why are the group elements shown as an orange field of arrows to the right left translations? What's translating to the left? Or does this simply mean that these operate on something from the left?
The orange arrows are elements of the Lie algebra, not the group. The Lie algebra can be defined as the tangent vectors to the group manifold at the identity, and left translation refers to the action of the group on itself, operating from the left. So you can view this action as transporting the Lie algebra vectors to a point on the group other than the identity. The Lie algebra is then a vector field over the manifold that is invariant under this action. These are concepts from differential geometry that apply here because Lie groups are manifolds as well as being groups. You can check the second video in this series, "Lie Algebras" for more clarification
Nice video
why are the group elements shown as an orange field of arrows to the right left translations? What's translating to the left? Or does this simply mean that these operate on something from the left?
The orange arrows are elements of the Lie algebra, not the group. The Lie algebra can be defined as the tangent vectors to the group manifold at the identity, and left translation refers to the action of the group on itself, operating from the left. So you can view this action as transporting the Lie algebra vectors to a point on the group other than the identity. The Lie algebra is then a vector field over the manifold that is invariant under this action. These are concepts from differential geometry that apply here because Lie groups are manifolds as well as being groups. You can check the second video in this series, "Lie Algebras" for more clarification
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But, can it be programmed in R?