Representation of a Lie group

In mathematics and theoretical physics, a representation of a Lie group is a way to see how a special group, called a Lie group, works on a space with arrows or directions, known as a vector space. This helps us understand how the group changes or moves things in a simple, straight way. Imagine it as a set of instructions showing how each part of the group can move or change the vectors.

Representations are important because they help us study continuous symmetry. Symmetry is found everywhere—like how a circle looks the same from any angle, or how objects move smoothly. By using representations, experts can break down complicated patterns into simpler parts that are easier to study.

One strong way to study these representations is by looking at their smaller, simpler versions, called representations of Lie algebras. These tiny versions show how the group behaves step by step, making it simpler to understand the whole idea. This concept is a key tool in many areas of advanced math and physics, helping people explore links between symmetry, shapes, and natural laws.

Finite-dimensional representations

Further information: Group representation

A representation of a Lie group is a way to show how the group works on a vector space. Think of a group of changes, like turning or flipping things. When we use these changes on a space of vectors (which are like arrows pointing in many directions), we can see how the group moves the vectors.

Simply put, a representation shows us how each part of the group shifts or changes the vectors. This idea is useful in physics, where the symmetries of space and time help us learn about natural laws. For instance, the group of turns in three dimensions has representations that explain how objects spin and change direction.

The General linear group helps us understand these changes as matrices, which are like grids of numbers that can stretch, turn, or flip vectors in set ways.

An example: The rotation group SO(3)

In quantum mechanics, the Schrödinger equation helps describe how tiny parts of nature, like particles, behave. When studying particles that can spin in three dimensions, scientists use a special group called SO(3) to understand this spinning. This group helps explain patterns in problems like the hydrogen atom.

The rotation group SO(3) has special ways to show how particles change when they spin. For example, when looking at particles with certain spins, these changes can be described using math that involves spinning in space. These ideas are important for understanding how atoms and particles work.

Main article: Representation theory of SU(2)

Main articles: hydrogen atom, mathematical analysis of hydrogen

half-integer spin

projective

Operations on representations

Main article: Tensor product of representations

Main article: Dual representation

Representations of groups can be combined in three basic ways. The direct sum combines two representations into a larger one by pairing their actions.

The tensor product combines representations by pairing their actions together. This is useful in physics.

The dual representation changes the way a representation acts, using a special "transpose" operation.

Lie group versus Lie algebra representations

Main article: Lie group–Lie algebra correspondence

Sophus Lie, the originator of Lie theory. The theory of manifolds was not discovered in Lie's time, so he worked locally with subsets of R n . {\displaystyle \mathbb {R} ^{n}.} The structure would today be called a local group.

Representations of Lie groups and Lie algebras help us understand continuous symmetry. We often study representations of a Lie group by looking at its related Lie algebra. But not every representation of the Lie algebra comes from the group. For example, this is why we see the difference between integer and half-integer spin in quantum mechanics. If the group is simply connected, there is a one-to-one match between group and Lie algebra representations.

To get group representations from the Lie algebra, we use a process called the Lie correspondence. This uses the matrix exponential of the Lie algebra representation. If the group is not simply connected, this can lead to projective representations. These are representations of the universal covering group of the original group. These ideas help explain some things in quantum physics, like fractional spin.

π ( X ) = d d t Π ( e t X ) | t = 0 , X ∈ g . {\displaystyle \pi (X)=\left.{\frac {d}{dt}}\Pi (e^{tX})\right|_{t=0},\quad X\in {\mathfrak {g}}.}

Π ( g = e X ) ≡ e π ( X ) , X ∈ g , g = e X ∈ im ⁡ ( exp ) , Π ( g = g 1 g 2 ⋯ g n ) ≡ Π ( g 1 ) Π ( g 2 ) ⋯ Π ( g n ) , g ∉ im ⁡ ( exp ) , g 1 , g 2 , … , g n ∈ im ⁡ ( exp ) . {\displaystyle {\begin{aligned}\Pi (g=e^{X})&\equiv e^{\pi (X)},&&X\in {\mathfrak {g}},\quad g=e^{X}\in \operatorname {im} (\exp ),\\\Pi (g=g_{1}g_{2}\cdots g_{n})&\equiv \Pi (g_{1})\Pi (g_{2})\cdots \Pi (g_{n}),&&g\notin \operatorname {im} (\exp ),\quad g_{1},g_{2},\ldots ,g_{n}\in \operatorname {im} (\exp ).\end{aligned}}}

Classification in the compact case

Main article: Compact group § Representation theory of a connected compact Lie group

When we study certain groups in mathematics, we can break them into simpler parts. These simpler parts are called "irreducible representations." For a special kind of group, we can sort these parts using a method called the "theorem of the highest weight." This helps us understand how these groups act in different situations.

One useful tool is called a "character." It gives us important information about a representation. The character helps us see if two representations are the same in a certain way. There is also a special formula, called the "Weyl character formula," that helps us find the character of a representation using its highest weight. This formula is very important for learning more about these groups.

Unitary representations on Hilbert spaces

A unitary representation of a Lie group is a special way to show how the group works on a vector space. It uses special operations called unitary operators. This helps us understand continuous symmetry in mathematics and physics.

When the vector space has a limited number of dimensions, there are simpler rules to follow. For example, if the Lie group is compact, we can change the space so that the representation becomes unitary. In more complex spaces with endless dimensions, studying these representations is challenging but important. It is especially useful in areas like quantum mechanics and Fourier analysis. For instance, the way the real numbers move functions left or right is an example of a unitary representation.

Projective representations

Main article: Projective representation

In quantum physics, scientists study something called projective unitary representations of a Lie group. This helps explain how particles behave in quantum systems.

These representations are different from regular ones. Changing an operator by a certain amount does not change the physical result. This helps explain things like fractional spin in particles. There are special rules and theorems, like Bargmann's theorem, that help us understand these representations better, especially for groups like the rotation group and the Poincaré group.

The commutative case

When we talk about special groups in mathematics, called commutative Lie groups, their representations can be very simple. These representations are one-dimensional, meaning they can be pictured as directions on a straight line.

For example, if we consider the group of real numbers, the representations look like waves that repeat their pattern. This idea connects to other areas of mathematics, such as Pontryagin duality.