nLab
super tangent bundle

Idea

The tangent bundle of an ordinary manifold is a vector bundle whose sheaf of sections? is given by the derivations of the structure sheaf. The same idea applies to a supermanifold to produce a super vector bundle.

Definition

The super tangent bundle $TX$ of a supermanifold $X$ is given by the sheaf $U\mapsto \mathrm{Der}{O}_X(U)$.

So a super tangent vector is a global section of this sheaf of derivations.

Example

On the supermanifold ${ℝ}^{1\mid 1}$ with its canonical coordinates

$$t\in C^{\mathrm{\infty }}({ℝ}^{1\mid 1}{)}^{\mathrm{ev}}$$t \in C^\infty(\mathbb{R}^{1|1})^{ev}

$$\theta \in C^{\mathrm{\infty }}({ℝ}^{1\mid 1}{)}^{\mathrm{odd}}$$\theta \in C^\infty(\mathbb{R}^{1|1})^{odd}

there is the odd vector field

$$D≔{\partial }_{\theta }+\theta \cdot {\partial }_t$$D \coloneqq \partial_\theta + \theta \cdot \partial_{t}

whose super Lie bracket with itself vanishes

$$[D,D]=0.$$[D, D] = 0 \,.

Claim

This odd vector field $D$ is left invariant with respect to the super translation group structure on ${ℝ}^{1\mid 1}$.

This means that $\mathrm{Lie}({ℝ}^{1\mid 1})$ is free on one odd generator.