Re: Identities in an algebra

From: Kevin Buzzard (buzzard_at_imperREMial.aOVEc.uk)
Date: 02/13/05

  • Next message: Simo Särkkä: "Re: How can you use a Stochastic Differential Equation" 
    Date: Sun, 13 Feb 2005 21:51:02 +0000 (UTC)

    Bill Dubuque <wgd@nestle.csail.mit.edu> wrote:

    > With D = d/dX it suffices to verify the following
    >
    > a 2a+2b a+2b 2b 2b a+2b 2a+2b a
    > D X D X = X D X D

    Setting D=d/dX and E=X^2 gives a representation of the universal
    enveloping algebra, and you're checking that the identity I want
    is true in the image. But I don't think that the representation is
    injective; it sends F to 2X and G to 2, so it sends 2EG-F^2 to zero.
    On the other hand setting D=d/dX and E=X gives another representation
    of the algebra in which 2EG-F^2 is not sent to zero, so 2EG-F^2
    is not zero in the algebra itself. Hence I don't see why it
    suffices to do this, unfortunately.

    I can quite believe that an argument of this nature might work though.
    Proving that the image of the identity in a representation
    is zero seems to frequently amount to verifying combinatorial
    identities; the example above results in a rather simple
    identity but I am not so sure that it's strong enough to prove
    what I need. I know of more complicated representations which I can
    prove are injective; however the resulting combinatorial identities
    are also quite complicated (double sums of products of 6 binomial
    coefficients!). I am sure that one could prove such identities
    nowadays following Zeilberger et al, but I didn't really want to
    go down that path, I am still optimistic that there is a simpler
    solution [At the end of the day the real reason I don't want
    to push this approach through is that the identities above
    are in some sense a theorem (hopefully) about Sp_4, and I would one day
    like to prove analogous ones for an arbitrary reductive group, when
    brute force calculations would be doomed.]

    Kevin

  • Next message: Simo Särkkä: "Re: How can you use a Stochastic Differential Equation"

    Relevant Pages

    • Re: Simple subtraction formula returning strange results = Excel g
      ... subtraction differ only in some number of the least significant bits ... The 64-bit floating-point representation is shown in the stylized hex form &hEEEMMMMM,M...M, where "E" is the biased exponent and "M" is the mantissa. ... always results in a zero in A3. ... A1 displays as 1.79769313486232E+308, but the first 30 digits of its exact representation are 179769313486231,57081452742373. ...
      (microsoft.public.excel.worksheet.functions)
    • Re: Casting an array to integer type
      ... its value when the sign bit is clear must be zero. ... succeed, in any of the three representation schemes, ... the result of the conversion is a trap representation, ... yield a trap representation and produce undefined behavior; ...
      (comp.lang.c)
    • Re: A ones complement sanity check, please
      ... representation of integer constant 0 and now how to zero memory ... errors in the 1's complement -1 and sign magnitude -1 representations. ... My questions are about 1's complement representation of the integer ...
      (comp.lang.c)
    • A ones complement sanity check, please
      ... representation of integer constant 0 and now how to zero memory ... errors in the 1's complement -1 and sign magnitude -1 representations. ... My questions are about 1's complement representation of the integer ...
      (comp.lang.c)
    • The Ghost of Von Neumann: Automata and Physics
      ... this details a synthesis of formal language and automata ... theory essentially within the setting of unversal algebra and category ... The Algebraic Representation of Formal Languages ... Varieties of Infinite State Automata ...
      (sci.physics.research)