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The cup product in the cohomology of algebras over quadratic operads has been studied in the general setting of Koszul duality for operads. We study the cup product on the cohomology of n-ary totally associative algebras with an operation of even (homological) degree. This cup product endows the cohomology with the structure of an n-ary partially associative algebra with an operation of even or odd degree depending on the parity of n. In the cases n=3 and n=4, we provide an explicit definition of this cup product and prove its basic properties.

We apply computer algebra, especially linear algebra over polynomial rings and Grobner bases, to classify inhomogeneous distributive laws between the operads for Lie algebras and commutative associative algebras.

Dotsenko and Vallette discovered an extension to nonsymmetric operads of Buchberger’s algorithm for Grobner bases of polynomial ideals. In the free nonsymmetric operad with one ternary operation \(({*}{*}{*})\), we compute a Grobner basis for the ideal generated by partial associativity \(((abc)de) + (a(bcd)e) + (ab(cde)\). In the category of \(\mathbb {Z}\)-graded vector spaces with Koszul signs, the (homological) degree of \(({*}{*}{*})\) may be even or odd. We use the Grobner bases to calculate the dimension formulas for these operads.

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,...

Using methods of computer algebra, especially Gr\"obner bases for submodules of free modules over polynomial rings, we solve a classification problem in theory of algebraic operads: we show that the only nontrivial (possibly inhomogeneous) distributive law between the operad of Lie algebras and the operad of commutative associative algebras is given by the Livernet-Loday formula deforming the Poisson operad into the associative operad., Comment: 8 pages

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., Comment: 25 pages, 15 tables

We study Lie brackets and Jordan products derived from associative operations ○, • satisfying the interchange identity (w•x) ○ (y•z) ≡ (w ○ y)•(x ○ z). 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.

A basic problem for any class of nonassociative algebras is to determine the polynomial identities satisfied by the symmetrization and the skew-symmetrization of the original product. We consider the symmetrization of the product in the class of special Jordan dialgebras. We use computational linear algebra to show that every polynomial identity of degree $n \le 5$ satisfied by the symmetrized Jordan diproduct in every diassociative algebra is a consequence of commutativity. We determine a complete set of generators for the polynomial identities in degree 6 which are not consequences of commutativity. We use a constructive version of the representation theory of the symmetric group to show that there exist further new identities in degree 7., 12 pages, 8 figures

Comtrans algebras, arising in web geometry, have two trilinear operations, commutator and translator. We determine a Gr\"obner 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 $\langle x, y, z \rangle = xyz-yzx$ 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 Gr\"obner bases to construct the universal associative envelope for the special comtrans algebra of $2 \times 2$ matrices., Comment: 18 pages, 4 figures. Sections 3.2 and 3.3 have been rewritten to emphasize the importance of the forgetful functor from symmetric operads to shuffle operads

Double semigroups have two associative operations $$\circ , \bullet $$ related by the interchange relation: $$( a \bullet b ) \circ ( c \bullet d ) \equiv ( a \circ c ) \bullet ( b \circ d )$$ . Kock (J Homotopy Relat Struct 2(2):217–228, 2007) discovered a commutativity property in degree 16 for double semigroups: associativity and the interchange relation combine to produce permutations of elements. We show that such properties can be expressed in terms of cycles in directed graphs with edges labelled by permutations. We use computer algebra to show that 9 is the lowest degree for which commutativity occurs, and we give self-contained proofs of the commutativity properties in degree 9.

We define Jordan quadruple systems by the polynomial identities of degrees 4 and 7 satisfied by the Jordan tetrad { a , b , c , d } = a b c d + d c b a as a quadrilinear operation on associative algebras. We find further identities in degree 10 which are not consequences of the defining identities. We introduce four infinite families of finite dimensional Jordan quadruple systems, and construct the universal associative envelope for a small system in each family. We obtain analogous results for the anti-tetrad [ a , b , c , d ] = a b c d − d c b a . Our methods rely on computer algebra, especially linear algebra on large matrices, the LLL algorithm for lattice basis reduction, representation theory of the symmetric group, noncommutative Grobner bases, and Wedderburn decompositions of associative algebras.

We use computer algebra to determine the Lie invariants of degree, 21 pages

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.

We consider arrays of size 2 × 2 × 2 and 2 × 2 × 2 × 2 over the fields with two and three elements. We use computer algebra to determine the canonical forms of these arrays with respect to the action of the semidirect product of general linear groups with the symmetric group. For each canonical form, we determine the size of its orbit and the rank of the arrays in its orbit.

