An application of lattice basis reduction to polynomial identities for algebraic structures

Linear and Multilinear Algebra

Elgendy

2018

Self Cite

Comtrans algebras, arising in web geometry, have two trilinear operations, commutator and translator. We determine a Gröbner basis for the comtrans operad, and state a conjecture on its dimension formula. We study multilinear polynomial identities for the special commutator [x, y, z] = xyz − yxz and special translator x, y, z = xyz − yzx in associative triple systems. In degree 3, the defining identities for comtrans algebras generate all identities. In degree 5, we simplify known identities for each operation and determine new identities relating the operations. In degree 7, we use representation theory of the symmetric group to show that each operation satisfies identities which do not follow from those of lower degree but there are no new identities relating the operations. We use noncommutative Gröbner bases to construct the universal associative envelope for the special comtrans algebra of 2 × 2 matrices.2010 Mathematics Subject Classification. Primary 17A40. Secondary 15-04, 15A21, 15A69, 15B36, 18D50, 20C30, 68W30.Key words and phrases. Comtrans algebras, trilinear operations, polynomial identities, lattice basis reduction, representation theory of the symmetric group, algebraic operads, Gröbner bases, universal associative enveloping algebras.The commutator alternates in the first two arguments (1), the translator satisfies the Jacobi identity (2), and the operations are related by the comtrans identity (3). Example 1.2. If T is a Lie triple system with bracket [x, y, z] then letting both commutator and translator equal [x, y, z] gives T the structure of a comtrans algebra T CT . If T is obtained from a Lie algebra L by [x, y, z] = [[x, y], z] then Shen & Smith [32, Theorem 3.2] have shown that L is simple if and only if T CT is simple. Example 1.3. Let A m,n denote the vector space of m × n matrices over F. Fix matrices p (n × n) and q (m × m) over F. Define a trilinear operation on A m,n by (x, y, z) = xpy t qz. Setting [x, y, z] = (x, y, z) − (y, x, z) and x, y, z = (x, y, z) − (y, z, x) gives A m,n the structure of a comtrans algebra [32, Example 2.1]. Definition 1.4. An associative triple system [25] is a vector space A with a trilinear operation xyz satisfying (vw(xyz)) = (v(wxy)z) = (vw(xyz)). The special commutator and special translator in A are [x, y, z] = xyz − yxz and x, y, z = xyz − yzx.

Section: Identities Of Degreementioning

confidence: 99%

Special identities for comtrans algebras

Linear and Multilinear Algebra

Elgendy

2018

Self Cite

“…Let Con(7) ⊂ SkewTS(7) be the S 7 -submodule generated by the consequences (8) with respect to operadic partial composition of the relation T T (a, b, c, d, e) displayed in Theorem 5, and let ConNew(7) ⊂ SkewTS(7) be the S 7 -submodule generated by those consequences together with the 60-term relation in Figure 2. The quotient module ConNew(7)/Con (7) has dimension 106 and the following multiplicity-free decomposition into the direct sum of irreducible representations: The dimensions of the irreducible summands are respectively 35, 21, 35, 15. Let All(7) ⊂ SkewTS(7) be the S 7 -module consisting of all relations satisfied by the tortkara triple product in arity 7.…”

Section: Relations Of Aritymentioning

confidence: 99%

On tortkara triple systems

Communications in Algebra

2017

The commutator [a, b] = ab − ba in a free Zinbiel algebra (dual Leibniz algebra) is an anticommutative operation which satisfies no new relations in arity 3. Dzhumadildaev discovered a relation T (a, b, c, d) which he called the tortkara identity and showed that it implies every relation satisfied by the Zinbiel commutator in arity 4. Kolesnikov constructed examples of anticommutative algebras satisfying T (a, b, c, d) which cannot be embedded into the commutator algebra of a Zinbiel algebra. We consider the tortkara triple product [a, b, c] = [[a, b], c] in a free Zinbiel algebra and use computer algebra to construct a relation T T (a, b, c, d, e) which implies every relation satisfied by [a, b, c] in arity 5. Thus, although tortkara algebras are defined by a cubic binary operad (with no Koszul dual), the corresponding triple systems are defined by a quadratic ternary operad (with a Koszul dual). We use computer algebra to construct a relation in arity 7 satisfied by [a, b, c] which does not follow from the relations of lower arity. It remains an open problem to determine whether there are further new identities in arity ≥ 9.

“…If the number of integer vectors is not too large, roughly < 500, then at this point we can apply the LLL algorithm for lattice basis reduction [6] to reduce the size of the coefficients. In any case, we then sort the integer vectors first by increasing number of nonzero components, then by increasing Euclidean norm, and finally by increasing maximum nonzero component (in absolute value).…”

Section: Two Jordan Productsmentioning

confidence: 99%

Lie and Jordan Products in Interchange Algebras

Communications in Algebra

Madariaga

2016

We study Lie brackets and Jordan products derived from associative opera-We use computational linear algebra, based on the representation theory of the symmetric group, to determine all polynomial identities of degree ≤ 7 relating (i) the two Lie brackets, (ii) one Lie bracket and one Jordan product, and (iii) the two Jordan products. For the Lie-Lie case, there are two new identities in degree 6 and another two in degree 7. For the Lie-Jordan case, there are no new identities in degree ≤ 6 and a complex set of new identities in degree 7. For the Jordan-Jordan case, there is one new identity in degree 4, two in degree 5, and complex sets of new identities in degrees 6 and 7.