A refined algorithm for testing the Leibniz n-algebra structure
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1 A refined algorithm for testing the Leibniz n-algebra structure J. M. Casas (1), M. A. Insua (1), M. Ladra (2) and S. Ladra (3) Universidade de Vigo - Dpto. de Matemática Aplicada I (1) Universidade de Santiago de Compostela (2) Universidade da Coruña (3) EACA 2016: XV Encuentro de Álgebra Computacional y Aplicaciones de Junio de 2016, Logroño Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 1 / 23
2 Contents 1 Backgrounds. Theoretical concepts involved. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 2 / 23
3 Contents 1 Backgrounds. Theoretical concepts involved. 1 Leibniz algebra and its Universal Enveloping Algebra. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 2 / 23
4 Contents 1 Backgrounds. Theoretical concepts involved. 1 Leibniz algebra and its Universal Enveloping Algebra. 2 Leibniz n-algebra and its Universal Enveloping Algebra. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 2 / 23
5 Contents 1 Backgrounds. Theoretical concepts involved. 1 Leibniz algebra and its Universal Enveloping Algebra. 2 Leibniz n-algebra and its Universal Enveloping Algebra. 3 Dataleskii-Takhtajan functor. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 2 / 23
6 Contents 1 Backgrounds. Theoretical concepts involved. 1 Leibniz algebra and its Universal Enveloping Algebra. 2 Leibniz n-algebra and its Universal Enveloping Algebra. 3 Dataleskii-Takhtajan functor. 4 Gröbner Basis in K X 1,..., X t. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 2 / 23
7 Contents 1 Backgrounds. Theoretical concepts involved. 1 Leibniz algebra and its Universal Enveloping Algebra. 2 Leibniz n-algebra and its Universal Enveloping Algebra. 3 Dataleskii-Takhtajan functor. 4 Gröbner Basis in K X 1,..., X t. 2 How do we get the algorithm? Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 2 / 23
8 Contents 1 Backgrounds. Theoretical concepts involved. 1 Leibniz algebra and its Universal Enveloping Algebra. 2 Leibniz n-algebra and its Universal Enveloping Algebra. 3 Dataleskii-Takhtajan functor. 4 Gröbner Basis in K X 1,..., X t. 2 How do we get the algorithm? 3 Constructing and refinning a computation program. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 2 / 23
9 Contents 1 Backgrounds. Theoretical concepts involved. 1 Leibniz algebra and its Universal Enveloping Algebra. 2 Leibniz n-algebra and its Universal Enveloping Algebra. 3 Dataleskii-Takhtajan functor. 4 Gröbner Basis in K X 1,..., X t. 2 How do we get the algorithm? 3 Constructing and refinning a computation program. 4 Some examples of computations. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 2 / 23
10 Backgrounds. Theoretical concepts involved Definition A Leibniz n-algebra is a K-vector space L equipped with an n-linear map, [,..., ]: L n L satisfying the Leibniz n-identity or Fundamental Identity [ ] n [ ] [x1,..., x n ], y 1,..., y n 1 = x1,..., x i 1, [x i, y 1,..., y n 1 ], x i+1,..., x n for all x 1,..., x n, y 1,..., y n 1 L. i=1 Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 3 / 23
11 Backgrounds. Theoretical concepts involved Definition A Leibniz n-algebra is a K-vector space L equipped with an n-linear map, [,..., ]: L n L satisfying the Leibniz n-identity or Fundamental Identity [ ] n [ ] [x1,..., x n ], y 1,..., y n 1 = x1,..., x i 1, [x i, y 1,..., y n 1 ], x i+1,..., x n for all x 1,..., x n, y 1,..., y n 1 L. i=1 A Leibniz 2-algebra is a Leibniz algebra. J-L. Loday, Une version non commutative del algébres de Lie: les algébres de Leibniz, Enseign. Math. (2) 39 (1993) J. M. Casas, J.-L. Loday and T Pirashvili, Leibniz n-algebras, Forum Math. 14 (2002) Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 3 / 23
