arXiv subjects
Publications and source records attributed to Murray Bremner.
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 resolve a ten year old open question of Loday of describing Koszul operads that act on the algebra of octonions. In fact, we obtain the answer by solving a more general classification problem: we find all Koszul operads among those encoding associator dependent algebras.
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.
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 basis properties.
Dotsenko and Vallette discovered an extension to nonsymmetric operads of Buchberger's algorithm for Gr\"obner bases of polynomial ideals. In the free nonsymmetric operad with one ternary operation $({\ast}{\ast}{\ast})$, we compute a Gr\"obner 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 $({\ast}{\ast}{\ast})$ may be even or odd. We use the Gr\"obner bases to calculate the dimension formulas for these operads.
By the Three Graces we refer, following J.-L. Loday, to the algebraic operads Ass, Com, and Lie, each generated by a single binary operation; algebras over these operads are respectively associative, commutative associative, and Lie. We classify all distributive laws (in the categorical sense of Beck) between these three operads. Some of our results depend on the computer algebra system Maple, especially its packages LinearAlgebra and Groebner.
We extend the work of Kock (2007) and Bremner & Madariaga (2016) on commutativity in double interchange semigroups (DIS) to relations with 10 arguments. Our methods involve the free symmetric operad generated by two binary operations with no symmetry, its quotient by the two associative laws, its quotient by the interchange law, and its quotient by all three laws. We also consider the geometric realization of free double interchange magmas by rectangular partitions of the unit square $I^2$. We define morphisms between these operads which allow us to represent elements of free DIS both algebraically as tree monomials and geometrically as rectangular partitions. With these morphisms we reason diagrammatically about free DIS and prove our new commutativity relations.
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 $TT(a,b,c,d,e)$ 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 $\ge 9$.
We establish a combinatorial model for the Boardman--Vogt tensor product of several absolutely free operads, that is free symmetric operads that are also free as $\mathbb{S}$-modules. Our results imply that such a tensor product is always a free $\mathbb{S}$-module, in contrast with the results of Kock and Bremner--Madariaga on hidden commutativity for the Boardman--Vogt tensor square of the operad of non-unital associative algebras.
We compute minimal sets of generators for the S_n-modules (n <= 4) of multilinear polynomial identities of arity n satisfied by the Jordan product and the Jordan diproduct (resp. pre-Jordan product) in every triassociative (resp. tridendriform) algebra. These identities define Jordan trialgebras and post-Jordan algebras: Jordan analogues of the Lie trialgebras and post-Lie algebras introduced by Dotsenko et al., Pei et al., Vallette & Loday. We include an extensive review of analogous structures existing in the literature, and their interrelations, in order to identify the gaps filled by our two new varieties of algebras. We use computer algebra (linear algebra over finite fields, representation theory of symmetric groups), to verify in both cases that every polynomial identity of arity <= 6 is a consequence of those of arity <= 4. We conjecture that in both cases the next independent identities have arity 8, imitating the Glennie identities for Jordan algebras. We formulate our results as a commutative square of operad morphisms, which leads to the conjecture that the squares in a much more general class are also commutative.
We consider nonsymmetric operads with two binary operations satisfying relations in arity 3; hence these operads are quadratic, and so we can investigate Koszul duality. We first consider operations which are nonassociative (not necessarily associative) and then specialize to the associative case. We obtain a complete classification of self-dual quadratic nonsymmetric operads with two (associative or nonassociative) binary operations. These operads generalize associativity for one operation to the setting of two operations.
