.

respective sets of structure constants {ck

ij } and {c′k ij}

Every isomorphism f : m → m′ is uniquely related to the square matrix Mf = (fij) such that f (ei) =

n

• j=1

fije′

j , for all i ≤ n.

Let Fp[f] be the polynomial ring over the finite field Fp in the set of n2 variables f = {f11, . . . , fnn}. Let m and m′ be the algebras over Fp[f] having the same structure constants as m and m′.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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

• k,m=1

ck

ij fkme′ m = n

• k,l,m=1

cm

kl fikfjle′ m.

It is related to the definition of isomorphism f (eiej) = f (ei)f (ej). Theorem The set of isomorphisms between m and m′ is identified with the algebraic set defined by the zero-dimensional radical ideal Im,m′ = pijm : i, j, m ≤ n + det(Mf )p−1 − 1 ⊂ Fp[f ].

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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´

• n, Ra´

ul Falc´

• n, 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 g ∼ = g6

abc ≡

         [e1, ei+1] = ei, for all i > 1, [e4, e5] = ae2, [e4, e6] = be2 + ae3, [e5, e6] = ce2 + be3 + ae4. Theorem There exist five isotopism classes of six-dimensional filiform Lie algebras over any field: g6

000, g6 001, g6 010, g6 100 and g6 110.

Theorem There exist six isomorphism classes of six-dimensional filiform Lie algebras over a field of characteristic two: g6

000, g6 001, g6 010, g6 011, g6 100 and g6 110.

If the base field has characteristic distinct of two, then there exist five isomorphism classes, which correspond to the previous ones, keeping in mind that now, g6

010 ∼

= g6

011. ´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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 g ∼ = g7

abcd ≡

             [e1, ei+1] = ei, for all i > 1, [e4, e7] = ae2, [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:

g ∼ = g7 a ≡                  [e1, e3] = [e4, e6] = [e5, e7] = e2, [e4, e7] = [e5, e6] = e3, [e1, e5] = e4, [e1, e6] = e5, [e1, e7] = e6, [e6, e7] = e3 + ae4, for some a ∈ {0, 1}.

• r

g ∼ = h7 a ≡                  [e3, e7] = [e4, e6] = e2, [e1, e4] = [e5, e6] = e3, [e5, e7] = e4, [e6, e7] = e5, [e1, e7] = e6, [e4, e7] = ae2, for some a ∈ {0, 1}. ´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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: g7

0000, g7 0001, g7 0010, g7 0100, g7 1000, g7 1100, g7 1110, g7 0, g7 1 and h7 0.

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: g7

0000, g7 0001, g7 0010, g7 0100, g7 1000, g7 1100, g7 1(−1)00 and g7 1(−1)10.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, 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: g7

0000, g7 0001, g7 0010, g7 0011, g7 0100, g7 0110, g7 1000, g7 1010,

g7

1011, g7 1100, g7 1110, g7 0, g7 1, h7 0 and h7 1.

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: g7

0000, g7 0001, g7 0010, g7 0011, g7 0100, g7 1001, g7 1b00, and g7 1(−1)10,

where b ∈ K. c) Given a prime p = 2, there exist p + 8 isomorphism classes of seven-dimensional filiform Lie algebras over the finite field Fp: g7

0000, g7 0001, g7 0010, g7 0011, g7 0100, g7 1001, g7 100q, g7 1b00 and g7 1(−1)10,

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

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

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• A. A. Albert. Non-associative algebras. I. Fundamental concepts and isotopy. Ann. of
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York, 1998.

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computational algebraic geometry and commutative algebra. Springer, New York, 2007.

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algebra system for polynomial computations. http://www.singular.uni-kl.de, 2014.

• M. G¨

unaydin and D. Minic. Nonassociativity, Malcev algebras and string theory. Fortschr. Phys., 61(10):873892, 2013.

• M. G¨

unaydin and B. Zumino. Magnetic Charge And Nonassociative Algebras. In L. B. L. report LBL 19200 (1985), editor, Proceedings of the conference Old and New Problems in Fundamental Physics: Symposium in honor of G. C. Wick, pages 4354, Scuola Normale Superiore Publication (Quaderni), Pisa, 1986.

• E. N. Kuzmin. Malcev algebras of dimension five over a field of characteristic zero. Algebra

Logika, 9:691700, 1971.

• E. N. Kuzmin. Structure theory of Malcev algebras. In Radical theory (Eger, 1982), volume

38 of Colloq. Math. Soc. Janos Bolyai, pages 229235. North-Holland, Amsterdam, 1985.

• A. I. Malcev. Analytic loops. Mat. Sb. N.S., 36(78):569576, 1955.
• A. A. Sagle. Malcev algebras. Trans. Amer. Math. Soc., 101:426458, 1961.
• A. A. Sagle. Simple Malcev algebras over fields of characteristic zero. Pacific J. Math.,

12:10571078, 1962.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields

THANK YOU!!!

Distribution of low-dimensional Malcev algebras

• ver finite fields into isomorphism and

isotopism classes

´ Oscar Falc´

• n1, Ra´

ul Falc´

• n2, Juan N´

u˜ nez1 rafalgan@us.es

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

University of Seville.

• Rota. July 6, 2015.

´ Oscar Falc´

• n, Ra´

ul Falc´

• n, Juan N´

u˜ nez Distribution of low-dimensional Malcev algebras over finite fields