12 Backgrounds. Theoretical concepts involved Definition A Lie n-algebra is a Leibniz n-algebra L whose n-bracket is skew-symmetric, that is, it satisfies the following identity: [x σ(1),..., x σ(n) ] = ( 1) ɛ(σ) [x 1,..., x n ] for all x 1,..., x n L, where σ S n and ɛ(σ) denotes the sign of the permutation σ. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 4 / 23
13 Backgrounds. Theoretical concepts involved Definition A Lie n-algebra is a Leibniz n-algebra L whose n-bracket is skew-symmetric, that is, it satisfies the following identity: [x σ(1),..., x σ(n) ] = ( 1) ɛ(σ) [x 1,..., x n ] for all x 1,..., x n L, where σ S n and ɛ(σ) denotes the sign of the permutation σ. A Lie 2-algebra is a Lie algebra. V. T. Filippov, n-lie algebras, Sibirsk. Mat. Zh. 26 (6) (1985) , 191. Y. Nambu, Generalized Hamiltonian dynamics, Phys. Rev. D 3 (7) (1973) Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 4 / 23
14 Backgrounds. Theoretical concepts involved Definition The universal enveloping algebra of the Leibniz algebra g is the associative and unital algebra UL(g) := T(gl g r ) I I is the two-sided ideal corresponding to the relations: r [x,y] = r x r y r y r x l [x,y] = l x r y r y l x (r y + l y ) l x = 0 J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296 (1993) Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 5 / 23
15 Backgrounds. Theoretical concepts involved Definition The universal enveloping algebra of the Leibniz n-algebra L is the associative and unital algebra U n L(L) := T((L (n 1) ) l (L (n 1) ) 1m... (L (n 1) ) n 2m (L (n 1) ) r ) I I is the two-sided ideal corresponding to the relations: k 1m [l1,...,l n] l n+1 l 2n 2 = l l2 l n k 1 m l1 l n+1 l 2n 2 + n 1 i 1m l1 ˆl i l n k 1 m li l n+1 l 2n 2 + r l1 l n 1 k 1 m ln l n+1 l 2n 2 i= J. M. Casas, M. A. Insua and M. Ladra, Poincaré-Birkhoff-Witt theorem for Leibniz n-algebras, J. Symb. Compt. 42 (2007) Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 6 / 23
16 Backgrounds. Theoretical concepts involved Definition The functor D n : n Leib Leib that assigns to a Leibniz n-algebra L the Leibniz algebra D n (L) = L (n 1) with the bracket: n 1 [a 1 a n 1, b 1 b n 1 ] := a 1 [a i, b 1,..., b n 1 ] a n 1 i=1 it is called the Dataletskii-Takhtajan functor. Y. L. Dataletskii, L. A. Takhtajan, Leibniz and Lie algebra structures for Nambu algebra, Lett. Math. Phys. 39 (2) (1997) Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 7 / 23
17 Backgrounds. Theoretical concepts involved Let be a given monomial order on the non-commutative polynomial ring K x 1,..., x t. For an arbitraty polynomial p K x 1,..., x t, we will use lm(p) to denote the leading monomial of p. Definition Let I be a two-sided ideal of K x 1,..., x t. A subset G I is called a Gröbner basis for I if for every 0 f I, there exists g G, such that lm(g) is a factor of lm(f ). T. Mora, An introduction to commutative and noncommutative Gröbner bases, Theoret. Comput. Sci. 134 (1994) Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 8 / 23
18 How do we get the algorithm? Using Göbner Basis Theory we wanted to demonstrate the Poincaré-Birkhoff-Witt theorem for the Universal Enveloping algebra of a Leibniz algebra g = {e 1,..., e d }. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 9 / 23
19 How do we get the algorithm? Using Göbner Basis Theory we wanted to demonstrate the Poincaré-Birkhoff-Witt theorem for the Universal Enveloping algebra of a Leibniz algebra g = {e 1,..., e d }. Identifying T(g l g r ) with K x 1,..., x d, z 1,..., z d, via the isomorphism Φ 2, Φ 2 (l ei ) = z i and Φ 2 (r ei ) = x i the previous relations are translated into: r [ei,e j] = r ei r ej r ej r ei Φ 2 (r [ei,e j]) = x i x j x j x i l [ei,e j] = l ei r ej r ej l ei Φ 2 (l [ei,e j]) = z i x j x j z i (r ei + l ei ) l ej = 0 (x i + z i ) z j = 0 Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 9 / 23
