Functions of class $C^\infty$ in non-commuting variablesin the context of triangular Lie algebras
- Авторлар: Aristov O.
- Шығарылым: Том 86, № 6 (2022)
- Беттер: 5-46
- Бөлім: Articles
- URL: https://journals.rcsi.science/1607-0046/article/view/133884
- DOI: https://doi.org/10.4213/im9236
- ID: 133884
Дәйексөз келтіру
Аннотация
We construct a certain completion $C^\infty_\mathfrak{g}$ of the universalenveloping algebra of a triangular real Lie algebra $\mathfrak{g}$.It is a Frechet–Arens–Michael algebra that consists of elementsof polynomial growth and satisfies to the following universal property:every Lie algebra homomorphism from $\mathfrak{g}$ to a real Banach algebraall of whose elements are of polynomial growth has an extensionto a continuous homomorphism with domain $C^\infty_\mathfrak{g}$.Elements of this algebracan be calledfunctions of class $C^\infty$ in non-commuting variables.The proof is based on representation theory and employsan ordered $C^\infty$-functional calculus. Beyond the general case,we analyze two simple examples. As an auxiliary material, the basicsof the general theory of algebras of polynomial growthare developed. We also consider local variants of the completion and obtaina sheaf of non-commutative functions on the Gelfand spectrumof $C^\infty_\mathfrak{g}$ in the case when $\mathfrak{g}$ is nilpotent.In addition, we discuss the theory of holomorphic functions in non-commutingvariables introduced by Dosi andapply our methods to prove theorems strengthening some his results.
Авторлар туралы
Oleg Aristov
Email: aristovoyu@inbox.ru
Candidate of physico-mathematical sciences, no status
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