Lie Algebra

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Lie algebra

Algebraic structure used in analysis

A Lie algebra — let's call it ${\mathfrak {g}}$ — isn't just some random collection of vectors. It's a vector space, yes, but it's burdened with an operation. This operation, the Lie bracket, denoted by $[x, y]$, takes two vectors from ${\mathfrak {g}}$ and spits out another vector within ${\mathfrak {g}}$. Think of it as a very specific, very particular kind of internal combustion.

This bracket isn't some free-for-all. It’s an alternating bilinear map. Bilinear means it respects scalar multiplication and addition: $[ax+by, z] = a[x,z] + b[y,z]$ and $[z, ax+by] = a[z,x] + b[z,y]$ for any scalars $a, b$ and vectors $x, y, z$ in ${\mathfrak {g}}$. Alternating means $[x, x] = 0$ for all $x$ in ${\mathfrak {g}}$. And on top of all that, it must satisfy the Jacobi identity: $[x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0$. It’s a rigid structure, like a perfectly constructed cage.

Essentially, a Lie algebra is an algebra over a field where multiplication, the Lie bracket, is alternating and adheres strictly to the Jacobi identity. The Lie bracket of $x$ and $y$ is written as $[x, y]$.

Lie algebras are typically non-associative algebras. Don't expect $(xy)z = x(yz)$ here. The bracket defines its own kind of multiplication, one that’s more about structure than simple sequential operation. However, and this is important, any associative algebra can be twisted into a Lie algebra. You just take the same vector space and define the Lie bracket as the commutator: $[x, y] = xy - yx$. It's like taking a perfectly functional tool and forcing it into a role it wasn't quite built for, but it can do it.

Lie groups and Lie algebras

The real meat of Lie algebras lies in their intimate connection to Lie groups. Think of Lie groups as smooth, continuous groups – like rotations or translations. Lie algebras are their infinitesimal counterparts, their tangent spaces at the identity element. Every Lie group has a Lie algebra, and this algebra’s bracket essentially quantifies how the Lie group fails to be commutative. It’s the first hint of non-commutativity, the subtle tremor before the seismic shift.

Conversely, and this is where it gets interesting, for any finite-dimensional Lie algebra over the real numbers or complex numbers, there exists a corresponding connected Lie group. This is Lie's third theorem. This correspondence is crucial. It allows us to understand the complex, often unwieldy structure of Lie groups by dissecting their simpler, linear algebraic counterparts – the Lie algebras. It’s like understanding a storm by studying the pressure systems that precede it.

To elaborate: near its identity element, any Lie group behaves like a vector space. That’s the Lie algebra, ${\mathfrak {g}}$, the tangent space at the identity. But the group operation itself, even near the identity, might not be commutative. The second-order terms, the subtle deviations from simple addition, are captured by the Lie bracket. These terms, the Lie algebra, are surprisingly powerful. They don't just hint at the group's structure; they completely determine it, at least locally. And often, globally, up to the nuances of covering spaces.

In physics, this relationship is fundamental. Lie groups represent the symmetry groups of physical systems. Their Lie algebras, the infinitesimal symmetries, are the "motions" or transformations that leave the system unchanged. Thus, Lie algebras and their representation theory are indispensable tools in fields like quantum mechanics and particle physics, where symmetries dictate the fundamental laws.

An elementary example

Let's consider a simple, concrete illustration. Imagine the 3-dimensional space ${\mathbb {R}}^{3}$. We can define a Lie bracket on this space using the cross product: $[x, y] = x \times y$.

This operation is inherently skew-symmetric: $x \times y = -y \times x$. This is a direct consequence of the alternating property: $[x, x] = x \times x = 0$.

And it satisfies the Jacobi identity: $x \times (y \times z) + y \times (z \times x) + z \times (x \times y) = 0$. This isn't just an algebraic curiosity; it reflects the geometric reality of how rotations in 3D space interact.

This particular Lie algebra, ${\mathbb {R}}^{3}$ with the cross product, is the Lie algebra of the 3D rotation group, SO(3). Each vector $v \in {\mathbb {R}}^{3}$ can be pictured as an infinitesimal rotation around the axis defined by $v$, with an angular speed equal to the magnitude of $v$. The Lie bracket, $[x, y]$, then measures the non-commutativity between two such infinitesimal rotations. If they commute, their bracket is zero.

