Question

Let id \(_{M}: M \rightarrow M\) be the identity map on \(M\). Prove that \(\mathrm{id}_{M *}\) is the identity map on \(T(M)\). Let \(I_{g}=R_{g^{-1}} \circ L_{g}\) be the inner automorphism of a Lie group \(G\). Show that if \(G\) is abelian, then \(I_{g}=\mathrm{id}_{G}\) for all \(g \in G\). Now show that \(A d_{g}=\mathrm{id}_{\mathfrak{g}}\).

Short answer

The solution consists in showing that the pushforward of the identity map is indeed the identity map on the tangent space, yielding a trivial action. Then, employing the fact that in an abelian group, conjugation does not change the element, we prove that the inner automorphism is identical for all elements in the group. Finally, we finish by demonstrating that the adjoint map must be the identity on the corresponding Lie algebra given the identical action of its generating inner automorphism.

Step-by-step solution

Understand definitions

Before even beginning, it is important to understand the elements at play in this exercise. \n 1. The identity map, \(\mathrm{id}_{M}\), is a map that sends any point in a set \(M\) to itself. 2. The pushforward, \(\mathrm{id}_{M *}\), of a smooth map between manifolds measures how the map pushes tangent vectors from one manifold to the other. 3. The inner automorphism \(I_{g}\) of a Lie group \(G\) associated to the element \(g\) is defined by conjugation, \(I_{g} = R_{g^{-1}} \circ L_{g}\), where \(R_{g^{-1}}\) and \(L_{g}\) are right and left translations by \(g\) and \(g^{-1}\) respectively. 4. A Lie group \(G\) is abelian if the group operation is commutative, which means that \(gh = hg\) for all \(g,h \in G\). 5. The adjoint map \(Ad_{g}\) of a Lie group \(G\) is the differential of the Inner automorphism at the identity element of the group.

Prove that \(\mathrm{id}_{M *}\) is the identity map on \(T(M)\)

To show this, suppose \(v\) is a tangent vector in \(T_p(M)\), the tangent space at a point \(p\). We are given that the pushforward of the identity map \(\mathrm{id}_{M *}\) applied to \(v\) equals \(v\). Since the identity map sends \(p\) to \(p\) and \(v\) to \(v\), we have shown that \(\mathrm{id}_{M *}\) is the identity map on \(T(M)\).

Prove that \(I_{g}=\mathrm{id}_{G}\) for all \(g \in G\)

If \(G\) is abelian, then the conjugation is trivial. That is, for any \(h \in G\), we have \((R_{g^{-1}} \circ L_{g})(h) = g^{-1} * (g * h) = g^{-1} * (h * g) = (g^{-1} * g) * h = e * h = h\), where \(e\) is the identity element in \(G\), and we have used the commutativity of group operation from the abelian property of \(G\). This proves that \(I_{g} = R_{g^{-1}} \circ L_{g} = \mathrm{id}_{G}\) for an abelian work \(G\).

Show that \(Ad_{g} = \mathrm{id}_{\mathfrak{g}}\)

The final claim rests on observating that, as per the definition, \(Ad_{g}\) is the differential of the map \(I_{g}\) at the identity element. From the above, we have shown that for any \(g \in G\), \(I_{g} = \mathrm{id}_{G}\). Hence, their differentials (pushforwards) must be identical as well, so we derive that \(Ad_{g} = \mathrm{id}_{\mathfrak{g}}\) where \(\mathfrak{g}\) is the Lie algebra of \(G\).

Key concepts

identity map

An identity map, often denoted as \( \mathrm{id}_{M} \), is a very simple but fundamental concept in mathematics. It is a function that maps every element of a set \( M \) to itself. In other words, if you take any element \( x \) in the set \( M \), the identity map will transport \( x \) right back to \( x \). It's like a mirror, showing you exactly what you started with.

In the context of differential geometry, when working with manifolds, the identity map's job is to leave the geometry unchanged. It not only applies to points in a set but can also act on tangent vectors, which leads us to the pushforward concept. The pushforward, denoted \( \mathrm{id}_{M *} \), specifically refers to how tangent vectors are 'moved' or 'transferred' along the map. The identity map's uniqueness is that it moves both points and tangent vectors to the exact position they originally occupied. This consistency makes it an essential tool in proving properties involving more complex transformations.

abelian group

An abelian group, named after the mathematician Niels Henrik Abel, is a type of algebraic structure where the order in which you perform operations doesn't matter. This property is known as commutativity, meaning for any two elements \( a \) and \( b \) in the group \( G \), the equation \( a * b = b * a \) always holds true.

This commutativity can simplify the treatment of problems significantly. In an abelian group, every element behaves very predictably when combined with others. When dealing with Lie groups, if the group happens to be abelian, the inner automorphisms become the identity map, implying that the transformation doesn't change any group element. The implications of this are rich in the study of symmetry and topological properties, often simplifying the approach to complex mathematical problems.

inner automorphism

Inner automorphisms of a group are transformations within the group that are defined through conjugation. For a Lie group \( G \), the inner automorphism associated with an element \( g \) is given as \( I_{g} = R_{g^{-1}} \circ L_{g} \), where \( R_{g^{-1}} \) is the right multiplication by the inverse of \( g \) and \( L_{g} \) is the left multiplication by \( g \). This combination forms a transformation where each element \( h \) in \( G \) is mapped to \( g^{-1}hg \).

In the context of abelian groups, since \( g \) and \( h \) commute, the inner automorphism simply results in the identity map. Therefore, no change is observed, because \( g^{-1}hg = h \). This effect demonstrates how inner automorphisms relate to the symmetry and intrinsic algebraic structure of the group. Understanding inner automorphisms is pivotal in group theory as these help in revealing the internal structure and symmetries of the group.

tangent space

The concept of a tangent space is essential in differential geometry. For any smooth manifold \( M \), a tangent space at a point \( p \) can be thought of as a plane that 'touches' the manifold at that point. More precisely, it consists of all the tangent vectors that indicate the directions in which you can tangentially pass through \( p \).

These vectors do not originate from an external source but are derived from curves passing through \( p \). The collection of all these tangent spaces, spread across the manifold, forms a structure called a tangent bundle. A convenient way to understand tangent spaces is to imagine flattening a part of a surface while keeping its orientation and properties intact. When we consider the identity map on a manifold \( M \), its role in tangent spaces is to ensure that these vectors are precisely preserved, maintaining the geometry of the manifold at that local point. This accommodation of tangent vectors is crucial in revealing how manifold sub-structures behave under transformations and maps.