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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeJun 12th 2011
   • (edited Jun 12th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI done quite a lot of work this weekend regarding our old series in Zagreb on the realization of Lie algebras via differential operators. One thing is that I proved a partial case of a conjecture I stated last year about the alternative descriptions of the Raševskii [[hyper-envelope of a Lie algebra]]. Namely, one can not equip consistently the completion of the underlying vector space of an enveloping algebra (with increasing filtration inducing topology) with a product extending the one from the enveloping algebra but there is another topology, coarser than the filtration topology (the latter in the commutative case gives as a completion the algebra of formal power series in $dim V$ generators), namely it is given by a countable family of norms $\| f\|_{\epsilon_i} = max_{s_1,\ldots, s_n}\epsilon_i^{-s} |f_{s_1\ldots s_n}|$ where $s_1\plus\cdots \plus s_N = s$, $\epsilon_i\to 0$ is a fixed sequence of strictly positive real numbers (say $1/i$-sequence), $f_{s_1,\ldots,s_n}$ is the Taylor coefficient in front of $x_1^{s_1}\cdots x_n^{s_n}$ of the commutative polynomial $F$ such that the symmetrization (coexponential) map sends $F$ to $f$. $x_1,\ldots, x_n$ is a fixed basis (the Lie algebra is finite dimensional) Now if one takes instead of a coexponential some other coalgebra isomorphism (fixing the degree 1 of the fitration) between the symmetric and enveloping algebra one can take analogous norms and ask if the completion is isomorphic to Raševskii hyper-envelope as a topological algebra (and in fact that the completion in those norms does make it a topological algebra, i.e. that the multiplication is continuous in that topology). I proved that on Saturday under some mild conditions on the coalgebra isomorphism (in fact I am not completely sure if I can produce a counterexample in general, but I have now ideas which convince me that it is very likely to exist). I hope to make this clear soon in entries. However there are related things which I am now trying to understand. One is maybe trivial and maybe well known to the community, so I will raise it here. First of all I would like to have convenient descriptions of all coalgebra automorphisms of the symmetric algebra $S(V)$ on a finite dimensional vector space $V$ over a field $k$ fixing $k\oplus V$. By transposition, they are in 1-1 correspondence with the automorphisms of the algebra of the formal power series $\hat{S}(V^*)\cong S(V)^*$. Now it is easy to see that the commutative Lie algebra whose underlying vector space is $V$ can be realized in many ways in the form $X_i = x_b \phi^b_i$ where $\phi^b_i$ is the formal power series in Weyl algebra generated by $x_1,\ldots, x_n$ and $\partial^1,\ldots,\partial^n$ such that $[X_i,X_j] = 0$ in the corresponding semicompleted Weyl algebra canonically isomorhic to $S(V)\otimes S(V)^*$ as a vector space. The action of polynomials in $X_i$-s on Fock vacuum gives a particular automorphism of $S(V)$. This has a generalization for any f.d. Lie algebra, but I do not quite understand this case yet. The claim of some people (which I would l like to verify!!) is that the automorphisms of semicompleted Weyl algebra of the form $$ y \mapsto exp(-\sum_i x_i U_i) y exp(\sum_j x_j U_j) $$ where $U_j$ is a formal power series in $\partial^1, \ldots, \partial^n$ starting with $1$, when applied to $x_1,\ldots,x_n$ gives all the sets of the form $\{ X_i\}$ as above. One can develop the automorphism when applied on the generators $x_i$ in the series based on the exp-ad formula but the combinatorics of higher partial derivatives of $U_i$-s becomes complicated. Does anybody here knows something about similar issues, that is such automorhisms of Weyl algebra, automorphisms of formal power series (modulo dilations and rotations), automorphisms of symmetric coalgebra and so on an has some hint or experience with similar questions ? I may post it to MO once, but now as I plan first to develop some $n$Lab entry I think some of you may know something.