20 How do we get the algorithm? Using Göbner Basis Theory we wanted to demonstrate the Poincaré-Birkhoff-Witt theorem for the Universal Enveloping algebra of a Leibniz algebra g = {e 1,..., e d }. Identifying T(g l g r ) with K x 1,..., x d, z 1,..., z d, via the isomorphism Φ 2, Φ 2 (l ei ) = z i and Φ 2 (r ei ) = x i the previous relations are translated into: r [ei,e j] = r ei r ej r ej r ei Φ 2 (r [ei,e j]) = x i x j x j x i l [ei,e j] = l ei r ej r ej l ei Φ 2 (l [ei,e j]) = z i x j x j z i (r ei + l ei ) l ej = 0 (x i + z i ) z j = 0 UL(g) = K x 1,..., x d, z 1,..., z d Φ 2 (I) Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 9 / 23
21 How do we get the algorithm? In the demonstration of this theorem, we could obtain an unexpected result: Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 10 / 23
22 How do we get the algorithm? In the demonstration of this theorem, we could obtain an unexpected result: Every non zero Leibniz relation gives us in the ideal Φ 2 (I) a linear polynomial in the z variables (and in the x variables too) Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 10 / 23
23 How do we get the algorithm? In the demonstration of this theorem, we could obtain an unexpected result: Every non zero Leibniz relation gives us in the ideal Φ 2 (I) a linear polynomial in the z variables (and in the x variables too) and conversely If linear polynomials in the z variables are found in Φ 2 (I) then there exists a non zero Leibniz relation Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 10 / 23
24 How do we get the algorithm? So, with the previous fact in mind, the algorithm appears in a natural way: Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 11 / 23
25 How do we get the algorithm? So, with the previous fact in mind, the algorithm appears in a natural way: Yes. It is not a Leibniz algebra Any z-linear polynomial in Φ 2 (I)? No. It is a Leibniz algebra Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 11 / 23
26 How do we get the algorithm?... and what about Lie algebras? Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 12 / 23
27 How do we get the algorithm?... and what about Lie algebras? During the demonstration of this theorem, we could check that every generator of g ann gives us a linear polynomial in the x variables in the ideal Φ 2 (I), so Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 12 / 23
28 How do we get the algorithm?... and what about Lie algebras? During the demonstration of this theorem, we could check that every generator of g ann gives us a linear polynomial in the x variables in the ideal Φ 2 (I), so For a given Leibniz algebra g, if there is not any polynomial of degree 1 in the x variables in the ideal Φ 2 (I) then g ann = 0 and so, the structure is a Lie algebra. For a given Leibniz algebra g, if there is some polynomial of degree 1 in the x variables in the ideal Φ 2 (I) then g ann 0 and so, the structure is a non-lie Leibniz algebra. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 12 / 23
29 How do we get the algorithm?... and what about Leibniz n-algebras? Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 13 / 23
30 How do we get the algorithm?... and what about Leibniz n-algebras? Using Göbner Basis Theory we wanted to demonstrate the Poincaré-Birkhoff-Witt theorem for the Universal Enveloping algebra of a Leibniz n-algebra L = {e 1,..., e d }. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 13 / 23
31 How do we get the algorithm?... and what about Leibniz n-algebras? Using Göbner Basis Theory we wanted to demonstrate the Poincaré-Birkhoff-Witt theorem for the Universal Enveloping algebra of a Leibniz n-algebra L = {e 1,..., e d }. Identifying T((L (n 1) ) l (L (n 1) ) 1m... (L (n 1) ) n 2m (L (n 1) ) r ) with K X s1,...,s n 1, k 1 Y s1,...,s n 1, Z s1,...,s n 1 s 1,..., s n 1 {1,..., d}, 1 k n 2 via the isomorphism Φ n, Φ n (l es1 e sn 1 ) = x s1,...,s n 1, Φ n ( k m es1 e sn 1 ) = k y s1,...,s n 1 and Φ n (r es1 e sn 1 ) = z s1,...,s n 1 it is possible to check: Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 13 / 23