The general linear Lie algebra

A cornerstone example, and one that doesn't necessarily derive from a geometric group initially, is the space of all linear maps from a vector space to itself. Let's call this space ${\mathfrak {gl}}(V)$. When $V$ has a finite dimension, say $n$, this becomes the space of $n \times n$ matrices. This is the general linear Lie algebra, ${\mathfrak {gl}}(n)$.

The Lie bracket here is the familiar commutator: $[X, Y] = XY - YX$. This is where the abstract definition meets tangible computation.

History

The concept of Lie algebras didn't just appear out of thin air. It was Sophus Lie who, in the 1870s, introduced them to explore the notion of infinitesimal transformations. He saw them as the algebraic underpinnings of continuous symmetries. Independently, Wilhelm Killing explored similar structures in the 1880s, particularly in his work on the classification of simple Lie groups. The name "Lie algebra" itself was later coined by Hermann Weyl in the 1930s, a testament to Lie's foundational contributions. Before that, they were often referred to as "infinitesimal groups."

Definition of a Lie algebra

Let’s formalize this. A Lie algebra, denoted by ${\mathfrak {g}}$, is a vector space over a field $F$. On this vector space, we define a binary operation, the Lie bracket, $[ \cdot, \cdot ] : {\mathfrak {g}} \times {\mathfrak {g}} \to {\mathfrak {g}}$. This operation must satisfy three axioms:

1. Bilinearity: $[ax+by, z] = a[x,z] + b[y,z]$ $[z, ax+by] = a[z,x] + b[z,y]$ for all scalars $a, b \in F$ and vectors $x, y, z \in {\mathfrak {g}}$. This ensures the bracket behaves predictably with respect to the vector space structure.

2. Alternating property: $[x, x] = 0$ for all $x \in {\mathfrak {g}}$. This is a crucial constraint, preventing trivial self-interaction.

3. Jacobi identity: $[x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0$ for all $x, y, z \in {\mathfrak {g}}$. This is the heart of the matter, the defining characteristic that distinguishes Lie brackets from other operations. It’s a constraint that, when you dig into it, reveals a profound structure.

From bilinearity and the alternating property, we can deduce anticommutativity: $[x, y] = -[y, x]$ for all $x, y \in {\mathfrak {g}}$. This holds unless the field $F$ has characteristic 2, in which case the alternating property implies anticommutativity anyway, as $[x, x] = -[x, x]$ forces $2[x, x] = 0$.

The Jacobi identity can also be rewritten using anticommutativity as a derivation property, often called the "Leibniz rule" for the adjoint map ${\rm {ad}}_{x}(y) = [x, y]$: $[x, [y, z]] = [[x, y], z] + [y, [x, z]]$ for all $x, y, z \in {\mathfrak {g}}$. This form highlights how the bracket acts as a derivative with respect to the bracket operation itself.

It’s customary to denote Lie algebras with lower-case fraktur letters like ${\mathfrak {g}}, {\mathfrak {h}}, {\mathfrak {b}}, {\mathfrak {n}}$. If a Lie algebra is associated with a Lie group, say SU(n), its Lie algebra is typically written as the fraktur version, ${\mathfrak {su}}(n)$.

Generators and dimension

The dimension of a Lie algebra refers to its dimension as a vector space over its base field. In the context of physics, a basis for the Lie algebra of a Lie group G is often called a set of generators for G. They are the "infinitesimal generators," the fundamental building blocks of the group's continuous transformations. In pure mathematics, a set of generators for a Lie algebra ${\mathfrak {g}}$ is a subset $S$ such that any Lie subalgebra containing $S$ must be the entire algebra ${\mathfrak {g}}$. This means any element of ${\mathfrak {g}}$ can be constructed from the generators using iterated brackets.

Basic examples

Abelian Lie algebras

A Lie algebra is called abelian if its Lie bracket is identically zero: $[x, y] = 0$ for all $x, y \in {\mathfrak {g}}$. Any vector space $V$ can be turned into an abelian Lie algebra by simply defining the bracket to be zero. Every one-dimensional Lie algebra is automatically abelian due to the alternating property.

