Schreier rewriting beyond the classical setting

Using actions of free monoids and free associative algebras, we establish some Schreiertype formulas involving ranks of actions and ranks of subactions in free actions or Grassmann-type relations for the ranks of intersections of subactions of free actions. The coset action of the free group is used to establish a generalization of the Schreier formula in the case of subgroups of infinite index. We also study and apply large modules over free associative and free group algebras. This work was supported by Natural Sciences and Engineering Research Council of Canada (Grant No. 227060-04), Yuri Bahturin, National Science Foundation (Grant No. DMS-0700811) and Russian Fund for Basic Research (Grant No. 08-01-00573), Alexander Olshanskii     

A Hilbert-Nagata theorem in noncommutative invariant theory

Nagata gave a fundamental sufficient condition on group actions on finitely generated commutative algebras for finite generation of the subalgebra of invariants. In this paper we consider groups acting on noncommutative algebras over a field of characteristic zero. We characterize all the T-ideals of the free associative algebra such that the algebra of invariants in the corresponding relatively free algebra is finitely generated for any group action from the class of Nagata. In particular, in the case of unitary algebras this condition is equivalent to the nilpotency of the algebra in Lie sense. As a consequence we extend the Hilbert-Nagata theorem on finite generation of the algebra of invariants to any finitely generated associative algebra which is Lie nilpotent. We also prove that the Hilbert series of the algebra of invariants of a group acting on a relatively free algebra with a non-matrix polynomial identity is rational, if the action satisfies the condition of Nagata.

     

Wild automorphisms of free metabelian algebras

We establish a necessary condition for the invertibility of an endomorphism of a free associative algebra. As an application, we offer examples of wild automorphisms of certain free metabelian algebras.     

Gröbner-Shirshov bases for associative algebras with multiple operators and free Rota-Baxter algebras

In this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Gröbner-Shirshov bases of free Rota-Baxter algebra, free λ-differential algebra and free λ-differential Rota-Baxter algebra, respectively. In particular, linear bases of these three free algebras are respectively obtained, which are essentially the same or similar to the recent results obtained by K. Ebrahimi-Fard-L. Guo, and L. Guo-W. Keigher by using other methods.     

Simple Poisson–Farkas Algebras and Ternary Filippov Algebras

We establish connection between the differentiably simple associative commutative algebras with unity and the simple Filippov algebras.     

Coorbits of Quantum Homogeneous Spaces

The notion of coorbits for spaces with quantum group actions is introduced. A space with a quantum group action is given by a pair of algebras: an associative algebra which is the analog of a classical topological space, and a Hopf algebra which is the analog of a classical topological group. The Hopf algebra acts on the associative algebra via a comodule structure mapping which is also an algebra homomorphism. For a space with a quantum group action, a coorbit is a pair of spaces given by the image and the kernel of an algebra homomorphism from the associative algebra to the Hopf algebra. The coorbits of several types of quantum homogeneous spaces are discussed. In the case when the associative algebra is the group algebra of a group and the Hopf algebra is a quotient of the group algebra, the connection between the set of coorbits and the character group is established.     

Spectral theory for non-associative complete normed algebras and automatic continuity

This paper concerns normed algebras whose product is free of the requirement of associativity. For these algebras, a very natural notion of spectrum is provided and basic spectral properties of (associative) Banach algebras are extended to this general non-associative setting. These results are applied to generalize Rickart's dense-range homomorphism theorem. Our approach sheds some light on the classical associative theory.     

The free product of algebras with standard base for ideals

A wide class of associative algebras admitting standard bases of ideals was introduced in previous papers of the author. They are referred to as algebras with strong filtration. This class includes all known examples of associative algebras whose ideals possess standard bases. It was proved by the author that the class of algebras with strong filtration is closed with respect to direct sums and tensorial products. In this paper it is shown that this class is also closed with respect to free products of algebras.     

Operator Algebraic Quotient Structures Induced by Directed Graphs

Our result is about inclusions for (finite or infinite) countable directed graphs. In the proof, we use Free Probability Theory, groupoids, and algebras of operators (von Neumann algebras). We show that inclusions of directed graphs induce quotients for associated groupoid actions. With the use of operator thechniques, we then establish a duality between inclusions of graphs on the one hand and quotients of algebras on the other. Our main result is that each connected graph induces a quotient generated by a free group. This is a generalization of the notion of induced representations in the context of unitary representations of groups, i.e., the induction from the representations of a subgroup of an ambient group. The analogue is to systems of imprimitivity based on the homogeneous space. The parallel of this is the more general context of graphs (extending from groups to groupoids): We first prove that inclusions for connected graphs correspond to free group quotients, and we then build up the general case via connected components of given graphs.     

