Algebraic Monoids Research Articles

Skew ring extensions and generalized monoid rings

A D-structure on a ring A with identity is a family of self-mappings indexed by the elements of a monoid G and subject to a long list of rather natural conditions. The mappings are used to define a generalization of the monoid algebra A[G]. We consider two of the simpler types of D-structure. The first is based on a homomorphism from G to End(A) and leads to a skew monoid ring. We also explore connections between these D-structures and normalizing and subnormalizing extensions. The second type of D-structure considered is built from an endomorphism of A. We use D-structures of this type to characterize rings which can be graded by a cyclic group of order 2.

Affinely spanned quasi-stochastic algebraic monoids

A linear algebraic monoid over an algebraically closed field [Formula: see text] of characteristic zero is called (row) quasi-stochastic if each row of each matrix element is of sum one. Any linear algebraic monoid over [Formula: see text] can be embedded as an algebraic submonoid of the maximum affinely spanned quasi-stochastic monoid of some degree [Formula: see text]. The affinely spanned quasi-stochastic algebraic monoids form a basic class of quasi-stochastic algebraic monoids. An initial study of structure of affinely spanned quasi-stochastic algebraic monoids is conducted. Among other things, it is proved that the Zariski closure of a parabolic subgroup of the unit group of an affinely spanned quasi-stochastic algebraic monoid is affinely spanned.

Gröbner-Shirshov basis for degenerate Ringel-Hall algebra of type $$D_4$$

In this paper, we first give a presentation of the degenerate Ringel-Hall algebra $$\mathfrak {H}_0(D_4)$$ by using the Frobenius morphism and the multiplication formulas of the generic extension monoid algebra. Then, by using the relations which are computed to give this presentation, we construct the Grobner-Shirshov basis for this algebra.

Invariant convex subcones of the Tits cone of a linear Coxeter group

We investigate the faces and the face lattices of arbitrary Coxeter group invariant convex subcones of the Tits cone of a linear Coxeter system as introduced by E.B. Vinberg. Particular examples are given by certain Weyl group invariant polyhedral cones, which underlie the theory of normal reductive linear algebraic monoids as developed by M.S. Putcha and L.E. Renner. We determine the faces and the face lattice of the Tits cone and the imaginary cone, generalizing some of the results obtained for linear Coxeter systems with symmetric root bases by M. Dyer, and for linear Coxeter systems with free root bases by E. Looijenga, P. Slodowy, and the author.

-THEORY OF MONOID ALGEBRAS AND A QUESTION OF GUBELADZE

We show that for any commutative Noetherian regular ring$R$containing$\mathbb{Q}$, the map$K_{1}(R)\rightarrow K_{1}\left(\frac{R[x_{1},\ldots ,x_{4}]}{(x_{1}x_{2}-x_{3}x_{4})}\right)$is an isomorphism. This answers a question of Gubeladze. We also compute the higher$K$-theory of this monoid algebra. In particular, we show that the above isomorphism does not extend to all higher$K$-groups. We give applications to a question of Lindel on the Serre dimension of monoid algebras.

Infinite-dimensional reductive monoids associated to highest weight representations of Kac–Moody groups

Starting with a highest weight representation of a Kac–Moody group over the complex numbers, we construct a monoid whose unit group is the image of the Kac–Moody group under the representation, multiplied by the nonzero complex numbers. We show that this monoid has similar properties to those of a J-irreducible reductive linear algebraic monoid. In particular, the monoid is unit regular and has a Bruhat decomposition, and the idempotent lattice of the generalized Renner monoid of the Bruhat decomposition is isomorphic to the face lattice of the convex hull of the Weyl group orbit of the highest weight.

Ideal class groups of monoid algebras

Let $A\subset B$ be an extension of commutative reduced rings and $M\subset N$ an extension of positive commutative cancellative torsion-free monoids. We prove that $A$ is subintegrally closed in $B$ and $M$ is subintegrally closed in $N$ if and only if the group of invertible $A$-submodules of $B$ is isomorphic to the group of invertible $A[M]$-submodules of $B[N]$ Theorem~\ref {6t2} (b), (d). In the case $M=N$, we prove the same without the assumption that the ring extension is reduced Theorem~\ref {6t2} (c), (d).

