Poisson triple systems Bremner, Murray R.; Elgendy, Hader A.
Linear & multilinear algebra,
05/2023, Volume:
71, Issue:
7
Journal Article
Peer reviewed
We introduce Poisson triple systems, which are vector spaces with 3 trilinear operations satisfying 9 polynomial identities of degree 5. We show that every Poisson triple system has a universal ...enveloping Poisson algebra. Finally, we briefly discuss operadic aspects of Poisson triple systems.
We study polynomial identities satisfied by the mutation product
x
p
y
-
y
q
x
on the underlying vector space of an associative algebra
A
, where
p
,
q
are fixed elements of
A
. We simplify known ...results for identities in degree 4, proving that only two identities are necessary and sufficient to generate them all; in degree 5, we show that adding one new identity suffices; in degree 6, we demonstrate the existence of a significant number of new identities, which induce us to conjecture that the variety generated by mutation algebras of associative algebras is not finitely based.
Livernet and Loday constructed a polarization of the nonsymmetric associative operad
with one operation into a symmetric operad
with two operations (the Lie bracket and Jordan product), and defined a ...one-parameter deformation of
, which includes Poisson algebras. We combine this with the dendriform splitting of an associative operation into the sum of two nonassociative operations, and use Koszul duality for quadratic operads, to construct one-parameter deformations of the nonsymmetric dendriform and diassociative operads into the category of symmetric operads.
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
and special translator
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
matrices.
We use computer algebra to demonstrate the existence of a multilinear polynomial identity of degree 8 satisfied by the bilinear operation in every Lie-Yamaguti algebra. This identity is a consequence ...of the defining identities for Lie-Yamaguti algebras, but is not a consequence of anticommutativity. We give an explicit form of this identity as an alternating sum over all permutations of the variables in a nonassociative polynomial with 8 terms. Our computations show that no such identities exist in degrees less than 8.
Velásquez and Felipe recently introduced quasi-Jordan algebras based on the product
in an associative dialgebra with operations ⊣ and ⊢. We determine the polynomial identities of degree ≤4 satisfied ...by this product. In addition to right commutativity and the right quasi-Jordan identity, we obtain a new associator-derivation identity.
We use computational linear algebra and commutative algebra to study spaces of relations satisfied by quadrilinear operations. The relations are analogues of associativity in the sense that they are ...quadratic (every term involves two operations) and nonsymmetric (every term involves the identity permutation of the arguments). We focus on determining those quadratic relations whose cubic consequences have minimal or maximal rank. We approach these problems from the point of view of the theory of algebraic operads.
We introduce a new approach to the classification of operator identities, based on basic concepts from the theory of algebraic operads together with computational commutative algebra applied to ...determinantal ideals of matrices over polynomial rings. We consider operator identities of degree 2 (the number of variables in each term) and multiplicity 1 or 2 (the number of operators in each term), but our methods apply more generally. Given an operator identity with indeterminate coefficients, we use partial compositions to construct a matrix of consequences, and then use computer algebra to determine the values of the indeterminates for which this matrix has submaximal rank. For multiplicity 1 we obtain six identities, including the derivation identity. For multiplicity 2 we obtain eighteen identities and two parametrized families, including the left and right averaging identities, the Rota-Baxter identity, the Nijenhuis identity, and some new identities which deserve further study.
We use computer algebra to determine all the multilinear polynomial identities of degree ≤7 satisfied by the trilinear operations (a·b)·c and a·(b·c) in the free dendriform dialgebra, where a·b is ...the pre-Lie or the pre-Jordan product. For the pre-Lie triple products, we obtain one identity in degree 3, and three independent identities in degree 5, and we show that every identity in degree 7 follows from the identities of lower degree. For the pre-Jordan triple products, there are no identities in degree 3, five independent identities in degree 5, and ten independent irreducible identities in degree 7. Our methods involve linear algebra on large matrices over finite fields, and the representation theory of the symmetric group.
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