Distribution of low-dimensional Malcev algebras over finite fields into isomorphism and isotopism classes ´ on 1 , Ra´ on 2 , Juan N´ nez 1 Oscar Falc´ u˜ ul Falc´ rafalgan@us.es 1 Department of of Geometry and Topology. 2 Department of Applied Mathematics I. University of Seville. Rota. July 6, 2015. ´ Oscar Falc´ on, Ra´ ul Falc´ on, Juan N´ u˜ nez Distribution of low-dimensional Malcev algebras over finite fields
CONTENTS 1 Preliminaries. 2 Algebraic sets related to M n , p . 3 Distribution into isotopism and isomorphism classes. ´ Oscar Falc´ on, Ra´ ul Falc´ 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´ ul Falc´ 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´ ul Falc´ 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´ ul Falc´ 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 ( v 1 , v 2 , v 3 ) = −− → ∇ · − → B , where − → B denotes the magnetic field. ´ Oscar Falc´ on, Ra´ ul Falc´ 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´ ul Falc´ 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 { e 1 , e 2 } defined by the product e 1 e 2 = e 1 ´ Oscar Falc´ on, Ra´ ul Falc´ 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 Cen m ( S ) = { u ∈ m | uv = 0 , for all v ∈ S } . The center of m is the ideal Z ( m ) = Cen m ( 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´ ul Falc´ 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 C 1 ( m ) = m ⊇ C 2 ( m ) = [ m , m ] ⊇ . . . ⊇ C k ( m ) = [ C k − 1 ( m ) , m ] ⊇ . . . If there exists a natural m such that C m ( m ) ≡ 0, then m is called nilpotent. If dim C k ( m ) = n − k , for all k ∈ { 2 , . . . , n } , then m is called filiform. ´ Oscar Falc´ on, Ra´ ul Falc´ 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 M n , p , L n , p , F n , p and A n , p of n -dimensional Malcev algebras, Lie algebras, filiform Lie algebras and alternative algebras over the finite field F p . 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´ on, Ra´ ul Falc´ 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´ ul Falc´ on, Juan N´ u˜ nez Distribution of low-dimensional Malcev algebras over finite fields
Isotopisms of algebras 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 . Abraham Adrian Albert 1905-1972 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´ on, Ra´ ul Falc´ 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´ ul Falc´ 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´ ul Falc´ on, Juan N´ u˜ nez Distribution of low-dimensional Malcev algebras over finite fields
Algebraic geometry Let F p [ x ] be the ring of polynomials in x = { x 1 , . . . , x n } over the finite field F p . A term order < on the set of monomials of F p [ x ] is a multiplicative well-ordering that has the constant monomial 1 as its smallest element. The largest monomial of a polynomial in F p [ 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¨ obner 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´ on, Ra´ ul Falc´ on, Juan N´ u˜ nez Distribution of low-dimensional Malcev algebras over finite fields
Algebraic geometry Let I be an ideal in F p [ x ]. The algebraic set defined by I is the set V ( I ) = { a ∈ F n p : f ( a ) = 0 for all f ∈ I } . I is zero-dimensional if V ( I ) is finite. In particular, |V ( I ) | ≤ dim F p F p [ x ] / I . I is radical if { f m ∈ I ⇒ q ∈ I } , for all f ∈ F p [ x ] and m ∈ N . Theorem If I is zero-dimensional and radical, then |V ( I ) | = dim F p F p [ x ] / I and coincides with the number of standard monomials of I. ´ Oscar Falc´ on, Ra´ ul Falc´ on, Juan N´ u˜ nez Distribution of low-dimensional Malcev algebras over finite fields
Recommend
More recommend