THE ANNIHILATOR OF TENSOR SPACE IN THE -ROOK MONOID ALGEBRA

In this paper, we give an explicit construction of a quasi-idempotent in the $q$-rook monoid algebra $R_{n}(q)$ and show that it generates the whole annihilator of the tensor space $U^{\otimes n}$ in $R_{n}(q)$.

Mesoprimary decomposition of binomial submodules

Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing “mesoprimary decompositions” determined by their underlying monoid congruences. Mesoprimary decompositions are highly combinatorial in nature, and are designed to parallel standard primary decomposition over Noetherean rings. In this paper, we generalize mesoprimary decomposition from binomial ideals to “binomial submodules” of certain graded modules over a monoid algebra, analogous to the way primary decomposition of ideals over a Noetherean ring R generalizes to R-modules. The result is a combinatorial method of constructing primary decompositions that, when restricting to the special case of binomial ideals, coincides with the method introduced by Kahle and Miller.

Another construction of the braided T-category

This paper introduces group-cograded monoidal Hom-Hopf algebras, and shows that this kind of group-cograded monoidal Hom-Hopf algebras are monoidal Hom-Hopf algebras in the Turaev category [Formula: see text] introduced by Canepeel and De Lombaerde. Then we define the [Formula: see text]-Yetter–Drinfeld category over a group-cograded monoidal Hom-Hopf algebra, and construct a new kind of braided [Formula: see text]-categories.

An algebra for distributed Big Data analytics

AbstractWe present an algebra for data-intensive scalable computing based on monoid homomorphisms that consists of a small set of operations that capture most features supported by current domain-specific languages for data-centric distributed computing. This algebra is being used as the formal basis of MRQL, which is a query processing and optimization system for large-scale distributed data analysis. The MRQL semantics is given in terms of monoid comprehensions, which support group-by and order-by syntax and can work on heterogeneous collections without requiring any extension to the monoid algebra. We present the syntax and semantics of monoid comprehensions and provide rules to translate them to the monoid algebra. We give evidence of the effectiveness of our algebra by presenting some important optimization rules, such as converting nested queries to joins.

On the radicals of linear algebraic monoids

In this paper we consider the Schwarz radical of linear algebraic semigroups as defined in semigroup theory. We give some new characterizations of the complete regularity, regularity and solvability of irreducible linear algebraic monoids in terms of Schwarz radical data. Moreover, we give a generalization about the results of the kernel to the results of completely regular mathscr {J}-classes.

The representations of nested composition algebras

ABSTRACTIn this paper, we investigate the basis graph of the monoid algebra of a submonoid of the monoid of mappings from N = {1,…,n} to itself, defined by a nested sequence of compositions of N. Each such monoid is a left regular band (LRB), that is, a semigroup S satisfying x2 = x and xyx = xy for all x,y∈S. This class is sufficiently rich that every path algebra of an acyclic quiver can be embedded in such a monoid algebra. The multiplication in the monoid algebra has a particularly simple quasi-multiplicative form, allowing definition over the integers. Combining this with a formula for Ext-groups for LRBs due to Margolis et al. [6], we get a simple criterion for the nested composition algebras to be hereditary.

K-regularity of locally convex algebras

The isomorphism of Karoubi–Villamayor K-groups with smooth K-groups for monoid algebras over quasi-stable locally convex algebras is established. We prove that the Quillen K-groups are isomorphic to smooth K-groups for monoid algebras over quasi-stable Fr $$\acute{\mathrm{e}}$$ chet algebras having a properly uniformly bounded approximate unit and not necessarily m-convex. Based on these results the K-regularity property for quasi-stable Fr $$\acute{\mathrm{e}}$$ chet algebras having a properly uniformly bounded approximate unit is established.

