# Left invariant vector fields and Lie brackets of the upper triangular matrices

This question was asked in my assignment on smooth manifolds and I am not able to solve it because vector fields has been 1 of my weak points. I have been following introduction to smooth manifolds by Lee along with my notes.

Question: Consider the group H of upper triangular matrices of the form $$\begin{bmatrix} 1&a&c \\ 0& 1& b\\ 0& 0& 1\end{bmatrix}$$ with the matrix multiplication being the group operation. Show that H is a Lie Group. Characterise the left invariant vector fields of H and describe the Lie bracket on the Lie algebra of H.

Attempt: I have proved that H is a lie group.

I am having troubles in using the definition of left invariant vector fields:

On pg 189 of John Lee the definition is given as: Suppose G is a Lie group. Recall that G acts smoothly and transitively on itself by left translation: $$L_g(h) = gh$$. A vector field X on G is said to be left-invariant if it is invariant under all left translations, in the sense that it is $$L_g$$ -related to itself for every $$g \in G.$$ More explicitly, this means that $$d(L_g)_{g'}(X_{g'}) =X_{gg'}$$ for all $$g,g' \in G$$.

Trying to use the definition the problem I am facing is that I am not sure how should I define $$X_{g'}$$ and $${X_{gg'}$$ Can you please help me with that

Describing Lie brackets on the Lie Algebra of H:

A Lie algebra (over R) is a real vector space $$g$$ endowed with a map called the bracket from $$g \times g \to g$$, usually denoted by $$(X,Y) \to [X,Y]$$ , that satisfies the following properties for all X,Y,Z $$\in g$$:(i) Bilinearity , Anti-symmetry, Jacobi Identity and one properties related to smooth functions.

Lie bracket is an operator $$[X,Y] : C^{\infty}(M) \to C^{\infty}(M)$$ is called a Lie bracket of X and Y. $$[X,Y]f = XYf -YXf$$.

The set of all upper triangular matrices is a vector space wrt +. Now, I think the operator will be if X,Y are Upper triangular matrices and f another matrix which is upper triangular then the operator will be $$XYf -YXf$$ with usual matrix multiplication. Have I defined the lie bracket correctly? It will satisfy all the properties of Lie brackets correctly.

Can you please verify the claim related to Lie Brackets and tell me how should I approach the part related to vector fields as I am not very comfortable in that.

## 1 Answer

When a vector field on $$H$$ is left-invariant, it is completely determined by its value at the identity of $$H$$. This is an immediate consequence of the definition.

The tangent space at the identity of $$H$$ is isomorphic to $$\mathbb R^3$$ as a vector space, because $$H$$ is three-dimensional. A basis is given by the matrices $$E_1 = \left( \begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \right), \quad E_2 = \left( \begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \right), \quad E_3 = \left( \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} \right).$$ Now computing $$(L_g)_*E_i$$ and taking linear combinations gives all possible left-invariant vector fields on $$H$$.

The commutator of two vector fields $$X$$ and $$Y$$ is defined as $$[X,Y] = XY-YX$$. Here you can easily compute \begin{align*} [E_1,E_2] & = E_1E_2-E_2E_1 = 0, \\ [E_1,E_3] & = E_1E_3-E_3E_1 = E_2 \\ [E_2,E_3] & = E_2E_3-E_3E_2 = 0. \end{align*} Since $$(L_g)_*([X,Y]) = [(L_g)_*X,(L_g)_*Y]$$, again the values taken by the commutators on $$H$$ are all determined by the above three identities. This operation gives the structure of Lie algebra to the space of left-invariant vector fields.

• Can you please give a proof of "When a vector field on H is left-invariant, it is completely determined by its value at the identity of H."?
– user775699
Commented May 12, 2022 at 12:57
• If $X$ is an $H$-invariant vector field on $M$, then $(L_g)_*X_p = X_{gp}$, where $L_g$ is left translation by $g \in H$. Now take $p = e$ (here $e$ is the identity of $H$). Then $X_g = (L_g)_*X_e$. This means $X_g$ is determined by the value of $X$ at $e$. Commented May 13, 2022 at 19:56
• I have asked many other questions on Manifolds and most of which are unanswered. Do you mind answering some of them.
– user775699
Commented Jul 5, 2022 at 7:40