The Lie algebra of matrices

As mentioned, any associative algebra $A$ over a field $F$ can be endowed with the Lie bracket $[x, y] = xy - yx$, turning it into a Lie algebra. The endomorphism ring of an $F$-vector space $V$, denoted ${\mathfrak {gl}}(V)$, is a prime example. For a field $F$ and a positive integer $n$, the space of $n \times n$ matrices over $F$, denoted ${\mathfrak {gl}}(n, F)$ or ${\mathfrak {gl}}_n(F)$, is a Lie algebra under the matrix commutator. This is a fundamental example, the general linear Lie algebra.

When $F$ is the set of real numbers, ${\mathfrak {gl}}(n, \mathbb {R})$ is the Lie algebra of the general linear group $\mathrm {GL} (n, \mathbb {R})$, the group of invertible $n \times n$ real matrices. Similarly, ${\mathfrak {gl}}(n, \mathbb {C})$ is the Lie algebra of the complex Lie group $\mathrm {GL} (n, \mathbb {C})$. The bracket $[X, Y] = XY - YX$ on ${\mathfrak {gl}}(n, \mathbb{R})$ precisely captures the failure of matrix multiplication to commute. For any field $F$, ${\mathfrak {gl}}(n,F)$ can also be viewed as the Lie algebra of the algebraic group $\mathrm {GL} (n)$ over $F$.

Definitions

Subalgebras, ideals and homomorphisms

The Lie bracket is not necessarily associative. That is, $[[x, y], z]$ might not equal $[x, [y, z]]$. This is a critical distinction. However, much of the terminology used for associative rings and algebras also applies to Lie algebras.

A Lie subalgebra is a linear subspace ${\mathfrak {h}} \subseteq {\mathfrak {g}}$ that is closed under the Lie bracket. Think of it as a smaller Lie algebra contained within a larger one.

An ideal ${\mathfrak {i}} \subseteq {\mathfrak {g}}$ is a linear subspace with a stronger condition: $[{\mathfrak {g}}, {\mathfrak {i}}] \subseteq {\mathfrak {i}}$. This means that bracketing any element of the entire algebra ${\mathfrak {g}}$ with an element of the ideal ${\mathfrak {i}}$ always results in an element that remains within ${\mathfrak {i}}$. In the context of Lie groups, ideals correspond to normal subgroups.

A Lie algebra homomorphism is a linear map $\phi \colon {\mathfrak {g}} \to {\mathfrak {h}}$ that respects the Lie bracket structure: $\phi([x, y]) = [\phi(x), \phi(y)]$ for all $x, y \in {\mathfrak {g}}$. An isomorphism is a bijective homomorphism – a perfect structural match.

Just like normal subgroups are the kernels of homomorphisms in group theory, ideals are precisely the kernels of Lie algebra homomorphisms. Given a Lie algebra ${\mathfrak {g}}$ and an ideal ${\mathfrak {i}}$, we can form the quotient Lie algebra ${\mathfrak {g}}/{\mathfrak {i}}$. This gives us a surjective homomorphism ${\mathfrak {g}} \to {\mathfrak {g}}/{\mathfrak {i}}$. The first isomorphism theorem holds here too: the image of a homomorphism $\phi$ is isomorphic to ${\mathfrak {g}}/{\text{ker}}(\phi)$.

For any Lie algebra, elements $x, y \in {\mathfrak {g}}$ are said to commute if their bracket vanishes: $[x, y] = 0$. This is a generalization of the concept of commuting elements in a group.

The centralizer of a subset $S \subseteq {\mathfrak {g}}$ is the set of elements that commute with everything in $S$: ${\mathfrak {z}}_{\mathfrak {g}}(S) = \{x \in {\mathfrak {g}} : [x, s] = 0 \text{ for all } s \in S\}$ The centralizer of the entire algebra ${\mathfrak {g}}$ is its center, ${\mathfrak {z}}({\mathfrak {g}})$. Similarly, the normalizer of a subspace $S$ is: ${\mathfrak {n}}_{\mathfrak {g}}(S) = \{x \in {\mathfrak {g}} : [x, s] \in S \text{ for all } s \in S\}$ If $S$ is a Lie subalgebra, ${\mathfrak {n}}_{\mathfrak {g}}(S)$ is the largest subalgebra containing $S$ for which $S$ is an ideal.

