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Binomial Skew Polynomial Rings, Artin-Schelter Regularity, and Binomial Solutions of the Yang-Baxter Equation

Gateva-Ivanova, Tatiana (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary 81R50, 16W50, 16S36, 16S37.Let k be a field and X be a set of n elements. We introduce and study a class of quadratic k-algebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual A! is Frobenius of dimension n, with a regular socle and for each x, y ∈ X an equality of the type xyy = αzzt, where α ∈ k {0, and z, t ∈ X is satisfied...

Composition-diamond lemma for modules

Yuqun Chen, Yongshan Chen, Chanyan Zhong (2010)

Czechoslovak Mathematical Journal

We investigate the relationship between the Gröbner-Shirshov bases in free associative algebras, free left modules and “double-free” left modules (that is, free modules over a free algebra). We first give Chibrikov’s Composition-Diamond lemma for modules and then we show that Kang-Lee’s Composition-Diamond lemma follows from it. We give the Gröbner-Shirshov bases for the following modules: the highest weight module over a Lie algebra s l 2 , the Verma module over a Kac-Moody algebra, the Verma module...

Exact sequences for mixed coproduct/ tensor-product ring constructions.

Warren Dicks, Ian J. Leary (1994)

Publicacions Matemàtiques

To a commutative ring K, and a family of K-algebras indexed by the vertex set of a graph, we associate a K-algebra obtained by a mixture of coproduct and tensor product constructions. For this, and related constructions, we give exact sequences and deduce homological properties.

Présentation jordanienne de l'algèbre de Weyl A₂

J. Alev, F. Dumas (2001)

Annales Polonici Mathematici

Let k be a commutative field. For any a,b∈ k, we denote by J a , b ( k ) the deformation of the 2-dimensional Weyl algebra over k associated with the Jordanian Hecke symmetry with parameters a and b. We prove that: (i) any J a , b ( k ) can be embedded in the usual Weyl algebra A₂(k), and (ii) J a , b ( k ) is isomorphic to A₂(k) if and only if a = b.

Some remarks on the Akivis algebras and the Pre-Lie algebras

Yuqun Chen, Yu Li (2011)

Czechoslovak Mathematical Journal

In this paper, by using the Composition-Diamond lemma for non-associative algebras invented by A. I. Shirshov in 1962, we give Gröbner-Shirshov bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an Akivis algebra, respectively. As applications, we show I. P. Shestakov’s result that any Akivis algebra is linear and D. Segal’s result that the set of all good words in X * * forms a linear basis of the free Pre-Lie algebra PLie ( X ) generated by the set X . For completeness,...

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