[Math] Exponential map for the Lie group of upper triangular matrices

lie-algebraslie-groupslinear algebra

Let $G$ be the Lie group of all upper triangular real matrices (over $\mathbb{R}$) with positive diagonal elements. Denote $\mathfrak{g}$ its Lie algebra.

  1. Do we have surjectivity of $\exp : \mathfrak{g}\to G$ ? and what about injectivity?

  2. Is $\mathfrak{g}$ equal to the algebra of all upper triangular matrices?

Best Answer

The exponential map $\exp\colon \mathfrak{n}^+\rightarrow N$ from the nilpotent Lie algebra of all stricly upper-triangular matrices to the nilpotent group of all upper uni-triangular matrices is given by polynomial maps with inverse given by matrix logarithm. Hence $\exp$ is bijective in this case. This is no longer true for upper-triangular matrices. The question when $\exp$ is injective is easier than the question when $\exp$ is surjective. For injectivity we have very nice criteria, given for example at this MSE-question:

If $G$ is a real Lie group with Lie algebra $\mathfrak{g}$, then the following are equivalent:

  1. $\exp$ is injective
  2. $\exp$ is bijective
  3. $\exp$ is a real analytic diffeomorphism
  4. $G$ is solvable, simply connected, and $\mathfrak{g}$ does not admit $\mathfrak{e}$ as subalgebra of a quotient.
  5. $G$ is solvable, simply connected, and $\mathfrak{g}$ does not admit $\mathfrak{e}$ or $\tilde{\mathfrak{e}}$ as subalgebra

Here $\mathfrak{e}$ is the 3-dimensional Lie algebra with basis $(H,X,Y)$ and bracket $[H,X]=Y$, $[H,Y]=-X$, $[X,Y]=0$. It is isomorphic to the Lie algebra of the group of isometries of the plane. Its central extension $\tilde{\mathfrak{e}}$ is defined as the 4-dimensional Lie algebra defined by adding a central generator $Z$ and the additional nonzero bracket $[X,Y]=Z$.

Concerning surjectivity the situation is more complicated, see On surjectivity of exponential map for Lie groups, or Exponential map is surjective for compact connected Lie group.

So, for 1. Yes, $\exp$ is surjective, because every upper-triangular real matrix with positive diagonal entries is invertible and is the square of an invertible matrix, see here.

For 2.) Yes.

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