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    • CommentRowNumber1.
    • CommentAuthorMirco Richter
    • CommentTimeOct 19th 2013
    • (edited Oct 19th 2013)

    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 𝔤\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 𝔤\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 𝔤\wedge\mathfrak{g}. Thats a common structure and a way to encode L L_\infty-algebras as well.

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

    d(a(x 1x 2))d((ax 1)x 2)d(x 1(ax 2))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’ aAa\in A and vectors x 1,x 2𝔤x_1,x_2\in\mathfrak{g}. Rinehard then defined another structure that has a well defined codifferential. This structure is U(g)𝔤U(g)\otimes \wedge\mathfrak{g}, where U(𝔤U(\mathfrak{g} is the univer. envel. algebra of 𝔤\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 AA-module 𝔤\wedge\mathfrak{g} itself?

    • CommentRowNumber2.
    • CommentAuthorMirco Richter
    • CommentTimeOct 19th 2013
    • (edited Oct 19th 2013)

    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 L_\infty-algebras isn’t sufficient anymore.

    • CommentRowNumber3.
    • CommentAuthorjim_stasheff
    • CommentTimeOct 22nd 2013
    Huebschmann has addressed this issue very thoroughly and there is an extensive literature on strong homotopy versions of Lie-Rinehart algebras. N.B. RINEHART
    • CommentRowNumber4.
    • CommentAuthorhuebschm
    • CommentTimeOct 22nd 2013
    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}},
    }