arXiv subjects

Murray R. Bremner

Publications and source records attributed to Murray R. Bremner.

At least 37 records · Page 2Linked to original sources

arXiv2012

Higher identities for the ternary commutator

We use computer algebra to study polynomial identities for the trilinear operation [a,b,c] = abc - acb - bac + bca + cab - cba in the free associative algebra. It is known that [a,b,c] satisfies the alternating property in degree 3, no new identities in degree 5, a multilinear identity in degree 7 which alternates in 6 arguments, and no new identities in degree 9. We use the representation theory of the symmetric group to demonstrate the existence of new identities in degree 11. The only irreducible representations of dimension < 400 with new identities correspond to partitions 2^5 1 and 2^4 1^3 and have dimensions 132 and 165. We construct an explicit new multilinear identity for partition 2^5 1 and we demonstrate the existence of a new non-multilinear identity in which the underlying variables are permutations of a^2 b^2 c^2 d^2 e^2 f.

math.RA↗

arXiv2012

Canonical forms of small tensors over F_2

We consider multidimensional arrays with at most 27 entries over the field with two elements, and their equivalence classes for the action of the direct product of general linear groups. The possible 3-dimensional formats are p x 2 x 2 (p = 2, ..., 6), p x 3 x 2 (p = 3, 4), and 3 x 3 x 3; the possible 4-dimensional formats are p x 2 x 2 x 2 (p = 2, 3). In each case, we compute the orbits for the group action, and then we determine the rank of each orbit. In particular, we determine the maximum rank for these arrays over F_2.

math.CO↗

arXiv2012

The fundamental invariants of 3 x 3 x 3 arrays

We determine the three fundamental invariants in the entries of a $3 \times 3 \times 3$ array over $\mathbb{C}$ as explicit polynomials in the 27 variables $x_{ijk}$ for $1 \le i, j, k \le 3$. By the work of Vinberg on $θ$-groups, it is known that these homogeneous polynomials have degrees 6, 9 and 12; they freely generate the algebra of invariants for the Lie group $SL_3(\mathbb{C}) \times SL_3(\mathbb{C}) \times SL_3(\mathbb{C})$ acting irreducibly on its natural representation $\mathbb{C}^3 \otimes \mathbb{C}^3 \otimes \mathbb{C}^3$. These generators have respectively 1152, 9216 and 209061 terms; we find compact expressions in terms of the orbits of the finite group $(S_3 \times S_3 \times S_3) \rtimes S_3$ acting on monomials of weight zero for the action of the Lie algebra $\mathfrak{sl}_3(\mathbb{C}) \oplus \mathfrak{sl}_3(\mathbb{C}) \oplus \mathfrak{sl}_3(\mathbb{C})$.

math.AC↗

arXiv2012

Algebras, dialgebras, and polynomial identities

This is a survey of some recent developments in the theory of associative and nonassociative dialgebras, with an emphasis on polynomial identities and multilinear operations. We discuss associative, Lie, Jordan, and alternative algebras, and the corresponding dialgebras; the KP algorithm for converting identities for algebras into identities for dialgebras; the BSO algorithm for converting operations in algebras into operations in dialgebras; Lie and Jordan triple systems, and the corresponding disystems; and a noncommutative version of Lie triple systems based on the trilinear operation abc-bca. The paper concludes with a conjecture relating the KP and BSO algorithms, and some suggestions for further research. Most of the original results are joint work with Raul Felipe, Luiz A. Peresi, and Juana Sanchez-Ortega.

math.RA↗

arXiv2011

Ranks of 0-1 arrays of size 2 x 2 x 2 and 2 x 2 x 2 x 2

We use computer algebra to determine the ranks of arrays of size 2 x 2 x 2 and 2 x 2 x 2 x 2 with entries in the set {0, 1} regarded as a field with two elements, as a Boolean algebra, and as non-negative integers. In the field case we also determine the canonical forms of the arrays with respect to the action of the direct product of the general linear groups.

math.CO↗

arXiv2011

The hyperdeterminant of 3 x 3 x 2 arrays, and the simplest invariant of 4 x 4 x 2 arrays

We use the representation theory of Lie algebras and computational linear algebra to obtain an explicit formula for the hyperdeterminant of a $3 \times 3 \times 2$ array: a homogeneous polynomial of degree 12 in 18 variables with 16749 monomials and 41 distinct integer coefficients; the monomials belong to 178 orbits under the action of $(S_3 \times S_3 \times S_2) \rtimes S_2$. We also obtain the simplest invariant for a $4 \times 4 \times 2$ array: a homogeneous polynomial of degree 8 in 32 variables with 14148 monomials and 13 distinct integer coefficients; the monomials belong to 28 orbits under $(S_4 \times S_4 \times S_2) \rtimes S_2$.

