Gröbner–Shirshov Bases Theory for Trialgebras

We establish a method of Gröbner–Shirshov bases for trialgebras and show that there is a unique reduced Gröbner–Shirshov basis for every ideal of a free trialgebra. As applications, we give a method for the construction of normal forms of elements of an arbitrary trisemigroup, in particular, A.V. Zhuchok’s (2019) normal forms of the free commutative trisemigroups are rediscovered and some normal forms of the free abelian trisemigroups are first constructed. Moreover, the Gelfand–Kirillov dimension of finitely generated free commutative trialgebra and free abelian trialgebra are calculated, respectively.

On finite GK-dimensional Nichols algebras over abelian groups

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GKdim \operatorname {GKdim} for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces over Z \mathbb {Z} with the generator acting as a single Jordan block and show that the corresponding Nichols algebra has finite GKdim \operatorname {GKdim} if and only if the size of the block is 2 and the eigenvalue is ± 1 \pm 1 ; when this is 1, we recover the quantum Jordan plane. We consider next a class of braided vector spaces that are direct sums of blocks and points that contains those of diagonal type. We conjecture that a Nichols algebra of diagonal type has finite GKdim \operatorname {GKdim} if and only if the corresponding generalized root system is finite. Assuming the validity of this conjecture, we classify all braided vector spaces in the mentioned class whose Nichols algebra has finite GKdim \operatorname {GKdim} . Consequently we present several new examples of Nichols algebras with finite GKdim \operatorname {GKdim} , including two not in the class alluded to above. We determine which among these Nichols algebras are domains.

Gröbner-Shirshov Bases Theory for Trialgebras

We establish a Gröbner-Shirshov bases theory for trialgebras and show that every ideal of a free trialgebra has a unique reduced Gröbner-Shirshov basis. As applications, we give a method for the construction of normal forms of elements of an arbitrary trisemigroup, in particular, A.V. Zhuchok’s (2019) normal forms of the free commutative trisemigroups are rediscovered and some normal forms of the free abelian trisemigroups are first constructed. Moreover, the Gelfand-Kirillov dimension of finitely generated free commutative trialgebra and free abelian trialgebra are calculated respectively.

Minimal varieties of associative algebras and transcendental series

A variety of associative algebras over a field of characteristic 0 is called minimal if the exponent of the variety which measures the growth of its codimension sequence is strictly larger than the exponent of any of its proper subvarieties, i.e., its codimension sequence grows much faster than the codimension sequence of its proper subvarieties. By the results of Giambruno and Zaicev it follows that the number [Formula: see text] of minimal varieties of given exponent [Formula: see text] is finite. Using methods of the theory of colored (or weighted) compositions of integers, we show that the limit [Formula: see text] exists and can be expressed as the positive solution of an equation [Formula: see text] where [Formula: see text] is an explicitly given power series. Similar results are obtained for the number of minimal varieties with a given Gelfand–Kirillov dimension of their relatively free algebras of rank [Formula: see text]. It follows from classical results on lacunary power series that the generating function of the sequence [Formula: see text], [Formula: see text], is transcendental. With the same approach we construct examples of free graded semigroups [Formula: see text] with the following property. If [Formula: see text] is the number of elements of degree [Formula: see text] of [Formula: see text], then the limit [Formula: see text] exists and is transcendental.

EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS IN CHARACTERISTIC 2

We present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

Combinatorics and structure of Hecke–Kiselman algebras

Hecke–Kiselman monoids [Formula: see text] and their algebras [Formula: see text], over a field [Formula: see text], associated to finite oriented graphs [Formula: see text] are studied. In the case [Formula: see text] is a cycle of length [Formula: see text], a hierarchy of certain unexpected structures of matrix type is discovered within the monoid [Formula: see text] and this hierarchy is used to describe the structure and the properties of the algebra [Formula: see text]. In particular, it is shown that [Formula: see text] is a right and left Noetherian algebra, while it has been known that it is a PI-algebra of Gelfand–Kirillov dimension one. This is used to characterize all Noetherian algebras [Formula: see text] in terms of the graphs [Formula: see text]. The strategy of our approach is based on the crucial role played by submonoids of the form [Formula: see text] in combinatorics and structure of arbitrary Hecke–Kiselman monoids [Formula: see text].