Example of Subalgebras and Ideals

Consider the Lie algebra ${\mathfrak {gl}}(n, F)$ of $n \times n$ matrices. The subspace ${\mathfrak {t}}_n$ of diagonal matrices forms an abelian Lie subalgebra. It's a Cartan subalgebra of ${\mathfrak {gl}}(n)$, analogous to a maximal torus in compact Lie groups. However, ${\mathfrak {t}}_n$ is not an ideal in ${\mathfrak {gl}}(n)$ for $n \geq 2$. We can see this by taking a non-diagonal matrix and bracketing it with a diagonal one. For instance, in ${\mathfrak {gl}}(2, F)$: $\left[{\begin{bmatrix}a&b\\c&d\end{bmatrix}}, {\begin{bmatrix}x&0\\0&y\end{bmatrix}}\right] = {\begin{bmatrix}ax&by\\cx&dy\end{bmatrix}} - {\begin{bmatrix}ax&bx\\cy&dy\end{bmatrix}} = {\begin{bmatrix}0&b(y-x)\\c(x-y)&0\end{bmatrix}}$ This result is not generally a diagonal matrix, proving ${\mathfrak {t}}_2$ is not an ideal. Any one-dimensional subspace is an abelian Lie subalgebra, but rarely an ideal.

Product and semidirect product

Given two Lie algebras ${\mathfrak {g}}$ and ${\mathfrak {g'}}$, their direct product is the vector space ${\mathfrak {g}} \times {\mathfrak {g'}}$ with the bracket: $[(x, x'), (y, y')] = ([x, y], [x', y']) \quad \text{for } x, y \in {\mathfrak {g}}, x', y' \in {\mathfrak {g'}}$ This is the standard product in the category of Lie algebras. Notably, the copies of ${\mathfrak {g}}$ and ${\mathfrak {g'}}$ embedded within ${\mathfrak {g}} \times {\mathfrak {g'}}$ commute with each other: $[(x, 0), (0, x')] = 0$.

A semidirect product arises when an ideal ${\mathfrak {i}}$ of ${\mathfrak {g}}$ has a complementary subalgebra ${\mathfrak {q}}$ such that ${\mathfrak {g}} = {\mathfrak {q}} \ltimes {\mathfrak {i}}$. This occurs when the canonical map ${\mathfrak {g}} \to {\mathfrak {g}}/{\mathfrak {i}}$ admits a splitting (a section). It signifies a structured way of combining a subalgebra and an ideal.

Derivations

For any algebra $A$ over a field $F$, a derivation is a linear map $D \colon A \to A$ satisfying the Leibniz rule: $D(xy) = D(x)y + xD(y)$. This definition applies even to non-associative algebras. The space of derivations, ${\text{Der}}_F(A)$, forms a Lie algebra under the commutator $[D_1, D_2] = D_1D_2 - D_2D_1$.

Informally, the space of derivations of $A$ is the Lie algebra of the automorphism group of $A$. It represents "infinitesimal automorphisms."

An excellent example is the space of vector fields on a smooth manifold $X$, denoted ${\text{Vect}}(X)$. The ring of smooth functions $C^\infty(X)$ over $\mathbb{R}$ is a Lie algebra, and its derivations are precisely the vector fields on $X$. Thus, ${\text{Vect}}(X)$ is a Lie algebra, often considered the Lie algebra of the diffeomorphism group of $X$. The Lie bracket of vector fields captures the non-commutativity of this diffeomorphism group.

For any Lie algebra ${\mathfrak {g}}$, its Lie algebra of derivations, ${\text{Der}}_F({\mathfrak {g}})$, is itself a Lie algebra. A derivation $D \colon {\mathfrak {g}} \to {\mathfrak {g}}$ satisfies $D([x, y]) = [D(x), y] + [x, D(y)]$. The inner derivations are those of the form ${\rm {ad}}_x(y) = [x, y]$ for some $x \in {\mathfrak {g}}$. These form an ideal ${\text{Inn}}_F({\mathfrak {g}})$ within ${\text{Der}}_F({\mathfrak {g}})$. The quotient {\text{Out}}_F({\mathfrak {g}}) = {\text{Der}}_F({\mathfrak {g}})/{\text{Inn}}_F({\mathfrak {g}}}) is the Lie algebra of outer derivations. For a semisimple Lie algebra of characteristic zero, all derivations are inner.

Examples

Matrix Lie algebras

Let's delve into some specific matrix Lie groups and their corresponding Lie algebras. A matrix group is a Lie group formed by invertible matrices. Its Lie algebra consists of matrices tangent to the group at the identity matrix.