math.RT↗

arXiv2011

Polynomial identities for tangent algebras of monoassociative loops

We introduce degree n Sabinin algebras, which are defined by the polynomial identities up to degree n in a Sabinin algebra. Degree 4 Sabinin algebras can be characterized by the polynomial identities satisfied by the commutator, associator and two quaternators in the free nonassociative algebra. We consider these operations in a free power associative algebra and show that one of the quaternators is redundant. The resulting algebras provide the natural structure on the tangent space at the identity element of an analytic loop for which all local loops satisfy monoassociativity, a^2 a = a a^2. These algebras are the next step beyond Lie, Malcev, and Bol algebras. We also present an identity of degree 5 which is satisfied by these three operations but which is not implied by the identities of lower degree.

math.RA↗

arXiv2011

The simple non-Lie Malcev algebra as a Lie-Yamaguti algebra

The simple 7-dimensional Malcev algebra $M$ is isomorphic to the irreducible $\mathfrak{sl}(2,\mathbb{C})$-module V(6) with binary product $[x,y] = α(x \wedge y)$ defined by the $\mathfrak{sl}(2,\mathbb{C})$-module morphism $α\colon Λ^2 V(6) \to V(6)$. Combining this with the ternary product $(x,y,z) = β(x \wedge y) \cdot z$ defined by the $\mathfrak{sl}(2,\mathbb{C})$-module morphism $β\colon Λ^2 V(6) \to V(2) \approx \s$ gives $M$ the structure of a generalized Lie triple system, or Lie-Yamaguti algebra. We use computer algebra to determine the polynomial identities of low degree satisfied by this binary-ternary structure.

math.RA↗

arXiv2011

Malcev dialgebras

We apply Kolesnikov's algorithm to obtain a variety of nonassociative algebras defined by right anticommutativity and a `noncommutative' version of the Malcev identity. We use computational linear algebra to verify that these identities are equivalent to the identities of degree <= 4 satisfied by the dicommutator in every alternative dialgebra. We extend these computations to show that any special identity for Malcev dialgebras must have degree at least 7. Finally, we introduce a trilinear operation which makes any Malcev dialgebra into a Leibniz triple system.

math.RA↗

arXiv2011

Leibniz triple systems

We define Leibniz triple systems in a functorial manner using the algorithm of Kolesnikov and Pozhidaev which converts identities for algebras into identities for dialgebras. We verify that Leibniz triple systems are the natural analogues of Lie triple systems in the context of dialgebras by showing that both the iterated bracket in a Leibniz algebra and the permuted associator in a Jordan dialgebra satisfy the defining identities for Leibniz triple systems. We construct the universal Leibniz envelopes of Leibniz triple systems and prove that every identity satisfied by the iterated bracket in a Leibniz algebra is a consequence of the defining identities for Leibniz triple systems. To conclude, we present some examples of 2-dimensional Leibniz triple systems and their universal Leibniz envelopes.

math.RA↗

arXiv2011

Cayley's hyperdeterminant: a combinatorial approach via representation theory

Cayley's hyperdeterminant is a homogeneous polynomial of degree 4 in the 8 entries of a 2 x 2 x 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 sl_2(C) to reduce the problem of finding the invariant polynomials for a 2 x 2 x 2 array to a combinatorial problem on the enumeration of 2 x 2 x 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 show how this approach can be applied to general multidimensional arrays.

math.RT↗

arXiv2011

A hyperdeterminant for 2 x 2 x 3 arrays

We use the representation theory of Lie algebras and computational linear algebra to determine the simplest nonconstant invariant polynomial in the entries of a general 2 x 2 x 3 array. This polynomial is homogeneous of degree 6 and has 66 terms with coefficients 1, -1, 2, -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 x 2 x 2 arrays.

