Lie algebra

Intuition [Beginner]

A Lie algebra is the infinitesimal version of a continuous group of symmetries. Take the rotations of three-dimensional space: they form a smooth surface of moves you can chain together, but the chain order matters — rotating about $x$ then about $y$ is not the same as rotating about $y$ then about $x$. The amount by which the order matters, examined infinitesimally near the identity rotation, is the Lie bracket.

Two infinitesimal rotations don't add and undo neatly: their commutator records what fails to cancel. The Lie algebra is the vector space of infinitesimal moves equipped with this bracket, and the bracket records all the rotation group's geometry up to first order around the identity.

The same picture works whenever a smooth space of transformations is built around an identity — rotations, translations on a curved manifold, gauge transformations in physics. The Lie algebra is the linear ledger of how those transformations refuse to commute.

Visual [Beginner]

Two arrows meeting at a point, each representing an infinitesimal flow. Following one then the other doesn't return to the same place as following them in the reverse order; the small gap between the two end-points is the bracket.

That third arrow is also an infinitesimal flow. The bracket lives inside the same linear space the original arrows do; that's what makes a Lie algebra a closed algebraic object.

Worked example [Beginner]

Take the three coordinate-axis rotations of three-dimensional space. Pick infinitesimal generators $L_x$, $L_y$, $L_z$ — small rotations about each axis. They live in a three-dimensional vector space.

Compute the bracket $[L_x,L_y]$. Operationally: rotate slightly about $x$, then slightly about $y$, then back about $x$, then back about $y$. To first order in the angles, the result is a tiny rotation. About which axis? About $z$.

The same computation gives

$$[L_x,L_y]=L_z,[L_y,L_z]=L_x,[L_z,L_x]=L_y.$$

These are the structure constants of the Lie algebra of rotations, $\mathfrak{so}(3)$. They tell you everything about how 3D rotations chain together infinitesimally — and through the exponential map, everything about the global rotation group too.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $K$ be a field. A Lie algebra over $K$ is a vector space $\mathfrak{g}$ over $K$ 01.01.03 equipped with a bilinear map

$$[\cdot ,\cdot ]:\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}$$

called the Lie bracket, satisfying:

1. Antisymmetry: $[X,X]=0$ for all $X\in \mathfrak{g}$.
2. Jacobi identity: $[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0$ for all $X,Y,Z\in \mathfrak{g}$.

When $\mathrm{char}K\ne 2$, antisymmetry is equivalent to the more familiar relation $[X,Y]=-[Y,X]$: substituting $X+Y$ for $X$ in $[X,X]=0$ and expanding gives $[X,Y]+[Y,X]=0$.

A Lie subalgebra of $\mathfrak{g}$ is a vector subspace $\mathfrak{h}\subseteq \mathfrak{g}$ closed under the bracket. An ideal is a subspace $\mathfrak{i}\subseteq \mathfrak{g}$ with $[\mathfrak{g},\mathfrak{i}]\subseteq \mathfrak{i}$. A homomorphism $\phi :\mathfrak{g}\to \mathfrak{h}$ of Lie algebras is a $K$-linear map preserving the bracket: $\phi ([X,Y])=[\phi (X),\phi (Y)]$.

The adjoint representation sends each $X\in \mathfrak{g}$ to the linear map

$${\mathrm{ad}}_X:\mathfrak{g}\to \mathfrak{g},{\mathrm{ad}}_X(Y)=[X,Y].$$

The Jacobi identity is precisely the statement that $\mathrm{ad}$ is a Lie-algebra homomorphism from $\mathfrak{g}$ to $\mathfrak{gl}(\mathfrak{g})$ — the bracket on the latter being the commutator of linear maps ^{[quantum-well Lie algebras.md]}.

Key theorem with proof [Intermediate+]

Theorem (commutator on an associative algebra). Let $A$ be an associative $K$-algebra. Define $[a,b]:=ab-ba$. Then $A$, viewed as a vector space with this bracket, is a Lie algebra.

Proof. Bilinearity and antisymmetry are immediate from distributivity and the definition. We verify the Jacobi identity. Expand each cyclic term:

$$[a,[b,c]]=a(bc-cb)-(bc-cb)a=abc-acb-bca+cba.$$

By cyclic permutation $a\to b\to c\to a$,

$$[b,[c,a]]=bca-bac-cab+acb,$$ $$[c,[a,b]]=cab-cba-abc+bac.$$

Add the three expressions. Each term on the right cancels with another:

• $abc$ in the first cancels $-abc$ in the third.
• $-acb$ in the first cancels $acb$ in the second.
• $-bca$ in the first cancels $bca$ in the second.
• $cba$ in the first cancels $-cba$ in the third.
• $-bac$ in the second cancels $bac$ in the third.
• $-cab$ in the second cancels $cab$ in the third.

So the sum is zero, which is the Jacobi identity. $\square$

Corollary. ${\mathfrak{gl}}_n(K):=M_n(K)$ with $[A,B]=AB-BA$ is a Lie algebra. Every Lie subalgebra of ${\mathfrak{gl}}_n(K)$ — for example ${\mathfrak{sl}}_n$ (traceless matrices), ${\mathfrak{so}}_n$ (skew-symmetric matrices), ${\mathfrak{u}}_n$ (skew-Hermitian matrices) — inherits the structure.

The corollary is the source of every Lie algebra arising in classical geometry. Ado's theorem (deeper, not proved here) says the converse: every finite-dimensional Lie algebra over a field of characteristic zero embeds into some ${\mathfrak{gl}}_n(K)$.

