Stably flat completions of universal enveloping algebras

Pirkovskii, A. Yu.

doi:10.4064/dm441-0-1

Cited by 26 publications

(40 citation statements)

References 54 publications

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“…We remark that the first (second) result is in contrast to (in accordance with) the complex situation [7].…”

Section: Introductioncontrasting

confidence: 46%

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Stable flatness of nonarchimedean hyperenveloping algebras

Schmidt

2010

Journal of Algebra

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Let L be a completely valued nonarchimedean field and g a finitedimensional Lie algebra over L. We show that its hyperenveloping algebra F(g) agrees with its Arens-Michael envelope and, furthermore, is a stably flat completion of its universal enveloping algebra. As an application, we prove that the Taylor relative cohomology for the locally convex algebra F(g) is naturally isomorphic to the Lie algebra cohomology of g.

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“…We remark that the first (second) result is in contrast to (in accordance with) the complex situation [7].…”

Section: Introductioncontrasting

confidence: 46%

“…[7]). Vaguely speaking, a continuous ring extension between topological algebras θ : A → B is stably flat if the restriction functor θ * identifies the category of topological B-modules with a full subcategory of topological A-modules in a way that leaves certain homological relations invariant.…”

Section: Introductionmentioning

confidence: 99%

Stable flatness of nonarchimedean hyperenveloping algebras

Schmidt

2010

Journal of Algebra

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“…Теперь следует показать, что теорема 3 в свою очередь влечет, что канони-ческое вложение U (g) → O g (D r ) -слабая локализация в смысле работы [15]; в частности, тривиальный O g (D r )-модуль обладает резольвентой Кошуля, что мы продемонстрируем в следующем параграфе. § 6.…”

Section: алгебры фрешеunclassified

Спектр Тейлора И Трансверсальность Для Операторной Алгебры Гейзенберга

Dosi¹,

Dosi²

2010

Матем. сб.

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Спектр Тейлора и трансверсальность для операторной алгебры ГейзенбергаРассмотрены проблемы некоммутативного голоморфного функцио-нального исчисления для банаховых модулей над конечномерными ниль-потентными алгебрами Ли. Основной результат устанавливает свойство трансверсальности алгебр некоммутативных голоморфных функций по отношению к спектру Тейлора семейства ограниченных линейных опера-торов, порождающего алгебру Гейзенберга.Библиография: 25 названий.Ключевые слова: голоморфная функция от элементов алгебры Ли, спектр Тейлора, свойство трансверсальности, обращающее пополнение Фреше.

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“…In the above situation, we say that ϕ extends ϕ (though ι A is not injective in general; see [5] or [11] for details).…”

Section: Preliminariesmentioning

confidence: 99%

“…If H is a bialgebra (resp., a Hopf algebra), then it is easy to show that the Arens-Michael envelope H is a ⊗-bialgebra (resp., a Hopf ⊗-algebra) in a natural way (for details, see [11]). …”

Section: Preliminariesmentioning

confidence: 99%

Arens-Michael enveloping algebras and analytic smash products

Pirkovskii

2006

Proc. Amer. Math. Soc.

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Abstract. Let g be a finite-dimensional complex Lie algebra, and let U (g) be its universal enveloping algebra. We prove that if U (g), the Arens-Michael envelope of U (g) is stably flat over U (g) (i.e., if the canonical homomorphism U (g) → U (g) is a localization in the sense of , then g is solvable.To this end, given a cocommutative Hopf algebra H and an H-module algebra A, we explicitly describe the Arens-Michael envelope of the smash product A#H as an "analytic smash product" of their completions w.r.t. certain families of seminorms.The Arens-Michael envelope of a complex associative algebra A is defined as the completion of A w.r.t. the family of all submultiplicative seminorms on A. This notion (under a different name) was introduced by Taylor [15], and the terminology "Arens-Michael envelope" is due to Helemskii [5]. An important example is the polynomial algebra C[t 1 , . . . , t n ] whose Arens-Michael envelope is isomorphic to the algebra O(C n ) of entire functions endowed with the compact-open topology. Thus the Arens-Michael envelope of a noncommutative finitely generated algebra can be viewed as an "algebra of noncommutative entire functions" (cf. [16,17]).Given an algebra A, it is natural to ask to what extent homological properties of its Arens-Michael envelope A (considered as a topological algebra) are related to those of A. To handle this problem, it is convenient to use the notion of localization. Roughly speaking, a topological algebra homomorphism A → B is a localization if it identifies the category of topological B-modules with a full subcategory of the category of topological A-modules, and if the homological relations between B-modules do not change when the modules are considered as A-modules. Localizations were introduced by Taylor [16] in connection with the functional calculus problem for several commuting Banach space operators. A purely algebraic counterpart of this notion was studied by Neeman and Ranicki [10]. (Note that their terminology differs from Taylor's terminology; namely, a homomorphism A → B is a localization in Taylor's sense precisely when B is stably flat over A in the sense of Neeman and Ranicki.)Thus a natural question is whether or not A is stably flat over A. Taylor [16] proved that this is the case for A = C[t 1 , . . . , t n ] and for A = F n , the free algebra on n generators. In the case where A = U (g), the universal enveloping algebra of a

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Stably flat completions of universal enveloping algebras

Cited by 26 publications

References 54 publications

Stable flatness of nonarchimedean hyperenveloping algebras

Stable flatness of nonarchimedean hyperenveloping algebras

Спектр Тейлора И Трансверсальность Для Операторной Алгебры Гейзенберга

Arens-Michael enveloping algebras and analytic smash products

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