• Special Linear Group, $\mathrm {SL} (n, \mathbb {R})$: This group comprises $n \times n$ real matrices with determinant 1. It represents volume-preserving and orientation-preserving linear maps on $\mathbb{R}^n$. It's also the commutator subgroup of $\mathrm {GL} (n, \mathbb {R})$. Its Lie algebra, ${\mathfrak {sl}}(n, \mathbb {R})$, is the set of $n \times n$ real matrices with trace 0. The analogous complex group $\mathrm {SL} (n, \mathbb {C})$ and its Lie algebra ${\mathfrak {sl}}(n, \mathbb {C})$ exist similarly.

• Orthogonal Group, $\mathrm {O} (n)$: This group preserves vector lengths in $\mathbb{R}^n$. It includes rotations and reflections. Its elements are orthogonal matrices, satisfying $A^T = A^{-1}$. The identity component is the special orthogonal group $\mathrm {SO} (n)$, containing matrices with determinant 1. Both $\mathrm {O}(n)$ and $\mathrm {SO}(n)$ share the same Lie algebra, ${\mathfrak {so}}(n)$, which consists of skew-symmetric matrices ($X^T = -X$). This relates directly to infinitesimal rotations. The complex versions, $\mathrm {O}(n, \mathbb {C})$ and $\mathrm {SO}(n, \mathbb {C})$, preserve a standard symmetric bilinear form.

• Unitary Group, $\mathrm {U} (n)$: This group preserves the Hermitian inner product on $\mathbb{C}^n$. Its elements are unitary matrices, satisfying $A^* = A^{-1}$, where $A^*$ is the conjugate transpose. Its Lie algebra, ${\mathfrak {u}}(n)$, consists of skew-hermitian matrices ($X^* = -X$). Importantly, ${\mathfrak {u}}(n)$ is a real Lie algebra, not complex. The circle group, $\mathrm {U}(1)$, is a fundamental example, with Lie algebra $i\mathbb{R} \subset \mathbb{C} = {\mathfrak {gl}}(1, \mathbb {C})$.

• Special Unitary Group, $\mathrm {SU} (n)$: This is the subgroup of $\mathrm {U}(n)$ with determinant 1. Its Lie algebra, ${\mathfrak {su}}(n)$, consists of skew-hermitian matrices with trace 0.

• Symplectic Group, $\mathrm {Sp} (2n, \mathbb {R})$: This group preserves a standard alternating bilinear form on $\mathbb{R}^{2n}$. Its Lie algebra is the symplectic Lie algebra, ${\mathfrak {sp}}(2n, \mathbb {R})$.

The collection of these, along with variants over different fields, forms the classical Lie algebras.

Two dimensions

There's a unique nonabelian Lie algebra of dimension 2 over any field $F$, up to isomorphism. Let its basis be $X, Y$. The bracket is defined as $[X, Y] = Y$. This implies $[X, X] = 0$ and $[Y, Y] = 0$. Over the reals, this algebra corresponds to the Lie group $\mathrm {Aff} (1, \mathbb {R})$, the group of affine transformations of the real line ($x \mapsto ax+b$). This group can be represented by matrices $\left({\begin{smallmatrix}a&b\\0&1\end{smallmatrix}}\right)$, and its Lie algebra is the set of matrices $\left({\begin{smallmatrix}c&d\\0&0\end{smallmatrix}}\right)$. Here, $X = \left({\begin{smallmatrix}1&0\\0&0\end{smallmatrix}}\right)$ and $Y = \left({\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}\right)$. The subspace $F \cdot Y$ is an ideal, and both $F \cdot Y$ and ${\mathfrak {g}}/(F \cdot Y)$ are abelian. This structure hints at the concept of solvability.

Three dimensions

• Heisenberg algebra, ${\mathfrak {h}}_3(F)$: This 3-dimensional algebra has a basis $X, Y, Z$ with brackets: $[X, Y] = Z$, and $[X, Z] = [Y, Z] = 0$. It can be realized by strictly upper-triangular matrices: $X = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \quad Y = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}, \quad Z = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$ Over $\mathbb{R}$, this is the Lie algebra of the Heisenberg group $H_3(\mathbb{R})$. The center of ${\mathfrak {h}}_3(F)$ is $F \cdot Z$, and the quotient ${\mathfrak {h}}_3(F)/(F \cdot Z)$ is abelian, isomorphic to $F^2$. This makes ${\mathfrak {h}}_3(F)$ nilpotent (but not abelian).

