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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 is a Lie algebra seen as -graded, but concentrated in degree one only (i.e using tensor grading), then its reduced(!) exterior power 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 . Thats a common structure and a way to encode -algebras as well.

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

   for ’scalars’ and vectors . Rinehard then defined another structure that has a well defined codifferential. This structure is , where is the univer. envel. algebra of 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 -module 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 -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}},
   }