Related papers: Groebner---Shirshov bases for replicated algebras

We show that the data needed for the method of the embedding tensor employed in gauging supergravity theories are precisely those of a Leibniz algebra (with one of its induced quotient Lie algebras embedded into a rigid symmetry Lie algebra…

Leibniz-type rules for Coifman-Meyer multiplier operators are studied in the settings of Triebel-Lizorkin and Besov spaces associated to weights in the Muckenhoupt classes. Even in the unweighted case, improvements on the currently known…

For a free presentation $0 \to R \to F \to G \to 0$ of a Leibniz algebra $G$, the Baer invariant ${\cal M}^{\sf Lie}(G) = \frac{R \cap [F, F]_{Lie}}{[F, R]_{Lie}}$ is called the Schur multiplier of $G$ relative to the Liezation functor or…

We describe a Schubert induction theorem, a tool for analyzing intersections on a Grassmannian over an arbitrary base ring. The key ingredient in the proof is the Geometric Littlewood-Richardson rule, described in a companion paper.…

Let T(x) in k[x] be a monic non-constant polynomial and write R=k[x] / (T) the quotient ring. Consider two bivariate polynomials a(x, y), b(x, y) in R[y]. In a first part, T = p^e is assumed to be the power of an irreducible polynomial p. A…

We extend a theorem, originally formulated by Blattner-Cohen-Montgomery for crossed products arising from Hopf algebras weakly acting on noncommutative algebras, to the realm of left Hopf algebroids. Our main motivation is an application to…

It is known that if $A$ is a finite-dimensional unital algebra equipped with a Rota-Baxter operator $R$ of weight $\lambda$, then spectrum of $R$ is a subset of $\{0,-\lambda\}$. We are interested on finding all consequences of the…

In this paper we prove the universal property of skew $PBW$ extensions generalizing this way the well known universal property of skew polynomial rings. For this, we will show first a result about the existence of this class of…

We use the Lie coalgebra and configuration pairing framework presented previously by Sinha and Walter to derive a new, left-normed monomial basis for free Lie algebras (built from associative Lyndon-Shirshov words), as well as a dual…

We study a question which can be roughly stated as follows: Given a (unital or nonunital) algebra $A$ together with a Gr\"obner-Shirshov basis $G$, consider the free operated algebra $B$ over $A$, such that the operator satisfies some…

We review the extent to which the universal enveloping algebra of a Lie-Rinehart algebra resembles a Hopf algebra, and refer to this structure as a Rinehart bialgebra. We then prove a Cartier-Milnor-Moore type theorem for such Rinehart…

In this paper, we establish the Composition-Diamond lemma for $\lambda$-differential associative algebras over a field $K $ with multiple operators. As applications, we obtain Gr\"{o}bner-Shirshov bases of free $\lambda$-differential…

We establish doubly-exponential degree bounds for Gr\"obner bases in certain algebras of solvable type over a field (as introduced by Kandri-Rody and Weispfenning). The class of algebras considered here includes commutative polynomial…

We construct a Gr\"obner-Shirshov basis of the Temperley-Lieb algebra $\mathfrak{T}(d,n)$ of the complex reflection group $G(d,1,n)$, inducing the standard monomials expressed by the generators $\{E_i\}$ of $\mathfrak{T}(d,n)$. This result…

Let $C(B,N)$ be the free associative conformal algebra generated by a set $B$ with a bounded locality $N$. Let $S$ be a subset of $C(B,N)$. A Composition-Diamond lemma for associative conformal algebras is firstly established by Bokut,…

Assuming sufficiently many terms of a n-dimensional table defined over a field are given, we aim at guessing the linear recurrence relations with either constant or polynomial coefficients they satisfy. In many applications, the table terms…

M.-P. Sch\"utzenberger asked to determine the support of the free Lie algebra ${\mathcal L}_{{\mathbb Z}_{m}}(A)$ on a finite alphabet $A$ over the ring ${\mathbb Z}_{m}$ of integers $\bmod m$ and all the corresponding pairs of twin and…

In this paper we will present the results of Artin--Markov on braid groups by using the Groebner--Shirshov basis. As a consequence we can reobtain the normal form of Artin--Markov--Ivanovsky as an easy corollary.

We develop versions of the Poincar\'e-Birkhoff-Witt and Cartier-Milnor-Moore theorems in the setting of braided Hopf algebras. To do so, we introduce new analogues of a Lie algebra in the setting of a braided monoidal category, using the…

The notion of Lie algebroids over a topological ringed space provides a unified framework to study various geometric structures. This geometric concept is intimately connected with well-known algebraic structures, including Gerstenhaber…