math.RT↗

arXiv2011

Jordan Triple Disystems

We take an algorithmic and computational approach to a basic problem in abstract algebra: determining the correct generalization to dialgebras of a given variety of nonassociative algebras. We give a simplified statement of the KP algorithm introduced by Kolesnikov and Pozhidaev for extending polynomial identities for algebras to corresponding identities for dialgebras. We apply the KP algorithm to the defining identities for Jordan triple systems to obtain a new variety of nonassociative triple systems, called Jordan triple disystems. We give a generalized statement of the BSO algorithm introduced by Bremner and Sanchez-Ortega for extending multilinear operations in an associative algebra to corresponding operations in an associative dialgebra. We apply the BSO algorithm to the Jordan triple product and use computer algebra to verify that the polynomial identities satisfied by the resulting operations coincide with the results of the KP algorithm; this provides a large class of examples of Jordan triple disystems. We formulate a general conjecture expressed by a commutative diagram relating the output of the KP and BSO algorithms. We conclude by generalizing the Jordan triple product in a Jordan algebra to operations in a Jordan dialgebra; we use computer algebra to verify that resulting structures provide further examples of Jordan triple disystems. For this last result, we also provide an independent theoretical proof using Jordan structure theory.

math.RA↗

arXiv2010

Special identities for quasi-Jordan algebras

Semispecial quasi-Jordan algebras (also called Jordan dialgebras) are defined by the polynomial identities $a(bc) = a(cb)$, $(ba)a^2 = (ba^2)a$, and $(b,a^2,c) = 2(b,a,c)a$. These identities are satisfied by the product $ab = a \dashv b + b \vdash a$ in an associative dialgebra. We use computer algebra to show that every identity for this product in degree $\le 7$ is a consequence of the three identities in degree $\le 4$, but that six new identities exist in degree 8. Some but not all of these new identities are noncommutative preimages of the Glennie identity.

math.RA↗

arXiv2010

Enveloping algebras of the nilpotent Malcev algebra of dimension five

Perez-Izquierdo and Shestakov recently extended the PBW theorem to Malcev algebras. It follows from their construction that for any Malcev algebra $M$ over a field of characteristic $\ne 2, 3$ there is a representation of the universal nonassociative enveloping algebra $U(M)$ by linear operators on the polynomial algebra $P(M)$. For the nilpotent non-Lie Malcev algebra $\mathbb{M}$ of dimension 5, we use this representation to determine explicit structure constants for $U(\mathbb{M})$; from this it follows that $U(\mathbb{M})$ is not power-associative. We obtain a finite set of generators for the alternator ideal $I(\mathbb{M}) \subset U(\mathbb{M})$ and derive structure constants for the universal alternative enveloping algebra $A(\mathbb{M}) = U(\mathbb{M})/I(\mathbb{M})$, a new infinite dimensional alternative algebra. We verify that the map $ι\colon \mathbb{M} \to A(\mathbb{M})$ is injective, and so $\mathbb{M}$ is special.

math.RA↗

arXiv2010

Lie invariants in two and three variables

We use computer algebra to determine the Lie invariants of degree <= 12 in the free Lie algebra on two generators corresponding to the natural representation of the simple 3-dimensional Lie algebra sl(2,C). We then consider the free Lie algebra on three generators, and compute the Lie invariants of degree <= 7 corresponding to the adjoint representation of sl(2,C), and the Lie invariants of degree <= 9 corresponding to the natural representation of sl(3,C). We represent the action of sl(2,C) and sl(3,C) on Lie polynomials by computing the coefficient matrix with respect to the basis of Hall words. We then use algorithms for linear algebra (row canonical form, Hermite normal form, lattice basis reduction) to compute a basis of the nullspace.

math.RA↗

arXiv2010

Enveloping algebras of Malcev algebras

We first discuss the construction by Perez-Izquierdo and Shestakov of universal nonassociative enveloping algebras of Malcev algebras. We then describe recent results on explicit structure constants for the universal enveloping algebras (both nonassociative and alternative) of the 4-dimensional solvable Malcev algebra and the 5-dimensional nilpotent Malcev algebra. We include a proof (due to Shestakov) that the universal alternative enveloping algebra of the real 7-dimensional simple Malcev algebra is isomorphic to the 8-dimensional division algebra of real octonions. We conclude with some brief remarks on tangent algebras of analytic Bol loops and monoassociative loops. This is a revised and expanded version of the survey talk given by the first author at the Second Mile High Conference on Nonassociative Mathematics at the University of Denver (Denver, Colorado, USA, June 22 to 26, 2009).

math.RA↗

arXiv2010

Universal associative envelopes of (n+1)-dimensional n-Lie algebras

For n even, we prove Pozhidaev's conjecture on the existence of associative enveloping algebras for simple n-Lie 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 from L to 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.

math.RA↗