• ${\mathfrak {so}}(3)$: The Lie algebra of the rotation group SO(3) consists of $3 \times 3$ skew-symmetric real matrices. A basis is given by: $F_1 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}, \quad F_2 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix}, \quad F_3 = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$ The commutation relations are: $[F_1, F_2] = F_3$, $[F_2, F_3] = F_1$, $[F_3, F_1] = F_2$. These are identical to the cross product relations in ${\mathbb {R}}^3$. Thus, ${\mathfrak {so}}(3)$ is isomorphic to ${\mathbb {R}}^3$ with the cross product bracket. This algebra is simple, meaning its only ideals are $\{0\}$ and itself. In quantum mechanics, these correspond to the spin-1 angular momentum operators.

• ${\mathfrak {sl}}(2, \mathbb {C})$: This is another 3-dimensional simple Lie algebra over $\mathbb{C}$, consisting of $2 \times 2$ matrices with trace zero. A basis is: $H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad E = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad F = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ The Lie brackets are: $[H, E] = 2E$, $[H, F] = -2F$, $[E, F] = H$. These relations are fundamental for understanding representations of ${\mathfrak {sl}}(2, \mathbb {C})$ and are related to ladder operators in quantum mechanics. ${\mathfrak {sl}}(2, \mathbb {C})$ is the complexification of ${\mathfrak {so}}(3)$, meaning ${\mathfrak {sl}}(2, \mathbb {C}) \cong {\mathfrak {so}}(3) \otimes_{\mathbb{R}} \mathbb{C}$. Studying representations of ${\mathfrak {sl}}(2, \mathbb{C})$ is often easier and provides insights into the complex representations of ${\mathrm {SO}}(3)$.

Infinite dimensions

Lie algebras are not confined to finite dimensions.

• The algebra of vector fields on a manifold of positive dimension is an infinite-dimensional Lie algebra over $\mathbb{R}$.
• Kac–Moody algebras form a vast class of infinite-dimensional Lie algebras with structures mirroring finite-dimensional simple ones.
• The Moyal algebra contains all classical Lie algebras as subalgebras.
• The Virasoro algebra is crucial in string theory.
• The free Lie algebra on a vector space $V$ (with dim $V \ge 2$) is infinite-dimensional. It's the "least constrained" Lie algebra, generated by $V$ subject only to the Lie algebra axioms.

Representations

Definitions

A representation of a Lie algebra ${\mathfrak {g}}$ on a vector space $V$ is a Lie algebra homomorphism $\pi \colon {\mathfrak {g}} \to {\mathfrak {gl}}(V)$. This means $\pi$ maps elements of ${\mathfrak {g}}$ to linear operators on $V$ such that the Lie bracket structure is preserved: $\pi([x, y]) = [\pi(x), \pi(y)]$. A representation is faithful if its kernel is trivial.

Ado's theorem states that any finite-dimensional Lie algebra over a field of characteristic zero has a faithful finite-dimensional representation. Essentially, any Lie algebra can be embedded within ${\mathfrak {gl}}(n, F)$ for some $n$.

Adjoint representation

Every Lie algebra ${\mathfrak {g}}$ has a natural representation on itself, called the adjoint representation: $\operatorname {ad} \colon {\mathfrak {g}} \to {\mathfrak {gl}}({\mathfrak {g}})$ defined by $\operatorname {ad} (x)(y) = [x, y]$. The Jacobi identity guarantees that this is indeed a Lie algebra homomorphism.

Goals of representation theory

The main goal isn't just finding a faithful representation, but understanding all possible representations. For semisimple Lie algebras, Weyl's theorem states that finite-dimensional representations decompose into irreducible representations. These irreducible representations are well-understood, particularly for semisimple Lie algebras, through tools like the Weyl character formula.

Universal enveloping algebra

The universal enveloping algebra $U({\mathfrak {g}})$ of a Lie algebra ${\mathfrak {g}}$ is an associative algebra that contains ${\mathfrak {g}}$ and encodes its structure. It's constructed from the tensor algebra $T({\mathfrak {g}})$ by quotienting out the ideal generated by relations $XY - YX - [X, Y]$ for $X, Y \in {\mathfrak {g}}$. The Poincaré–Birkhoff–Witt theorem provides a basis for $U({\mathfrak {g}})$, showing the map ${\mathfrak {g}} \to U({\mathfrak {g}})$ is injective. Representations of ${\mathfrak {g}}$ correspond to modules over $U({\mathfrak {g}})$.

