Generators of a Lie Group from One-Parameter Subgroups

1. The Generator: Tangent to the Group

Imagine a curve $g(t)$ on the manifold of a Lie group $G$, where $t$ is a real parameter. Let's ensure this curve passes through the identity element, say at $t=0$, so $g(0)=e$. Since the group is a smooth manifold, we can ask for the tangent vector to this curve at the identity. This tangent vector is precisely what we call a generator. It describes the "direction" of an infinitesimal transformation away from the identity.

Mathematically, the tangent vector is found by taking the derivative of the curve with respect to its parameter and evaluating it at the point corresponding to the identity:

This matrix $X$ is an element of the Lie algebra $\mathfrak{g}$. This simple procedure is the core recipe for finding generators.

Let's see this in action for a familiar case: a rotation in SO(3) about the z-axis. The one-parameter family of rotation matrices is given by:

This is a curve in the SO(3) manifold parameterized by $\theta$, passing through the identity matrix at $\theta =0$. To find the generator of these rotations, we differentiate and evaluate at $\theta =0$:

This matrix $X_z$ is the generator of rotations about the z-axis. It is an element of the Lie algebra $\mathfrak{so}(3)$. The following video provides a clear and intuitive walkthrough of this idea, starting from the inverse perspective of building the group from the generator.

2. The Lie Algebra as the Tangent Space

We've defined a generator as the tangent vector to a curve at the identity. This points to a deeper geometric meaning. For any point on a manifold, the set of all possible tangent vectors at that point forms a vector space, known as the tangent space.

The Lie algebra $\mathfrak{g}$ of a Lie group $G$ is formally defined as the tangent space at the identity element, $\mathfrak{g}=T_{e}G$.

Each generator we calculate is simply one vector in this space. The set of all such generators, which can be obtained by considering all possible smooth curves through the identity, spans the entire vector space $\mathfrak{g}$. This is a profound simplification: we have moved from a curved manifold $G$ to a flat vector space $\mathfrak{g}$.

This geometric picture is fundamental. The structure of the group in an entire neighborhood of the identity is captured by this linear space of "infinitesimal" transformations.

3. One-Parameter Subgroups and the Exponential Map

The curves we've considered, like $R_z(\theta )$, are special. They satisfy the property $R_z({\theta }_1)R_z({\theta }_2)=R_z({\theta }_1+{\theta }_2)$. This means that composing two transformations is equivalent to adding their parameters. Such a curve is called a one-parameter subgroup. Formally, it is a Lie group homomorphism $\gamma :(\mathbb{R},+)\to G$.

There is a fundamental theorem in Lie theory that connects generators and one-parameter subgroups:
For every generator $X$ in the Lie algebra $\mathfrak{g}$, there exists a unique one-parameter subgroup ${\gamma }_X(t)$ in $G$ such that its tangent vector at the identity is $X$.

This establishes a one-to-one correspondence between elements of the Lie algebra and one-parameter subgroups of the Lie group. This unique curve is generated by the exponential map:

For matrix Lie groups, this is simply the standard matrix exponential, defined by its Taylor series: $\exp (A)={\sum }_{k=0}^{\infty }\frac{A^k}{k!}$.

This closes the loop beautifully:

1. Start with a one-parameter subgroup $g(t)$ in $G$.
2. Find its generator $X\in \mathfrak{g}$ by differentiating: $X=\frac{d}{dt}g(t){|}_{t=0}$.
3. Recover the entire one-parameter subgroup by exponentiating the generator: $g(t)=\exp (tX)$.

The following reading provides an excellent physicist's perspective on this connection, bridging the intuitive notion of infinitesimal transformations with the more formal structure of one-parameter subgroups.

Given your background in differential geometry from courses like General Relativity, it's worth noting that this structure can be described even more abstractly. The generator $X\in T_{e}G$ can be used to define a left-invariant vector field over the entire manifold $G$. The one-parameter subgroup ${\gamma }_X(t)$ is then the unique integral curve of this vector field that passes through the identity. This powerful formulation, detailed in resources like the lecture notes by Frederic Schuller or Gregory Moore, guarantees the existence of the exponential map for any Lie group, not just matrix groups.

4. Deriving Lie Algebras for the Classical Groups

We can now put this formalism to work to derive the Lie algebras for the groups we met in the last lesson. The strategy is to consider a curve $g(t)=\exp (tX)$ and enforce the defining conditions of the group on $g(t)$. This will reveal the constraints that $X$ must satisfy, thereby defining the Lie algebra.

The Orthogonal Algebra $\mathfrak{so}(n)$

Let $G=\text{SO}(n)$. The defining property is that its elements $g$ are real matrices satisfying $g^{T}g=I$ and $\det (g)=1$.
Consider a curve $g(t)=\exp (tX)$ near the identity. We must have:

Using the property $(e^A{)}^T=e^{A^T}$, this becomes:

Differentiating with respect to $t$ at $t=0$ using the product rule gives:

Since the right-hand side is constant ($I$), its derivative is 0. Thus, we must have:

So, the Lie algebra $\mathfrak{so}(n)$ consists of all $n\times n$ real anti-symmetric matrices. The determinant condition $\det (g)=1$ imposes an additional constraint, which for matrices near the identity is $\text{Tr}(X)=0$. For anti-symmetric matrices, the diagonal elements are zero, so their trace is automatically zero. Thus, the condition is satisfied.

This method is extremely powerful. The reading below applies it to derive the algebras for the other main matrix groups.

Conclusion

In this lesson, we have forged the critical link between a Lie group and its Lie algebra. We established a concrete procedure for deriving the generators of the group and gave this procedure a deep geometric meaning.

Key Takeaways:

• The Lie algebra $\mathfrak{g}$ of a Lie group $G$ is the tangent space to the group at the identity element, $\mathfrak{g}=T_{e}G$. Its elements are called generators.
• A generator $X$ can be derived from a one-parameter subgroup $g(t)$ by taking the derivative at the identity: $X=g^{\prime }(0)$.
• Conversely, a one-parameter subgroup can be reconstructed from its generator via the exponential map: $g(t)=\exp (tX)$.
• This procedure allows us to determine the properties of the Lie algebra. For example:
  • $\mathfrak{so}(n)$ is the algebra of real anti-symmetric matrices.
  • $\mathfrak{su}(n)$ is the algebra of complex traceless anti-hermitian matrices.

We have successfully linearized the problem of continuous symmetries. Instead of the curved group manifold, we now have a flat vector space of generators to work with. However, this vector space is not just a set of vectors; it has its own rich algebraic structure, defined by a product called the Lie bracket. In our next lesson, we will explore how this bracket arises from the group multiplication law and how to compute the commutation relations that define the Lie algebra.