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First, we provide an introduction to the theory and algorithms for noncommutative Gröbner bases for ideals in free associative algebras. Second, we explain how to construct universal associative envelopes for nonassociative structures defined by multilinear operations. Third, we extend the work of Elgendy (2012) for nonassociative structures on the 2-dimensional simple associative triple system to the 4- and 6-dimensional systems.

Gelfand-Kirillov dimension in some crossed products.

Guédénon, Thomas (2002)

Beiträge zur Algebra und Geometrie

Localization of cliques and model theory. (Localisation des cliques et théorie des modèles.)

Aribaud, François, Malliavin, Marie Paule (1994)

Portugaliae Mathematica

Modèle de Whittaker et ideaux primitifs complètement premiers dans les algèbres enveloppantes des algèbres de Lie semisimples complexes II.

C. Moeglin (1988)

Mathematica Scandinavica

Modules simples sur une algèbre de Lie nilpotente contenant un vecteur propre pour une sous-algèbre

Yves Benoist (1990)

Annales scientifiques de l'École Normale Supérieure

On Lie algebras in braided categories

Bodo Pareigis (1997)

Banach Center Publications

The category of group-graded modules over an abelian group $G$ is a monoidal category. For any bicharacter of $G$ this category becomes a braided monoidal category. We define the notion of a Lie algebra in this category generalizing the concepts of Lie super and Lie color algebras. Our Lie algebras have $n$ -ary multiplications between various graded components. They possess universal enveloping algebras that are Hopf algebras in the given category. Their biproducts with the group ring are noncommutative...

On the $K$ -theory and Hattori-Stallings traces of minimal primitive factors of enveloping algebras of semisimple Lie algebras : the singular case

Patrick Polo (1995)

Annales de l'institut Fourier

Let $G$ be a semisimple complex algebraic group and $X$ its flag variety. Let $𝔤 = Lie (G)$ and let $U$ be its enveloping algebra. Let $𝔥$ be a Cartan subalgebra of $𝔤$ . For $μ \in 𝔥^{*}$ , let $J_{μ}$ be the corresponding minimal primitive ideal, let $U_{μ} = U / J_{μ}$ , and let $𝒯_{U_{μ}} : K_{0} (U_{m} u) \to ℂ$ be the Hattori-Stallings trace. Results of Hodges suggest to study this map as a step towards a classification, up to isomorphism or Morita equivalence, of the $ℂ$ -algebras $U_{μ}$ . When $μ$ is regular, Hodges has shown that $K_{0} (U_{μ}) ≅ K_{0} (X)$ . In this case $K_{0} (U_{μ})$ is generated by the classes corresponding to...

Orthogonal Gelfand-Zetlin algebras. I.

Mazorchuk, Volodymyr (1999)

Beiträge zur Algebra und Geometrie

Quantization of the Marsden-Weinstein reduction for extended Dynkin quivers

Martin P. Holland (1999)

Annales scientifiques de l'École Normale Supérieure

Serre functors for Lie algebras and superalgebras

Volodymyr Mazorchuk, Vanessa Miemietz (2012)

Annales de l’institut Fourier

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category $𝒪$ associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category $𝒪$ and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic...

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