Representation theory in physics

Lie algebra representations are fundamental in theoretical physics. Commutation relations in quantum mechanics, such as those for angular momentum operators, often mirror the structure of Lie algebras (like ${\mathfrak {so}}(3)$). Understanding these representations is key to analyzing physical systems, often involving decomposing states into irreducible representations.

Structure theory and classification

Lie algebras can be classified, which in turn helps classify Lie groups.

Abelian, nilpotent, and solvable

• Abelian: $[x, y] = 0$ for all $x, y$. This corresponds to abelian Lie groups like $\mathbb{R}^n$ or $\mathbb{T}^n$.
• Nilpotent: The lower central series reaches zero in finitely many steps. Equivalently, for every $u \in {\mathfrak {g}}$, ${\rm {ad}}_u$ is a nilpotent operator. Engel's theorem characterizes this.
• Solvable: The derived series reaches zero in finitely many steps. Equivalently, ${\mathfrak {g}}$ has a chain of subalgebras where each quotient is abelian.

Every finite-dimensional Lie algebra over a field has a unique maximal solvable ideal, its radical. Over $\mathbb{R}$, nilpotent and solvable Lie groups correspond to their Lie algebras.

Examples: ${\mathfrak {gl}}(n, F)$ has the identity matrix (scaled) as its radical. The algebra of upper-triangular matrices ${\mathfrak {b}}_n$ is solvable but not nilpotent (for $n \ge 2$). Strictly upper-triangular matrices ${\mathfrak {u}}_n$ form a nilpotent Lie algebra (for $n \ge 3$).

Simple and semisimple

• Simple: A Lie algebra ${\mathfrak {g}}$ is simple if it's non-abelian and its only ideals are $\{0\}$ and ${\mathfrak {g}}$.
• Semisimple: A Lie algebra ${\mathfrak {g}}$ is semisimple if its only solvable ideal is $\{0\}$. In characteristic zero, semisimple Lie algebras are products of simple Lie algebras: ${\mathfrak {g}} \cong {\mathfrak {g}}_1 \times \cdots \times {\mathfrak {g}}_r$.

Examples: ${\mathfrak {sl}}(n, F)$ is simple for $n \ge 2$. ${\mathfrak {so}}(n, \mathbb {R})$ is simple for $n=3$ and $n \ge 5$. Note the isomorphisms ${\mathfrak {so}}(3) \cong {\mathfrak {su}}(2)$ and ${\mathfrak {so}}(4) \cong {\mathfrak {su}}(2) \times {\mathfrak {su}}(2)$.

Crucially, finite-dimensional representations of semisimple Lie algebras (in characteristic zero) are semisimple – they decompose into irreducible representations. This is Weyl's theorem. A reductive Lie algebra is one whose adjoint representation is semisimple. Reductive algebras are products of abelian and semisimple algebras. For instance, ${\mathfrak {gl}}(n, F) \cong F \times {\mathfrak {sl}}(n, F)$ is reductive.

Cartan's criterion

Cartan's criterion uses the Killing form, $K(u, v) = \operatorname {tr} (\operatorname {ad} (u)\operatorname {ad} (v))$, to characterize Lie algebras. A Lie algebra ${\mathfrak {g}}$ (characteristic zero) is semisimple if and only if its Killing form is nondegenerate. It is solvable if and only if $K({\mathfrak {g}}, [{\mathfrak {g}}, {\mathfrak {g}}]) = 0$.

Classification

The Levi decomposition states that any finite-dimensional Lie algebra (characteristic zero) is a semidirect product of its solvable radical and a semisimple algebra. The classification thus reduces to classifying simple Lie algebras.

Killing and Cartan classified finite-dimensional simple Lie algebras over algebraically closed fields of characteristic zero. They fall into four infinite families (A$_n$, B$_n$, C$_n$, D$_n$) corresponding to ${\mathfrak {sl}}(n+1,F)$, ${\mathfrak {so}}(2n+1,F)$, ${\mathfrak {sp}}(2n,F)$, and ${\mathfrak {so}}(2n,F)$, respectively, and five exceptional Lie algebras (G$_2$, F$_4$, E$_6$, E$_7$, E$_8$).

