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.
The paper is devoted to give a complete classification of all n-dimensional non-associative Jordan algebras with (n−3)-dimensional annihilator over an algebraically closed field of characteristic ≠2. ...We also give a complete classification of all n-dimensional Jordan algebras with (n−1)- and (n−2)-dimensional annihilator.
We study Frobenius extensions which are free-filtered by a totally ordered, finitely generated abelian group, and their free-graded counterparts. First we show that the Frobenius property passes up ...from a free-graded extension to a free-filtered extension, then also from a free-filtered extension to the extension of their Rees algebras. Our main theorem states that, under some natural hypotheses, a free-filtered extension of algebras is Frobenius if and only if the associated graded extension is Frobenius. In the final section we apply this theorem to provide new examples and non-examples of Frobenius extensions.
In this paper we give the classification of 5-dimensional complex nilpotent associative algebras. We use
- the isomorphism invariant. We prove that there are only two possible isomorphism invariants
...or
for any 5-dimensional complex nilpotent associative algebra
with the condition
and we give the classification (up to isomorphism) of such type algebras. 5-dimensional complex nilpotent associative algebras with the conditions
and
have already been classified. Thus, we complete the classification of 5-dimensional complex nilpotent associative algebras.
We extend the results about left-invariant Codazzi tensor fields on Lie groups equipped with left-invariant Riemannian metrics obtained by d’Atri in 1985 to the setting of reductive homogeneous ...spaces
G
/
H
, where the curvature of the canonical connection of second kind associated with the fixed reductive decomposition
g
=
h
⊕
m
enters the picture. In particular, we show that invariant Codazzi tensor fields on a naturally reductive homogeneous space are parallel.
Several constructive homological methods based on noncommutative Gröbner bases are known to compute free resolutions of associative algebras. In particular, these methods relate the Koszul property ...for an associative algebra to the existence of a quadratic Gröbner basis of its ideal of relations. In this article, using a higher-dimensional rewriting theory approach, we give several improvements of these methods. We define polygraphs for associative algebras as higher-dimensional linear rewriting systems that generalise the notion of noncommutative Gröbner bases, and allow more possibilities of termination orders than those associated to monomial orders. We introduce polygraphic resolutions of associative algebras, giving a categorical description of higher-dimensional syzygies for presentations of algebras. We show how to compute polygraphic resolutions starting from a convergent presentation, and how these resolutions can be linked with the Koszul property.
A linear locally nilpotent derivation of the polynomial algebra KXm in m variables over a field K of characteristic 0 is called a Weitzenböck derivation. It is well known from the classical theorem ...of Weitzenböck that the algebra of constants KXmδ of a Weitzenböck derivation δ is finitely generated. Assume that δ acts on the polynomial algebra KX2d in 2d variables as follows: δ(x2i)=x2i−1, δ(x2i−1)=0, i=1,…,d. The Nowicki conjecture states that the algebra KX2dδ is generated by x1,x3.…,x2d−1, and x2i−1x2j−x2ix2j−1, 1≤i<j≤d. The conjecture was proved by several authors based on different techniques. We apply the same idea to two relatively free algebras of rank 2d. We give the infinite set of generators of the algebra of constants in the free metabelian associative algebras F2d(A), and finite set of generators in the free algebra F2d(G) in the variety determined by the identities of the infinite dimensional Grassmann algebra.
We propose an algebraic viewpoint of the problem of deformation quantization of the so-called almost Poisson algebras, which are algebras with a commutative associative product and an antisymmetric ...bracket which is a biderivation but does not necessarily satisfy the Jacobi identity. From that viewpoint, the main result of the paper asserts that, by contrast with Poisson algebras, the only reasonable category of algebras in which almost Poisson algebras can be quantized is isomorphic to the category of almost Poisson algebras itself, and the trivial two-term quantization formula already gives a solution to the quantization problem.
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