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1. • CommentRowNumber1.
   • CommentAuthorMirco Richter
   • CommentTimeOct 19th 2013
   • (edited Oct 19th 2013)
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   Author: Mirco Richter
   Format: MarkdownItexI was wondering, if the presentation of Lie algebras in terms of (co)differantial graded coalgebras can be carried over to the setting of Lie Rineard pairs. To be more precise: If $\mathfrak{g}$ is a Lie algebra seen as $\mathbb{Z}$-graded, but concentrated in degree one only (i.e using tensor grading), then its reduced(!) exterior power $\wedge\mathfrak{g}$ can be seen as a locally nilpotent graded symmetric coalgebra, with the shuffle coproduct and the Lie bracket can be enoded into a (co)differential on $\wedge\mathfrak{g}$. Thats a common structure and a way to encode $L_\infty$-algebras as well. However if $A$ is a commutative, associative algebra with unit and $(A,\mathfrak{g})$ is a Lie Rinehard pair, such that the action of $\mathfrak{g}$ on $A$ is not trivial, then on a first sight this doesn't work anymore, since the (co)differential of the coalgebra $\wedge\mathfrak{g}$ does not behave well with respect to the $A$-module structure of $\wedge\mathfrak{g}$, since in general $d(a\cdot (x_1\wedge x_2))\neq d((a\cdot x_1)\wedge x_2)\neq d(x_1\wedge(a\cdot x_2))$ for 'scalars' $a\in A$ and vectors $x_1,x_2\in\mathfrak{g}$. Rinehard then defined another structure that has a well defined codifferential. This structure is $U(g)\otimes \wedge\mathfrak{g}$, where $U(\mathfrak{g}$ is the univer. envel. algebra of $\mathfrak{g}$ and in case of the Lie Rinehard pair of vector fields and functions, its dual is the usual DeRahm complex. However, does anyone know, if there is a (co)differential on the $A$-module $\wedge\mathfrak{g}$ itself?

   I was wondering, if the presentation of Lie algebras in terms of (co)differantial graded coalgebras can be carried over to the setting of Lie Rineard pairs.

   To be more precise:

   If $\mathfrak{g}$ is a Lie algebra seen as $\mathbb{Z}$-graded, but concentrated in degree one only (i.e using tensor grading), then its reduced(!) exterior power $\wedge\mathfrak{g}$ can be seen as a locally nilpotent graded symmetric coalgebra, with the shuffle coproduct and the Lie bracket can be enoded into a (co)differential on $\wedge\mathfrak{g}$. Thats a common structure and a way to encode $L_\infty$-algebras as well.

   However if $A$ is a commutative, associative algebra with unit and $(A,\mathfrak{g})$ is a Lie Rinehard pair, such that the action of $\mathfrak{g}$ on $A$ is not trivial, then on a first sight this doesn’t work anymore, since the (co)differential of the coalgebra $\wedge\mathfrak{g}$ does not behave well with respect to the $A$-module structure of $\wedge\mathfrak{g}$, since in general

   $d(a\cdot (x_1\wedge x_2))\neq d((a\cdot x_1)\wedge x_2)\neq d(x_1\wedge(a\cdot x_2))$

   for ’scalars’ $a\in A$ and vectors $x_1,x_2\in\mathfrak{g}$. Rinehard then defined another structure that has a well defined codifferential. This structure is $U(g)\otimes \wedge\mathfrak{g}$, where $U(\mathfrak{g}$ is the univer. envel. algebra of $\mathfrak{g}$ and in case of the Lie Rinehard pair of vector fields and functions, its dual is the usual DeRahm complex.

   However, does anyone know, if there is a (co)differential on the $A$-module $\wedge\mathfrak{g}$ itself?

2. • CommentRowNumber2.
   • CommentAuthorMirco Richter
   • CommentTimeOct 19th 2013
   • (edited Oct 19th 2013)
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   Author: Mirco Richter
   Format: MarkdownItexMaybe the coalgebraic formalism has to be extended to not necessarily locally nilpotent cofree coalgebras, since we now have a non trivial 'degree zero' part and the REDUCED symmetric coalgebra as in the case of $L_\infty$-algebras isn't sufficient anymore.

   Maybe the coalgebraic formalism has to be extended to not necessarily locally nilpotent cofree coalgebras, since we now have a non trivial ’degree zero’ part and the REDUCED symmetric coalgebra as in the case of $L_\infty$-algebras isn’t sufficient anymore.

