-”this is just the lie algebra, and is why elements of it are always invertible.”
First of all, how did we move from talking about numbers to talking about Lie algebras? What is the Lie group here? The only way I can make sense of your statement is if you are considering the case of a Lie subgroup of GL(n,R) for some n, and letting 1 denote the identity matrix (rather than the number 1) [1]. But then...
Shouldn’t the Lie algebra be the monad of 0, rather than the monad of 1? Because usually Lie algebras are defined in terms of being equipped with two operations, addition and the Lie bracket. But neither the sum nor the Lie bracket of two elements of the monad of 1 are in the monad of 1.
[1] Well, I suppose you could be considering just the special case n=1, in which case 1 the identity matrix and 1 the number are the same thing. But then why bother talking about Lie algebras? The group is commutative, so the formalism does not appear to be necessary.
-”this is just the lie algebra, and is why elements of it are always invertible.”
First of all, how did we move from talking about numbers to talking about Lie algebras? What is the Lie group here? The only way I can make sense of your statement is if you are considering the case of a Lie subgroup of GL(n,R) for some n, and letting 1 denote the identity matrix (rather than the number 1) [1]. But then...
Shouldn’t the Lie algebra be the monad of 0, rather than the monad of 1? Because usually Lie algebras are defined in terms of being equipped with two operations, addition and the Lie bracket. But neither the sum nor the Lie bracket of two elements of the monad of 1 are in the monad of 1.
[1] Well, I suppose you could be considering just the special case n=1, in which case 1 the identity matrix and 1 the number are the same thing. But then why bother talking about Lie algebras? The group is commutative, so the formalism does not appear to be necessary.