32 How do we get the algorithm?... and what about Leibniz n-algebras? Using Göbner Basis Theory we wanted to demonstrate the Poincaré-Birkhoff-Witt theorem for the Universal Enveloping algebra of a Leibniz n-algebra L = {e 1,..., e d }. Identifying T((L (n 1) ) l (L (n 1) ) 1m... (L (n 1) ) n 2m (L (n 1) ) r ) with K X s1,...,s n 1, k 1 Y s1,...,s n 1, Z s1,...,s n 1 s 1,..., s n 1 {1,..., d}, 1 k n 2 via the isomorphism Φ n, Φ n (l es1 e sn 1 ) = x s1,...,s n 1, Φ n ( k m es1 e sn 1 ) = k y s1,...,s n 1 and Φ n (r es1 e sn 1 ) = z s1,...,s n 1 it is possible to check: U n L(L) = K X s 1,...,s n 1, k 1 Y s1,...,s n 1, Z s1,...,s n 1 Φ n (I) Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 13 / 23
33 How do we get the algorithm? During the demonstration of this theorem, we could check that every linear polynomial in the z variables comes from a non zero Leibniz n-identity in L or a non zero Leibniz identity in L (n 1) : L: Every non zero Leibniz n-identity gives a linear polynomial in the z variables in Φ n (I), Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 14 / 23
34 How do we get the algorithm? During the demonstration of this theorem, we could check that every linear polynomial in the z variables comes from a non zero Leibniz n-identity in L or a non zero Leibniz identity in L (n 1) : L: Every non zero Leibniz n-identity gives a linear polynomial in the z variables in Φ n (I), and the existence of, at least, one linear polynomial in the z variables in Φ n (I) guaranties that L is not a Leibniz n-algebra Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 14 / 23
35 How do we get the algorithm? Theorem The following statements hold: (L, [,..., ]) is a Leibniz n-algebra if and only if (L (n 1), [, ]) is a Leibniz algebra. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 15 / 23
36 How do we get the algorithm? Theorem The following statements hold: (L, [,..., ]) is a Leibniz n-algebra if and only if (L (n 1), [, ]) is a Leibniz algebra. If (L (n 1), [, ]) is a Lie algebra then (L, [,..., ]) is a Lie n-algebra. J. M. Casas, M. A. Insua, M. Ladra, S. Ladra, Test for Leibniz n-algebra structure, Linear Algebra and its Applications, 494 (2016) Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 15 / 23
37 How do we get the algorithm? The previous theorem allows us to go back to the case of Leibniz algebras and work in (L (n 1), [, ]) instead of (L, [,..., ]). Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 16 / 23
38 How do we get the algorithm? The previous theorem allows us to go back to the case of Leibniz algebras and work in (L (n 1), [, ]) instead of (L, [,..., ]).... and what about Lie n-algebras, how can we check whether the bracket is skew-symmetric or not?: Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 16 / 23
39 How do we get the algorithm? The previous theorem allows us to go back to the case of Leibniz algebras and work in (L (n 1), [, ]) instead of (L, [,..., ]).... and what about Lie n-algebras, how can we check whether the bracket is skew-symmetric or not?: The solution of this question lies, again, on the set of linear polynomials in the x variables in the ideal Φ 2 (I). Searching for a determinate type of linear polynomials in the x variables it is possible to check whether a structure is a Lie n-algebra or not. J. M. Casas, M. A. Insua, M. Ladra, S. Ladra, Test for Leibniz n-algebra structure, Linear Algebra and its Applications, 494 (2016) Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 16 / 23
40 How do we get the algorithm? Conclusion To test Leibniz/Lie n-algebra structure in L Work in D n (L) = L (n 1). Leibniz n-algebra structure Check linear polynomials in the z variables of Φ 2 (I). Lie n-algebra structure Check linear polynomials in the x variables of Φ 2 (I). Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 17 / 23
