On a filtered multiplicative basis of group algebras

Graded Monomial Ordering for $\mathbb{N}$-graded and $\mathbb{N}$-filtered Solvable Polynomial Algebras of $({\cal B},d(~))$-type

Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a solvable polynomial algebra in the sense of [K-RW, {\it J. Symbolic Comput.}, 9(1990), 1--26]. It is shown that if $A$ is an $\mathbb{N}$-graded algebra of $({\cal B},d(~))$-type, then $A$ has a graded monomial ordering $\prec_{gr}$. It is also shown that $A$ is an $\mathbb{N}$-filtered algebra of $({\cal B},d(~))$-type if and only if $A$ has a graded momomial ordering, where ${\cal B}$ is the PBW basis of $A$.

Computation of Minimal Filtered Free Resolutions over $\mathbb{N}$-Filtered Solvable Polynomial Algebras

Let $A=K[a_1,\ldots,a_n]$ be a weighted $\mathbb{N}$-filtered solvable polynomial algebra with filtration $FA=\{ F_pA\}_{p\in\mathbb{N}}$, where solvable polynomial algebras are in the sense of (A. Kandri-Rody and V. Weispfenning, Non-commutative Gröbner bases in algebras of solvable type. {\it J. Symbolic Comput.}, 9(1990), 1--26), and $FA$ is constructed with respect to a positive-degree function $d(~)$ on $A$. By introducing minimal F-bases and minimal standard bases respectively for left $A$-modules and their submodules with respect to good filtrations, minimal filtered free resolutions fo…

Generic Variables in Acyclic Cluster Algebras

Let $Q$ be an acyclic quiver. We introduce the notion of generic variables for the coefficient-free acyclic cluster algebra $\mathcal A(Q)$. We prove that the set $\mathcal G(Q)$ of generic variables contains naturally the set $\mathcal M(Q)$ of cluster monomials in $\mathcal A(Q)$ and that these two sets coincide if and only if $Q$ is a Dynkin quiver. We establish multiplicative properties of these generic variables analogous to multiplicative properties of Lusztig's dual semicanonical basis. This allows to compute explicitly the generic variables when $Q$ is a quiver of affine type. When $Q$…

Multiplicative structures for Koszul algebras

Let $Λ=kQ/I$ be a Koszul algebra over a field $k$, where $Q$ is a finite quiver. An algorithmic method for finding a minimal projective resolution $\mathbb{F}$ of the graded simple modules over $Λ$ is given in Green-Solberg. This resolution is shown to have a "comultiplicative" structure in Green-Hartman-Marcos-Solberg, and this is used to find a minimal projective resolution $\mathbb{P}$ of $Λ$ over the enveloping algebra $Λ^e$. Using these results we show that the multiplication in the Hochschild cohomology ring of $Ł$ relative to the resolution $\mathbb{P}$ is given as a cup product and als…

A-infinity-algebras, spectral sequences and exact couples

We study in this article a possible further structure of homotopic nature on multiplicative spectral sequences. More precisely, since Kadeishvili's theorem asserts that, given a dg (or A-infinity-)algebra, its cohomology has also a structure of A-infinity-algebra such that both become quasi-isomorphic, and in a multiplicative spectral sequence one considers the cohomology of dg algebras when moving from a term to the next one, a natural problem that arises is to study how this two possible structures intertwine. We give such a homotopic structure proposal, called A-infinity-enhancement of mult…

The configuration basis of a Lie algebra and its dual

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 monomial basis for Lie coalgebras. Our focus is on computational dexterity gained by using the configuration framework and basis. We include several explicit examples using the dual coalgebra basis and configuration pairing to perform Lie algebra computations. As a corollary of our work, we get a new multiplicative basis for the shuffle algebra.

极小对合基

Minimal Involutive Bases

本文提出构造极小对合多项式基的算法，定义两种新对合除法并证明其性质，确保算法正确终止，且极小对合基在给定序下唯一，可作典范基。

Looking for Groebner Basis Theory for (Almost) Skew 2-Nomial Algebras

In this paper, we introduce (almost) skew 2-nomial algebras and look for a one-sided or two-sided Gröbner basis theory for such algebras at a modest level. That is, we establish the existence of a skew multiplicative $K$-basis for every skew 2-nomial algebra, and we explore the existence of a (left, right, or two-sided) monomial ordering for an (almost) skew 2-nomial algebra. As distinct from commonly recognized algebras holding a Gröbner basis theory (such as algebras of the solvable type [K-RW] and some of their homomorphic images), a subclass of skew 2-nomial algebras that have a left Gröbn…

Yang–Baxter 方程集合论解的结构群的 Garside 性质

Garsideness properties of structure groups of set-theoretic solutions of the Yang-Baxter equation

本文类比辫群到 Temperley–Lieb 代数的同态，研究非退化对合集合论 Yang–Baxter 解的结构群，并利用其 Garside 性质为群像的底层向量空间构造有限基。

On (De)homogenized Gröbner Bases

Let $K$ be a field and $R=\oplus_{p\in\mathbb{N}}R_p$ an $\mathbb{N}$-graded $K$-algebra, which has an SM $K$-basis (i.e. a skew multiplicative $K$-basis) such that $R$ holds a Gröbner basis theory. It is proved that there is a one-to-one correspondence between the set of Gröbner bases in $R$ and the set of dh-closed homogeneous Gröbner bases in the polynomial algebra $R[t]$; and that the similar result holds true if $R$ and $R[t]$ are replaced respectively by the free algebra $K< X_1,...,X_n>$ and the free algebra $K< X_1,...,X_n,T>$. Moreover, it is shown that dh-closed graded ideals in $R[t…

On multiplying curves in the Kauffman bracket skein algebra of the thickened four-holed sphere

Based on the presentation of the Kauffman bracket skein module of the torus given by the third author in previous work, Charles D. Frohman and Răzvan Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure of the Kauffman bracket skein algebra of the thickened four-holed sphere. We present an algorithm to compute the product of any two elements of the algebra, and give an explicit formula for some families of curves. We surmise that the algorithm has quasi-polynomial growth with re…