Pluriassociative algebras I: The pluriassociative operad

Diassociative algebras form a category of algebras recently introduced by Loday. A diassociative algebra is a vector space endowed with two associative binary operations satisfying some very natural relations. Any diassociative algebra is an algebra over the diassociative operad, and, among its most notable properties, this operad is the Koszul dual of the dendriform operad. We introduce here, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter γ of diassociative algebras, called γ-pluriassociative algebras, so that 1-pluriassociative algebras are diassociative algebras. Pluriassociative algebras are vector spaces endowed with 2γ associative binary operations satisfying some relations. We provide a complete study of the γ-pluriassociative operads, the underlying operads of the category of γ-pluriassociative algebras. We exhibit a realization of these operads, establish several presentations by generators and relations, compute their Hilbert series, show that they are Koszul, and construct the free objects in the corresponding categories. We also study several notions of units in γ-pluriassociative algebras and propose a general way to construct such algebras. This paper ends with the introduction of an analogous generalization of the triassociative operad of Loday and Ronco.     

Gröbner-Shirshov bases for Rota-Baxter algebras

We establish the composition-diamond lemma for associative nonunitary Rota-Baxter algebras of weight λ. To give an application, we construct a linear basis for a free commutative and nonunitary Rota-Baxter algebra, show that every countably generated Rota-Baxter algebra of weight 0 can be embedded into a two-generated Rota—Baxter algebra, and prove the 1-PBW theorems for dendriform dialgebras and trialgebras.     

The commutator width of some relatively free lie algebras and nilpotent groups

We determine the exact values of the commutator width of absolutely free and free solvable Lie rings of finite rank, as well as free and free solvable Lie algebras of finite rank over an arbitrary field. We calculate the values of the commutator width of free nilpotent and free metabelian nilpotent Lie algebras of rank 2 or of nilpotency class 2 over an arbitrary field. We also find the values of the commutator width for free nilpotent and free metabelian nilpotent Lie algebras of finite rank at least 3 over an arbitrary field in the case that the nilpotency class exceeds the rank at least by 2. In the case of free nilpotent and free metabelian nilpotent Lie rings of arbitrary finite rank, as well as free nilpotent and free metabelian nilpotent Lie algebras of arbitrary finite rank over the field of rationals, we calculate the values of commutator width without any restrictions. It follows in particular that the free or nonabelian free solvable Lie rings of distinct finite ranks, as well as the free or nonabelian free solvable Lie algebras of distinct finite ranks over an arbitrary field are not elementarily equivalent to each other. We also calculate the exact values of the commutator width of free ?-power nilpotent, free nilpotent, free metabelian, and free metabelian nilpotent groups of finite rank.     

Rota-Baxter algebras and dendriform algebras

In this paper we study the adjoint functors between the category of Rota-Baxter algebras and the categories of dendriform dialgebras and trialgebras. In analogy to the well-known theory of the adjoint functor between the category of associative algebras and Lie algebras, we first give an explicit construction of free Rota-Baxter algebras and then apply it to obtain universal enveloping Rota-Baxter algebras of dendriform dialgebras and trialgebras. We further show that free dendriform dialgebras and trialgebras, as represented by binary planar trees and planar trees, are canonical subalgebras of free Rota-Baxter 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.     

Identities of metabelian alternative algebras

We study metabelian alternative (in particular, associative) algebras over a field of characteristic 0. We construct additive bases of the free algebras of mentioned varieties, describe some centers of these algebras, compute the values of the sequence of codimensions of corresponding T-ideals, and find unitarily irreducible components of the decomposition of mentioned varieties into a union and their bases of identities. In particular, we find a basis of identities for the metabelian alternative Grassmann algebra. We prove that the free algebra of a variety that is generated by the metabelian alternative Grassmann algebra possesses the zero associative center.     

On the topological stable rank of non-selfadjoint operator algebras

We provide a negative solution to a question of M. Rieffel who asked if the right and left topological stable ranks of a Banach algebra must always agree. Our example is found amongst a class of nest algebras. We show that for many other nest algebras, both the left and right topological stable ranks are infinite. We extend this latter result to Popescu’s non-commutative disc algebras and to free semigroup algebras as well. K. R. Davidson, L. W. Marcoux, H. Radjavi’s research was supported in part by NSERC (Canada). An erratum to this article can be found at     

Local and subquotient inheritance of the heart¶of a Jordan system

We establish the inheritance of the heart by local algebras and subquotients of Jordan and associative systems. Received: 8 May 2001     

The Category and Operad of Matching Dialgebras

This paper gives a systematic study of matching dialgebras corresponding to the operad As ⁽²⁾ in Zinbiel (2012) as the only Koszul self dual operad there other than the operads of associative algebras and Poisson algebras. The close relationship of matching dialgebras with semi-homomorphisms and matched pairs of associative algebras are established. By anti-symmetrizing, matching dialgerbas are also shown to give compatible Lie algebras, pre-Lie algebras and PostLie algebras. By the rewriting method, the operad of matching dialgebras is shown to be Koszul and the free objects are constructed in terms of tensor algebras. The operadic complex computing the homology of the matching dialgebras is made explicit.     

Classification of constant solutions of the associative Yang-Baxter equation on Mat3

We find all nonequivalent constant solutions of the classical associative Yang-Baxter equation for Mat ₃ . New examples found in the classification yield the corresponding Poisson brackets on traces, double Poisson brackets on a free associative algebra with three generators, and anti-Frobenius associative algebras.