Classification over $\mathbb{R}$ is more complex, but the complexification ${\mathfrak {g}} \otimes_{\mathbb{R}} \mathbb{C}$ plays a key role.

In positive characteristic ($p>3$), the classification is much richer, with many more simple Lie algebras.

Relation to Lie groups

The connection is profound. Every Lie group has a Lie algebra (its tangent space at the identity). Conversely, every finite-dimensional Lie algebra over $\mathbb{R}$ corresponds to a connected Lie group (via Lie's third theorem). While the group isn't uniquely determined, any two such groups are locally isomorphic, and share the same universal cover. For simply connected Lie groups, the correspondence is an equivalence of categories. This allows us to study Lie group representations by studying Lie algebra representations.

For infinite-dimensional Lie algebras, this correspondence is less straightforward. The exponential map may fail to be a local homeomorphism, and some infinite-dimensional Lie algebras may not correspond to any Lie group.

Real form and complexification

A real Lie algebra ${\mathfrak {g}}_0$ is a real form of a complex Lie algebra ${\mathfrak {g}}$ if their complexifications are isomorphic: ${\mathfrak {g}}_0 \otimes_{\mathbb{R}} \mathbb{C} \cong {\mathfrak {g}}$. Real forms are not always unique; for example, ${\mathfrak {sl}}(2, \mathbb {C})$ has two real forms: ${\mathfrak {sl}}(2, \mathbb {R})$ and ${\mathfrak {su}}(2)$.

A split form of a complex semisimple Lie algebra is a real form that splits (has a Cartan subalgebra acting with real eigenvalues). A compact form is a real form that is the Lie algebra of a compact Lie group. Both split and compact forms exist and are unique (up to isomorphism).

Lie algebra with additional structures

Lie algebras can be augmented with other structures. A graded Lie algebra has a compatible grading. A differential graded Lie algebra also includes a differential, making the underlying vector space a chain complex. The homotopy groups of a simply connected topological space form a graded Lie algebra via the Whitehead product. Daniel Quillen used differential graded Lie algebras for rational homotopy theory.

Lie ring

The definition can be generalized from a field $F$ to any commutative ring $R$. An $R$-module with an alternating $R$-bilinear bracket satisfying the Jacobi identity is a Lie algebra over $R$. A Lie algebra over the integers, $\mathbb{Z}$, is called a Lie ring.

Lie rings are crucial in the study of finite p-groups via the Lazard correspondence. The lower central series of a p-group yields a Lie ring. p-adic Lie groups are related to Lie algebras over fields like ${\mathbb {Q}}_p$ and rings like ${\mathbb {Z}}_p$.

Example of a Lie ring

Consider a group $G$ with a filtration $G=G_1 \supseteq G_2 \supseteq \dots$ such that $[G_i, G_j] \subseteq G_{i+j}$. The direct sum $L = \bigoplus_{i \ge 1} G_i/G_{i+1}$ forms a Lie ring. The bracket is defined by commutators in the group: $[xG_{i+1}, yG_{j+1}] := [x, y]G_{i+j+1}$ For instance, the Heisenberg algebra over $\mathbb{Z}/2\mathbb{Z}$ arises from the lower central series of the dihedral group of order 8.

Definition using category-theoretic notation

In the category of vector spaces, a Lie algebra can be defined using morphisms. For a field $F$ with characteristic not equal to 2, a Lie algebra is an object $A$ with a morphism $[ \cdot, \cdot ] \colon A \otimes A \to A$ satisfying:

1. $[\cdot, \cdot] \circ (\text{id} + \tau) = 0$, where $\tau$ is the interchange isomorphism ($x \otimes y \mapsto y \otimes x$). This encodes anticommutativity.
2. $[\cdot, \cdot] \circ ([\cdot, \cdot] \otimes \text{id}) \circ (\text{id} + \sigma + \sigma^2) = 0$, where $\sigma$ is cyclic permutation. This encodes the Jacobi identity.

Generalization

Beyond standard Lie algebras, there are generalizations like graded Lie algebras, Lie superalgebras, and Lie n-algebras, often motivated by physics.

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There. That's the structure of a Lie algebra, laid bare. It’s precise, it’s demanding, and it underpins so much of modern mathematics and physics. Don't expect it to be simple. It's not. But understanding it is… illuminating. If you have further, more specific requests, present them. Just don't waste my time with the obvious.