41 Constructing and refinning a computation program Following the previous ideas, a computer program was made, but doing things in this way, we could check that the average computation time was high. So we had to do something extra to decrease computation time. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 18 / 23
42 Constructing and refinning a computation program Following the previous ideas, a computer program was made, but doing things in this way, we could check that the average computation time was high. So we had to do something extra to decrease computation time. [, (D n (L)) ann ] = 0. If one of the previous brackets is not equal to zero, then the structure cannot be a Leibniz n-algebra and it is possible to stop the program to avoid unnecessary computations. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 18 / 23
43 Constructing and refinning a computation program Following the previous ideas, a computer program was made, but doing things in this way, we could check that the average computation time was high. So we had to do something extra to decrease computation time. [, (D n (L)) ann ] = 0. If one of the previous brackets is not equal to zero, then the structure cannot be a Leibniz n-algebra and it is possible to stop the program to avoid unnecessary computations. Work with a subideal of Φ 2 (I) to reduce computation time: {x i x j x j x i Φ 2 (r [ei,e j])} i,j {1,...,m},i<j {zi x j x j z i Φ(l [ei,e j])} i {1,...,d},j {1,...,m} Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 18 / 23
44 Constructing and refinning a computation program Following the previous ideas, a computer program was made, but doing things in this way, we could check that the average computation time was high. So we had to do something extra to decrease computation time. [, (D n (L)) ann ] = 0. If one of the previous brackets is not equal to zero, then the structure cannot be a Leibniz n-algebra and it is possible to stop the program to avoid unnecessary computations. Work with a subideal of Φ 2 (I) to reduce computation time: {x i x j x j x i Φ 2 (r [ei,e j])} i,j {1,...,m},i<j {zi x j x j z i Φ(l [ei,e j])} i {1,...,d},j {1,...,m} At the end of this process we obtained nleibnizq, a Mathematica R procedure to test these kind of structures. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 18 / 23
45 Some examples of computations Example Let L be the C-vector space with basis {e 1, e 2, e 3 } and the 3-linear map [e 3, e 2, e 3 ] = e 2 = (0, 1, 0), [e 3, e 3, e 2 ] = e 2 = (0, 1, 0) and 0 otherwise. BracketEqZero[3, 3] LeibnizBracket[{3, 2, 3}, {0, 1, 0}] LeibnizBracket[{3, 3, 2}, {0, 1, 0}] nleibnizq[3, 3] It is a non-lie Leibniz 3-algebra. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 19 / 23
46 Some examples of computations Example Let L be the C-vector space with basis {e 1, e 2, e 3, e 4 } and the 3-linear map [e σ(1), e σ(2), e σ(3) ] = [e σ(1), e σ(2), e σ(4) ] = ( 1) ɛ(σ) e 3, σ S 3, and 0 otherwise. BracketEqZero[3, 4] AntiSymBrackets[{1, 2, 3}, {0, 0, 1, 0}] AntiSymBrackets[{1, 2, 4}, {0, 0, 1, 0}] nleibnizq[3, 4] It is a Lie 3-algebra. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 20 / 23
47 Some examples of computations Example Let L be the C-vector space with basis {e 1, e 2 } and the 5-linear map [e 1, e 2, e 2, e 1, e 1 ] = e 2 and 0 otherwise. BracketEqZero[5, 2] LeibnizBracket[{1, 2, 2, 1, 1}, {0, 1}] nleibnizq[5, 2] It is not a Leibniz 5-algebra. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 21 / 23
48 Some examples of computations Timing... Initial Algorithm Refined Algorithm Reduction Example s s. 87% Example s s. 90% Example s s. 99% Table: Time comparison nleibnizq package was ran on a Intel(R) Core(TM) i GHz, 16 GB RAM, running Windows 7 (64 bits) with Mathematica R 10. Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 22 / 23
49 Thanks for your attention! Avelino Insua (Universidade de Vigo) Testing Leibniz n-algebra structure 23 / 23
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