In the first part, we consider 3 × 3 × 3 arrays with real or complex entries, and provide a self-contained proof of Kruskal’s theorem that the maximum rank is 5. In the second part, we provide a complete classification of the canonical forms of 3 × 3 × 3 arrays over the field F 2 with two elements; in particular, we obtain explicit examples of such arrays with rank 6.

We adapt the algorithm of Kolesnikov and Pozhidaev, which converts a polynomial identity for algebras into the corresponding identities for dialgebras, to the Cayley–Dickson doubling process. We obtain a generalization of this process to the setting of dialgebras, establish some of its basic properties, and construct dialgebra analogues of the quaternions and octonions.

We compute minimal sets of generators for the S_n-modules (n, 30 pages, 2 tables, 1 figure, 54 references

Livernet and Loday constructed a polarization of the nonsymmetric associative operad A with one operation into a symmetric operad SA with two operations (the Lie bracket and Jordan product), and defined a one-parameter deformation of SA 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., Comment: 8 pages

We use the representation theory of Lie algebras and computational linear algebra to determine the simplest non-constant invariant polynomial in the entries of a general 2 × 2 × 3 array. This polynomial is homogeneous of degree 6 and has 66 terms with coefficients ±1, ±2 in the 12 indeterminates x ijk where i, j = 1, 2 and k = 1, 2, 3. This invariant can be regarded as a natural generalization of Cayley's hyperdeterminant for 2 × 2 × 2 arrays.

Cayley’s hyperdeterminant is a homogeneous polynomial of degree 4 in the 8 entries of a 2 × 2 × 2 array. It is the simplest (nonconstant) polynomial which is invariant under changes of basis in three directions. We use elementary facts about representations of the 3-dimensional simple Lie algebra s l 2 ( C ) to reduce the problem of finding the invariant polynomials for a 2 × 2 × 2 array to a combinatorial problem on the enumeration of 2 × 2 × 2 arrays with non-negative integer entries. We then apply results from linear algebra to obtain a new proof that Cayley’s hyperdeterminant generates all the invariants. In the last section we discuss the application of our methods to general multidimensional arrays.

For n even, we prove Pozhidaev's conjecture on the existence of associative enveloping algebras for simple n-Lie (Filippov) algebras. More generally, for n even and any (n + 1)-dimensional n-Lie algebra L, we construct a universal associative enveloping algebra U(L) and show that the natural map L → U(L) is injective. We use noncommutative Grobner bases to present U(L) as a quotient of the free associative algebra on a basis of L and to obtain a monomial basis of U(L). In the last section, we provide computational evidence that the construction of U(L) is much more difficult for n odd.

We study the universal enveloping algebras of the one-parameter family of solvable 5-dimensional non-Lie Malcev algebras. We explicitly determine the universal nonassociative enveloping algebras (in the sense of Perez-Izquierdo and Shestakov) and the centers of the universal enveloping algebras. We also determine the universal alternative enveloping algebras., Comment: 18 pages

Velasquez and Felipe recently introduced quasi-Jordan algebras based on the product $a \triangleleft b = \tfrac12 ( a \dashv b + b \vdash a )$ in an associative dialgebra with operations $\dashv$ and $\vdash$. We determine the polynomial identities of degree $\le 4$ satisfied by this product. In addition to right commutativity and the right quasi-Jordan identity, we obtain a new associator-derivation identity., Comment: 8 pages, 5 tables

We determine the multiplicity of the irreducible representation V ( n ) of the simple Lie algebra sl 2 ( C ) as a direct summand of its fourth exterior power Λ 4 V ( n ) . The multiplicity is 1 (resp. 2) if and only if n = 4 , 6 (resp. n = 8 , 10 ). For these n we determine the multilinear polynomial identities of degree ⩽ 7 satisfied by the sl 2 ( C ) -invariant alternating quaternary algebra structures obtained from the projections Λ 4 V ( n ) → V ( n ) . We represent the polynomial identities as the nullspace of a large integer matrix and use computational linear algebra to find the canonical basis of the nullspace.

The semigroup of binary relations on {1,…, n} with the relative product is isomorphic to the semigroup B n of n × n zero-one matrices with the Boolean matrix product. Over any field F, we prove that the semigroup algebra FB n contains an ideal K n of dimension (2 n − 1)2, and we construct an explicit isomorphism of K n with the matrix algebra M 2 n −1(F).