   I done quite a lot of work this weekend regarding our old series in Zagreb on the realization of Lie algebras via differential operators. One thing is that I proved a partial case of a conjecture I stated last year about the alternative descriptions of the Raševskii hyper-envelope of a Lie algebra. Namely, one can not equip consistently the completion of the underlying vector space of an enveloping algebra (with increasing filtration inducing topology) with a product extending the one from the enveloping algebra but there is another topology, coarser than the filtration topology (the latter in the commutative case gives as a completion the algebra of formal power series in $dim V$ generators), namely it is given by a countable family of norms $\| f\|_{\epsilon_i} = max_{s_1,\ldots, s_n}\epsilon_i^{-s} |f_{s_1\ldots s_n}|$ where $s_1\plus\cdots \plus s_N = s$, $\epsilon_i\to 0$ is a fixed sequence of strictly positive real numbers (say $1/i$-sequence), $f_{s_1,\ldots,s_n}$ is the Taylor coefficient in front of $x_1^{s_1}\cdots x_n^{s_n}$ of the commutative polynomial $F$ such that the symmetrization (coexponential) map sends $F$ to $f$. $x_1,\ldots, x_n$ is a fixed basis (the Lie algebra is finite dimensional)

   Now if one takes instead of a coexponential some other coalgebra isomorphism (fixing the degree 1 of the fitration) between the symmetric and enveloping algebra one can take analogous norms and ask if the completion is isomorphic to Raševskii hyper-envelope as a topological algebra (and in fact that the completion in those norms does make it a topological algebra, i.e. that the multiplication is continuous in that topology). I proved that on Saturday under some mild conditions on the coalgebra isomorphism (in fact I am not completely sure if I can produce a counterexample in general, but I have now ideas which convince me that it is very likely to exist).

   I hope to make this clear soon in entries. However there are related things which I am now trying to understand. One is maybe trivial and maybe well known to the community, so I will raise it here.

   First of all I would like to have convenient descriptions of all coalgebra automorphisms of the symmetric algebra $S(V)$ on a finite dimensional vector space $V$ over a field $k$ fixing $k\oplus V$. By transposition, they are in 1-1 correspondence with the automorphisms of the algebra of the formal power series $\hat{S}(V^*)\cong S(V)^*$. Now it is easy to see that the commutative Lie algebra whose underlying vector space is $V$ can be realized in many ways in the form $X_i = x_b \phi^b_i$ where $\phi^b_i$ is the formal power series in Weyl algebra generated by $x_1,\ldots, x_n$ and $\partial^1,\ldots,\partial^n$ such that $[X_i,X_j] = 0$ in the corresponding semicompleted Weyl algebra canonically isomorhic to $S(V)\otimes S(V)^*$ as a vector space. The action of polynomials in $X_i$-s on Fock vacuum gives a particular automorphism of $S(V)$. This has a generalization for any f.d. Lie algebra, but I do not quite understand this case yet. The claim of some people (which I would l like to verify!!) is that the automorphisms of semicompleted Weyl algebra of the form

   $$y \mapsto exp(-\sum_i x_i U_i) y exp(\sum_j x_j U_j)$$

   where $U_j$ is a formal power series in $\partial^1, \ldots, \partial^n$ starting with $1$, when applied to $x_1,\ldots,x_n$ gives all the sets of the form $\{ X_i\}$ as above. One can develop the automorphism when applied on the generators $x_i$ in the series based on the exp-ad formula but the combinatorics of higher partial derivatives of $U_i$-s becomes complicated.

   Does anybody here knows something about similar issues, that is such automorhisms of Weyl algebra, automorphisms of formal power series (modulo dilations and rotations), automorphisms of symmetric coalgebra and so on an has some hint or experience with similar questions ? I may post it to MO once, but now as I plan first to develop some $n$Lab entry I think some of you may know something.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeJun 12th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThe entry on Raševskii's associative [[hyper-envelope of a Lie algebra]] is created. I recommend reading it before the contribution in numero 1.

   The entry on Raševskii’s associative hyper-envelope of a Lie algebra is created. I recommend reading it before the contribution in numero 1.