Representation of a Lie group — Safekipedia Discoverer

In mathematics and theoretical physics, a representation of a Lie group is a way to understand how a special kind of group, called a Lie group, acts on a space filled with arrows or directions, known as a vector space. This action helps us see how the group changes or moves things in a straightforward, linear way. Think of it like a set of instructions that tells us exactly how each part of the group can move or transform the vectors in the space.

Representations are very important because they help us study continuous symmetry. Symmetry is everywhere—in the way a circle looks the same from any angle, or how objects move smoothly without sudden jumps. By using representations, mathematicians and physicists can break down complex symmetrical patterns into simpler parts that are easier to analyze.

One powerful method to study these representations is by looking at their "infinitesimal" versions, called representations of Lie algebras. These smaller, simpler versions capture the essence of how the group behaves in tiny steps, making it easier to understand the whole picture. This idea is a key tool in many areas of advanced math and physics, helping experts explore the deep connections between symmetry, geometry, and the laws of nature.

Finite-dimensional representations

Further information: Group representation

A representation of a Lie group is a way to describe how the group acts on a vector space. Imagine a group of transformations, like rotating or flipping objects. When we apply these transformations to a space of vectors (which are like arrows pointing in different directions), we can study how the group changes the vectors.

In simple terms, a representation tells us how each element of the group moves or changes the vectors in the space. This idea is important in physics, where symmetries of space and time help us understand the laws of nature. For example, the group of rotations in three dimensions has representations that describe how objects spin and change orientation.

General linear group helps us understand these transformations as matrices, which are like grids of numbers that can stretch, rotate, or flip vectors in predictable ways.

An example: The rotation group SO(3)

In quantum mechanics, the Schrödinger equation helps describe how 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 the symmetry 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. For two representations, the direct sum creates a new space where the group acts separately on each part.

The tensor product combines representations in a way that pairs their actions together. This operation is important in physics, especially in studying how particles combine their properties.

The dual representation flips the way a representation acts, using a special "transpose" operation. This helps in understanding how representations relate to each other.

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 are important in studying continuous symmetry. Often, we study representations of a Lie group by looking at the related Lie algebra. However, not every representation of the Lie algebra comes from the group. For instance, 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 correspondence between group and Lie algebra representations.

To get group representations from the Lie algebra, we use a process called the Lie correspondence. This involves using the matrix exponential of the Lie algebra representation. If the group is not simply connected, this can lead to projective representations, which are representations of the universal covering group of the original group. These concepts help explain certain phenomena in quantum physics, such as 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 down their representations into simpler parts. These simpler parts are called "irreducible representations." For a special kind of group, we can classify these parts using a method called the "theorem of the highest weight." This helps us understand how these groups behave in various situations.

One useful tool in this study is called a "character," which 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 understanding these groups better.

Unitary representations on Hilbert spaces

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

When the vector space is finite-dimensional, there are simpler rules to follow. For example, if the Lie group is compact, we can always adjust the space so that the representation becomes unitary. In more complex, infinite-dimensional spaces, studying these representations becomes more challenging but is very important in areas like quantum mechanics and Fourier analysis. For instance, the way the real numbers act on a space of functions by shifting them 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 is important because the way particles behave in quantum systems can sometimes only be described using these special representations.

These representations are different from regular ones because changing an operator by a certain amount does not change the physical result. This helps explain things like fractional spin in particles, which cannot be explained using regular representations alone. 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 certain special groups in mathematics, called commutative Lie groups, something interesting happens with their representations. These representations can be very simple — in fact, they 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, following a simple rule. This idea connects to other areas of mathematics, such as Pontryagin duality.

Main article: Pontryagin duality