Jean-Louis Loday introduced a class of symmetric operads generated by one bilinear operation subject to one relation making each left-normed product of three elements equal to a linear combination of right-normed products: \[ (a_1a_2)a_3=\sum_{\sigma\in S_3}x_\sigma\, a_{\sigma(1)}(a_{\sigma(2)}a_{\sigma(3)})\ ; \] such an operad is called a parametrized one-relation operad. For a particular choice of parameters $\{x_\sigma\}$, this operad is said to be regular if each of its components is the regular representation of the symmetric group; equivalently, the corresponding free algebra on a vector space $V$ is, as a graded vector space, isomorphic to the tensor algebra of $V$. We classify, over an algebraically closed field of characteristic zero, all regular parametrized one-relation operads. In fact, we prove that each such operad is isomorphic to one of the following five operads: the left-nilpotent operad defined by the identity $((a_1a_2)a_3)=0$, the associative operad, the Leibniz operad, the dual Leibniz (Zinbiel) operad, and the Poisson operad. Our computational methods combine linear algebra over polynomial rings, representation theory of the symmetric group, and Gr\"obner bases for determinantal ideals and their radicals.
We study Lie brackets and Jordan products derived from associative operations $\circ, \bullet$ satisfying the interchange identity $(w \bullet x ) \circ ( y \bullet z ) \equiv (w \circ y ) \bullet ( x \circ z )$. We use computational linear algebra, based on the representation theory of the symmetric group, to determine all polynomial identities of degree $\le 7$ 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 $\le 6$ 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.
In part 1, we review the structure theory of $\mathbb{F} S_n$, the group algebra of the symmetric group $S_n$ over a field of characteristic 0. We define the images $\psi(E^\lambda_{ij})$ of the matrix units $E^\lambda_{ij}$ ($1 \le i, j \le d_\lambda$), where $d_\lambda$ is the number of standard tableaux of shape $\lambda$, and obtain an explicit construction of Young's isomorphism $\psi\colon \bigoplus_\lambda M_{d_\lambda}(\mathbb{F}) \to \mathbb{F} S_n$. We then present Clifton's algorithm for the construction of the representation matrices $R^\lambda(p) \in M_{d_\lambda}(\mathbb{F})$ for all $p \in S_n$, and obtain the reverse isomorphism $\phi\colon \mathbb{F} S_n \to \bigoplus_\lambda M_{d_\lambda}(\mathbb{F})$. In part 2, we apply the structure theory of $\mathbb{F} S_n$ to the study of multilinear polynomial identities of degree $n \le 7$ for the algebra $\mathbb{O}$ of octonions over a field of characteristic 0. We compare our results with earlier work of Racine, Hentzel & Peresi, and Shestakov & Zhukavets on the identities of degree $n \le 6$. We use computational linear algebra to verify that every identity in degree 7 is a consequence of the known identities of lower degrees: there are no new identities in degree 7. We conjecture that the known identities of degree $\le 6$ generate all octonion identities in characteristic 0.
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 \cite{Kock2007} (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} = abcd + dcba 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] = abcd - dcba. 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.
On the set H_n(K) of symmetric n by n matrices over the field K we can define various binary and ternary products which endow it with the structure of a Jordan algebra or a Lie or Jordan triple system. All these non-associative structures have the orthogonal Lie algebra so(n,K) as derivation algebra. This gives an embedding of so(n,K) into so(N,K) for N = n(n+1)/2 - 1. We obtain a sequence of reductive pairs (so(N,K), so(n,K)) that provides a family of irreducible Lie-Yamaguti algebras. In this paper we explain in detail the construction of these Lie-Yamaguti algebras. In the cases n < 5, we use computer algebra to determine the polynomial identities of degree < 7; we also study the identities relating the bilinear Lie-Yamaguti product with the trilinear product obtained from the Jordan triple product.
We briefly review previous work on the invariant theory of 3 x 3 x 3 arrays. We then recall how to generate arrays of arbitrary size m_1 x ... x m_k with hyperdeterminant 0. Our main result is an explicit formula for the 3 x 3 x 3 hyperdeterminant as a polynomial in the fundamental invariants of degrees 6, 9 and 12 for the action of the Lie group SL(3,C) x SL(3,C) x SL(3,C). We apply our calculations to Nurmiev's classification of normal forms for 3 x 3 x 3 arrays.