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Let $\mathfrak a$ be a Lie algebra graded by the abelian semigroup $S$, then the universal enveloping algebra $U(\mathfrak a)$ of $\mathfrak a$ is $S \sqcup \{0\}$ graded. I have the following questions.

  1. What is the definition of infinite fold tensor product ($U(\mathfrak a)^{\otimes \infty}$) of $U(\mathfrak a)$ and is this also $S \sqcup \{0\}$ graded?
  2. If so, how to express the grade spaces of this infinite tensor product in terms of grade spaces of $U(\mathfrak a)$?
  3. Is it a good notation $U(\mathfrak a)^{\otimes \infty}$?

Thank you.

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  • $\begingroup$ I am rolling back the title to what the OP originally put, rather than @user64494's doctrinaire attempt to apply style guides, because MathOverflow is not being curated to have a consistent style guide, unless we are willing to do it properly in a way that will require geographic if not astronomic amounts of effort and enforcement $\endgroup$
    – Yemon Choi
    Commented Jun 29, 2019 at 13:52
  • $\begingroup$ Why are you so certain that there should be a reasonable definition? $\endgroup$
    – Mark Wildon
    Commented Jun 29, 2019 at 17:33
  • $\begingroup$ @Yemon Choi: My English handbook says it should be "Infinite" in the title. Can you kindly give a reference to a grammar which allows such spelling? TIA. $\endgroup$
    – user64494
    Commented Jun 29, 2019 at 17:53

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Assuming $\infty$= the cardinality of some set $I$, then consider the Lie algebra $\mathfrak a^{(I)}$= the direct sum of $I$-copies of $\mathfrak a$, with bracket coordinatewise. Then take its universal envelopping algebra, it realizes your desired Infinity tensor product. Of course it is $S$- graded (it is actually $S^{(I)}$- graded).

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  • $\begingroup$ Yes. By PBW for example. $\endgroup$
    – Marco Farinati
    Commented Jun 30, 2019 at 12:32
  • $\begingroup$ Sorry. I wrongly delete my comment. Even if it is infinite direct sum still the universal algebra would be the tensor product of the individual universal enveloping algebras? In this case, how to express the grade spaces. any help please? $\endgroup$
    – GA316
    Commented Jun 30, 2019 at 12:38
  • $\begingroup$ Again using a PBW basis un the enveloping álgebra, comming from a homogéneos basis in the Lie algebra, you will get homogeneous monomials $\endgroup$
    – Marco Farinati
    Commented Jun 30, 2019 at 14:53

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