Center construction and duality of category of Hom-Yetter-Drinfeld modules over monoidal Hom-Hopf algebras

Let (H,α) be a monoidal Hom-Hopf algebra. In this paper, we will study the category of Hom-Yetter-Drinfeld modules. First, we show that the category of left-left Hom-Yetter-Drinfeld modules HHH Y D is isomorphic to the center of the category of left (H,α)-Hom-modules. Also, by the center construction, we get that the categories of left-left, left-right, right-left, and right-right Hom-Yetter-Drinfeld modules are isomorphic as braided monoidal categories. Second, we prove that the category of finitely generated projective left-left Hom-Yetter-Drinfeld modules has left and right duality.

The structure and construction of bi-Frobenius Hom-algebras

ABSTRACTBi-Frobenius Hom-algebras are introduced in this paper. They provide a common generalization of finite dimensional monoidal Hom-Hopf algebras and of bi-Frobenius algebras introduced by Doi and Takeuchi. The different conditions for finite dimensional monoidal Hom-algebras to be bi-Frobenius Hom-algebras are discussed. The substructures, quotient structures as well as morphisms of bi-Frobenius Hom-algebras are also studied. In addition, the Radford’s formula for S4 of a bi-Frobenius Hom-algebra is shown. The semisimplicity and separability for a special class of finite dimensional bi-Frobenius Hom-algebras are researched finally, which presents a version of Maschke’s theorem for this family.

Irreducible representations of the Chinese monoid

All irreducible representations of the Chinese monoid Cn, of any rank n, over a nondenumerable algebraically closed field K, are constructed. It turns out that they have a remarkably simple form and they can be built inductively from irreducible representations of the monoid C2. The proof shows also that every such representation is monomial. Since Cn embeds into the algebra K[Cn]/J(K[Cn]), where J(K[Cn]) denotes the Jacobson radical of the monoid algebra K[Cn], a new representation of Cn as a subdirect product of the images of Cn in the endomorphism algebras of the constructed simple modules follows.

Gröbner bases and their application to the Cauchy problem on finitely generated affine monoids

For finitely generated submonoids of the integer lattice and submodules over the associated monoid algebra, we investigate Gröbner bases with respect to generalised term orders. Up to now, this theory suffered two disadvantages: The algorithm for computing the Gröbner bases was slow and it was not known whether there existed generalised term orders for arbitrary finitely generated submonoids. This limited the applicability of the theory. Here, we describe an algorithm which transports the problem of computing the Gröbner bases to one over a polynomial ring and use the conventional Gröbner theory to solve it, thus making it possible to apply known, optimised algorithms to it. Furthermore, we construct generalised term orders for arbitrary finitely generated submonoids. As an application we solve the Cauchy problem (initial value problem) for systems of linear partial difference equations over finitely generated submonoids.

On Twisted Smash Products of Monoidal Hom–Hopf Algebras

Let (H, α) be a monoidal Hom-Hopf algebra and (A, β) be an (H, α)-Hom-bimodule algebra. In this article, we first introduce the notion of a twisted Hom-smash product A☆H and then find some sufficient and necessary conditions making it into a monoidal Hom-bialgebra. Furthermore, we give a Maschke-type theorem for the twisted Hom-smash product over a semisimple monoidal Hom-Hopf algebra (H, α). Finally, we form an associated Morita context [AbiH, AbiHAA☆H, A☆HAAbiH, A☆H] in the category .

Some Results about Projective Modules over Monoid Algebras

(1) Let R be a 1-dimensional commutative Noetherian anodal ring with finite seminormalization and M a commutative cancellative torsion-free monoid. Let P be a projective R[M]-module of rank r. Then P ≅ ∧rP ⊕ R[M]r−1.(2) Murthy and Pedrini [11] proved K0 homotopy invariance of polynomial extension of some affine normal surfaces. We extend this result to a monoid extension (see 1.5).