What's a rigorous way to connect the Lie derivative of a generic finite-dimensional group to its Lie algebra? I mean showing that Lie derivatives can be represented as matrices.
>>16535160what do you mean by connect? Isnt the algebra something about the infinitesimal elements of the group? Like group elements close to the identity?You mean how to write down the elements of the algebra, if given some group?
>>16535189>what do you mean by connect?Any finite dimensional algebra admits two distinct representations. One is an embedding into matrices over a field of characteristic zero with the bracket given by the commutator. There other is the so-called local representation given by the Lie derivative. I'm looking for an explicit bijection or at least a construction that leads to it.
>>16535195well i dont know.. i have a lie algebra book nearby but.. im going out now. I once asked myself how to recognized is some given algebra was "the same" as some other algebra, given how you can write the basis vectors differently as linear combinations. Never understood ways to out them in canonical forms
>>16535210>some given algebra was "the same" as some other algebraI sense a physicist
>>16535160please first define what you mean by "lie derivative of a generic finite-dimensional group" and "lie algebra of a generic finite-dimensional group"
>>16535160You have to lie on your back
>>16535160Duck tape.
>>16536545Eh… The standard definitions? Is this some advanced trolling?
>>16535160The Lie algebra of a Lie group is the space of left-invariant vector fields on the Lie group together with the Lie bracket of vector fields.The Lie derivative and Lie bracket of vector fields are the same thing.
>>16536662no, I don't understand your question because with standard definitions it's trivial because lie algebra of a general lie group is *defined* using lie derivative, see >>16536675
>>16535160Let X be a matrix in the Lie algebra. You can define a curve on the Lie group which passes through an arbitrary element g at t=0 by the exponential map g e^(tX). This family of curves is the same thing as the flow associated to the Lie derivative.
>>16537324Ok, thank you. Finally something constructive. >>16536675 >>16537174 doesn't even answer my OP.
>>16537601Your op is retarded. The Lie algebra of a Lie group is DEFINED as the tangent space of the Lie group at the identity, where the lie algebra structure on the tangent space comes from the lie bracket of vector fields. There is no additional connection to be made.
>>16537953>The Lie algebra of a Lie group is DEFINED as the tangent space of the Lie groupYou can define a Lie algebra as a purely algebraic structure. You can conjure up the wildest Lie algebra there is without referencing a Lie group. The Lie algebra - Lie group correspondence isn’t one-to-one either. The logarithm map from a connected Lie group to a Lie algebra has its center as the kernel and disconnected Lie groups don’t even have all their elements map to a Lie algebra.
>>16538056that's not what you've asked though.
>>16538056I said the "Lie algebra of a Lie group", which is what the original question asked about.Obviously other Lie algebras exist, but they are irrelevant to the question.
