Nijenhuis Algebra Research Articles

Nijenhuis operators and associative D-bialgebras

Firstly, we prove that a quasitriangular associative D-bialgebra together with a coquasitriangular structure can induce a Nijenhuis algebra. This shows that Nijenhuis operators and associative D-bialgebras are closely related. Secondly, we investigate some properties of a representation of a Nijenhuis algebra and introduce the notion of a corepresentation of a Nijenhuis coalgebra, which also can be seen as an infinitesimal deformation of compatible bicomodule. Thirdly, a Nijenhuis associative D-bialgebra can be established by a Nijenhuis algebra and a Nijenhuis coalgebra satisfying some compatible conditions. Furthermore, we find that a Nijenhuis associative D-bialgebra is equivalent to a matched pair of Nijenhuis algebras or a double construction of a Nijenhuis Frobenius algebra. Lastly, we provide some methods to construct Nijenhuis associative D-bialgebras through admissible associative Yang-Baxter equations (aYBes for short) and O-operators on Nijenhuis algebras.

Left counital Hopf algebra structures on free commutative Nijenhuis algebras

Motivated by the Hopf algebra structures established on free commutative Rota-Baxter algebras, we explore Hopf algebra related structures on free commutative Nijenhuis algebras. Applying a cocycle condition, we first prove that a free commutative Nijenhuis algebra on a left counital bialgebra (in the sense that the right-sided counicity needs not hold) can be enriched to a left counital bialgebra. We then establish a general result that a connected graded left counital bialgebra is a left counital right antipode Hopf algebra in the sense that the antipode is also only right-sided. We finally apply this result to show that the left counital bialgebra on a free commutative Nijenhuis algebra on a connected left counital bialgebra is connected and graded, hence is a left counital right antipode Hopf algebra.

Left counital Hopf algebras on free Nijenhuis algebras

ABSTRACTFactorization in algebra is an important problem. In this paper, we first obtain a unique factorization in free Nijenhuis algebras. By using of this unique factorization, we then define a coproduct and a left counital bialgebraic structure on a free Nijenhuis algebra. Finally, we prove that this left counital bialgebra is connected and hence obtain a left counital Hopf algebra on a free Nijenhuis algebra.

Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases

In this paper we apply the methods of rewriting systems and Grobner-Shirshov bases to give a unified approach to a class of linear operators on associative algebras. These operators resemble the classic Rota-Baxter operator, and they are called Rota-Baxter type operators. We characterize a Rota-Baxter type operator by the convergency of a rewriting system associated to the operator. By associating such an operator to a Grobner-Shirshov basis, we obtain a canonical basis for the free algebras in the category of associative algebras with that operator. This construction include as special cases several previous ones for free objects in similar categories, such as those of Rota-Baxter algebras and Nijenhuis algebras.

Nijenhuis algebras, NS algebras, and N-dendriform algebras

In this paper, we study (associative) Nijenhuis algebras, with emphasis on the relationship between the category of Nijenhuis algebras and the categories of NS algebras and related algebras. This is in analogy to the well-known theory of the adjoint functor from the category of Lie algebras to that of associative algebras, and the more recent results on the adjoint functor from the categories of dendriform and tridendriform algebras to that of Rota-Baxter algebras. We first give an explicit construction of free Nijenhuis algebras and then apply it to obtain the universal enveloping Nijenhuis algebra of an NS algebra. We further apply the construction to determine the binary quadratic nonsymmetric algebra, called the N-dendriform algebra, that is compatible with the Nijenhuis algebra. As it turns out, the N-dendriform algebra has more relations than the NS algebra.

Differential type operators and Gröbner–Shirshov bases

A long standing problem of Gian-Carlo Rota for associative algebras is the classification of all linear operators that can be defined on them. In the 1970s, there were only a few known operators, for example, the derivative operator, the difference operator, the average operator, and the Rota–Baxter operator. A few more appeared after Rota posed his problem. However, little progress was made to solve this problem in general. In part, this is because the precise meaning of the problem is not so well understood. In this paper, we propose a formulation of the problem using the framework of operated algebras and viewing an associative algebra with a linear operator as one that satisfies a certain operated polynomial identity. This framework also allows us to apply theories of rewriting systems and Gröbner–Shirshov bases. To narrow our focus more on the operators that Rota was interested in, we further consider two particular classes of operators, namely, those that generalize differential or Rota–Baxter operators. These two classes of operators correspond to those that possess Gröbner–Shirshov bases under two different monomial orderings. Using this framework and computer algebra, we are able to come up with a list of these two classes of operators, and provide some evidence that these lists may be complete. Our search has revealed quite a few new operators of these types whose properties are expected to be similar to the differential operator and Rota–Baxter operator respectively.Recently, a more unified approach has emerged in related areas, such as difference algebra and differential algebra, and Rota–Baxter algebra and Nijenhuis algebra. The similarities in these theories can be more efficiently explored by advances on Rotaʼs problem.

Some spacetimes with higher rank Killing–Stäckel tensors

By applying the lightlike Eisenhart lift to several known examples of low-dimensional integrable systems admitting integrals of motion of higher-order in momenta, we obtain four- and higher-dimensional Lorentzian spacetimes with irreducible higher-rank Killing tensors. Such metrics, we believe, are first examples of spacetimes admitting higher-rank Killing tensors. Included in our examples is a four-dimensional supersymmetric pp-wave spacetime, whose geodesic flow is superintegrable. The Killing tensors satisfy a non-trivial Poisson–Schouten–Nijenhuis algebra. We discuss the extension to the quantum regime.

Generalized Shuffles Related to Nijenhuis and TD-Algebras

Shuffle type products are well known in mathematics and physics. They are intimately related to Loday's dendriform algebras and were extensively used to give explicit constructions of free Rota–Baxter algebras. In the literature there exist at least two other Rota–Baxter type algebras, namely, the Nijenhuis algebra and the so-called TD-algebra. The explicit construction of the free unital commutative Nijenhuis algebra uses a modified quasi-shuffle product, called the right-shift shuffle. We show that another modification of the quasi-shuffle, the so-called left-shift shuffle, can be used to give an explicit construction of the free unital commutative TD-algebra. We explore some basic properties of TD-operators. Our construction is related to Loday's unital commutative tridendriform algebra, including the involutive case. The concept of Rota–Baxter, Nijenhuis and TD-bialgebras is introduced at the end, and we show that any commutative bialgebra provides such objects.

On the Associative Nijenhuis Relation

We give the construction of a free commutative unital associative Nijenhuis algebra on a commutative unital associative algebra based on an augmented modified quasi-shuffle product.

Schouten–Nijenhuis brackets

The Poisson and graded Poisson Schouten–Nijenhuis algebras of symmetric and antisymmetric contravariant tensor fields, respectively, on an n-dimensional manifold M are shown to be n-symplectic. This is accomplished by showing that both brackets may be defined in a unified way using the n-symplectic structure on the bundle of linear frames LM of M. New results in n-symplectic geometry are presented and then used to give globally defined representations of the Hamiltonian operators defined by the Schouten–Nijenhuis brackets.