3. • CommentRowNumber3.
   • CommentAuthorjim_stasheff
   • CommentTimeOct 22nd 2013
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   Author: jim_stasheff
   Format: TextHuebschmann has addressed this issue very thoroughly and there is an extensive literature on strong homotopy versions of Lie-Rinehart algebras. N.B. RINEHART

   Huebschmann has addressed this issue very thoroughly and there is an extensive literature on strong homotopy versions of Lie-Rinehart algebras. N.B. RINEHART
4. • CommentRowNumber4.
   • CommentAuthorhuebschm
   • CommentTimeOct 22nd 2013
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   Author: huebschm
   Format: TextThis kind of structure does presumably not even exist on U(g)otimes Lambda[g] unless the g-action on A is trivial. I change to my standard notation (A,L). In the case of a Lie-Rinehart algebra (A,L) with non-trivial L-action on A, the exterior algebra Lambda[L] is over A and the tensor product U(L)otimes Lambda[L] is over A. But the differential on U(L)otimes Lambda[L], while it preserves A-multilinearity (a non-trivial fact which makes the Lie-Rinehart approach work), is linear over the ground ring only. The standard construction turns U(L)otimes Lambda[L] into an A-coalgebra, not into a coalgebra over the ground ring. I doubt that the Rinehart differential on U(L)otimes Lambda[L] is compatible with the A-coalgebra structure but don't have the time to check right now. A parallel situation: Given the commutative R-algebra A, let B be a commutative R-algebra and U a commutative subalgebra of B that is actually an A-algebra. Suppose that B is the dual of an R-coalgebra C and that U is the dual of an A-coalgebra D, and that the injection of U into A is induced by an R-linear map from C to D. This map cannot be a morphism of coalgebras since C is over R and D over A. Anyway a good way to understand the situation is in terms of Maurer-Cartan algebras. Some remarks are in my paper @article {bv, AUTHOR = {Huebschmann, Johannes}, TITLE = {Lie-{R}inehart algebras, {G}erstenhaber algebras and {B}atalin-{V}ilkovisky algebras}, JOURNAL = {Ann. Inst. Fourier (Grenoble)}, FJOURNAL = {Universit\'e de Grenoble. Annales de l'Institut Fourier}, VOLUME = {48}, YEAR = {1998}, NUMBER = {2}, PAGES = {425--440}, ISSN = {0373-0956}, CODEN = {AIFUA7}, MRCLASS = {17B55 (17B56 17B66 17B81)}, MRNUMBER = {1625610 (99b:17021)}, MRREVIEWER = {J. Stasheff}, URL = {http://www.numdam.org/item?id=AIF_1998__48_2_425_0}, note = {{\tt math.dg/9704005}}, }

   This kind of structure does presumably not even exist
   on U(g)otimes Lambda[g] unless the g-action on A is trivial.
   
   I change to my standard notation (A,L).
   In the case of a Lie-Rinehart algebra (A,L) with non-trivial
   L-action on A, the exterior algebra Lambda[L] is over A and the
   tensor product U(L)otimes Lambda[L] is over A.
   But the differential
   on U(L)otimes Lambda[L], while it preserves A-multilinearity
   (a non-trivial fact which makes the Lie-Rinehart approach work),
   is linear over the ground ring only.
   The standard construction turns U(L)otimes Lambda[L]
   into an A-coalgebra, not into a coalgebra over the ground ring.
   
   I doubt that the Rinehart differential on U(L)otimes Lambda[L]
   is compatible with the A-coalgebra structure but don't have
   the time to check right now.
   
   A parallel situation: Given the commutative R-algebra A,
   let B be a commutative R-algebra and U a commutative
   subalgebra of B that is actually an A-algebra.
   
   
   Suppose that B is the dual of an R-coalgebra C and
   that U is the dual of an A-coalgebra D,
   and that the injection of U into A is induced by an R-linear map
   from C to D. This map cannot be a morphism of coalgebras
   since C is over R and D over A.
   
   Anyway a good way to understand the situation is in terms of
   Maurer-Cartan algebras.
   
   Some remarks are in my paper
   
   @article {bv,
   AUTHOR = {Huebschmann, Johannes},
   TITLE = {Lie-{R}inehart algebras, {G}erstenhaber algebras and
   {B}atalin-{V}ilkovisky algebras},
   JOURNAL = {Ann. Inst. Fourier (Grenoble)},
   FJOURNAL = {Universit\'e de Grenoble. Annales de l'Institut Fourier},
   VOLUME = {48},
   YEAR = {1998},
   NUMBER = {2},
   PAGES = {425--440},
   ISSN = {0373-0956},
   CODEN = {AIFUA7},
   MRCLASS = {17B55 (17B56 17B66 17B81)},
   MRNUMBER = {1625610 (99b:17021)},
   MRREVIEWER = {J. Stasheff},
   URL = {http://www.numdam.org/item?id=AIF_1998__48_2_425_0},
   note = {{\tt math.dg/9704005}},
   }