Bridge. The construction here builds toward 03.06.05 (invariant polynomial on a lie algebra), where the same data is upgraded, and the symmetry side is taken up in 03.06.06 (chern-weil homomorphism). The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has LieAlgebra as a typeclass and proves the basic structural results. The companion module records the conventions used in this unit.

[object Promise]

Mathlib uses ⁅·, ·⁆ for the bracket and LieAlgebra for the structure. The companion module re-exports the definitions for downstream Codex units.

Advanced results [Master]

A Lie algebra is simple if it is nonabelian and has no proper nonzero ideals. Semisimple means the radical (largest solvable ideal) is zero; equivalently, $\mathfrak{g}$ is a direct sum of simple Lie algebras.

The finite-dimensional simple Lie algebras over $\mathbb{C}$ are classified by Killing and Cartan into four infinite families (the classical types $A_n,B_n,C_n,D_n$ corresponding to ${\mathfrak{sl}}_{n+1},{\mathfrak{so}}_{2n+1},{\mathfrak{sp}}_{2n},{\mathfrak{so}}_{2n}$) plus five exceptional cases ($G_2,F_4,E_6,E_7,E_8$). The classification proceeds via the Killing form

$$B(X,Y):=\mathrm{tr}({\mathrm{ad}}_X{\mathrm{ad}}_Y),$$

a symmetric invariant bilinear form whose nondegeneracy characterises semisimplicity (Cartan's criterion).

Every finite-dimensional Lie algebra $\mathfrak{g}$ over a field of characteristic zero has a unique maximal solvable ideal (the radical), and Levi's theorem gives a canonical decomposition $\mathfrak{g}=\mathrm{rad}(\mathfrak{g})\oplus \mathfrak{s}$ with $\mathfrak{s}$ semisimple — the Levi subalgebra. Solvable Lie algebras over $\mathbb{C}$ are upper-triangularisable in any finite-dimensional representation (Lie's theorem), generalising the classical result that a single nilpotent matrix can be put in Jordan form.

The exponential map $\exp :\mathfrak{g}\to G$ for a Lie group $G$ realises the Lie algebra as the tangent space at the identity, and the Baker-Campbell-Hausdorff formula expresses $\exp (X)\exp (Y)$ as $\exp$ of a series in iterated brackets. The Lie bracket is therefore the first nontrivial term in the failure of the group product to be commutative.

Infinite-dimensional Lie algebras play a role too. The Witt algebra $\mathfrak{W}$ of holomorphic vector fields on the punctured disk, and its central extension the Virasoro algebra 03.11.03, are infinite-dimensional Lie algebras whose representation theory underlies two-dimensional conformal field theory 03.10.02. Affine Lie algebras (Kac-Moody algebras) extend the classical classification into the infinite-dimensional setting.

Synthesis. This construction generalises the pattern fixed in 01.01.03 (vector space), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

Antisymmetry from $[X,X]=0$. Expand $0=[X+Y,X+Y]$ using bilinearity:

$$0=[X,X]+[X,Y]+[Y,X]+[Y,Y]=[X,Y]+[Y,X].$$

So $[X,Y]=-[Y,X]$. The argument needs $2\ne 0$ in $K$ to invert no factor; the implication holds in any characteristic for this direction.

Adjoint is a homomorphism. Compute, for $Z\in \mathfrak{g}$,

$${\mathrm{ad}}_{[X,Y]}(Z)=[[X,Y],Z].$$

The Jacobi identity says

$$[[X,Y],Z]+[[Y,Z],X]+[[Z,X],Y]=0,$$

equivalently

$$[[X,Y],Z]=[X,[Y,Z]]-[Y,[X,Z]].$$

The right-hand side is ${\mathrm{ad}}_X({\mathrm{ad}}_{Y}Z)-{\mathrm{ad}}_Y({\mathrm{ad}}_{X}Z)=[{\mathrm{ad}}_X,{\mathrm{ad}}_Y](Z)$. Both sides agree for all $Z$, so ${\mathrm{ad}}_{[X,Y]}=[{\mathrm{ad}}_X,{\mathrm{ad}}_Y]$.

Killing form is invariant. For $X,Y,Z\in \mathfrak{g}$,

$$B([X,Y],Z)=\mathrm{tr}({\mathrm{ad}}_{[X,Y]}{\mathrm{ad}}_Z)=\mathrm{tr}([{\mathrm{ad}}_X,{\mathrm{ad}}_Y]{\mathrm{ad}}_Z).$$

Using the trace identity $\mathrm{tr}([A,B]C)=\mathrm{tr}(A[B,C])$,

$$\mathrm{tr}([{\mathrm{ad}}_X,{\mathrm{ad}}_Y]{\mathrm{ad}}_Z)=\mathrm{tr}({\mathrm{ad}}_X[{\mathrm{ad}}_Y,{\mathrm{ad}}_Z])=\mathrm{tr}({\mathrm{ad}}_X{\mathrm{ad}}_{[Y,Z]})=B(X,[Y,Z]).$$

So $B([X,Y],Z)=B(X,[Y,Z])$, the invariance of the Killing form under the adjoint action.

Subalgebra of ${\mathfrak{gl}}_n$ generated by skew-symmetric matrices is closed. If $A^T=-A$ and $B^T=-B$, then $(AB{)}^T=B^T{A}^T=(-B)(-A)=BA$, so $[A,B{]}^T=(AB-BA{)}^T=BA-AB=-[A,B]$. The commutator of two skew-symmetric matrices is skew-symmetric, so ${\mathfrak{so}}_n$ is a Lie subalgebra of ${\mathfrak{gl}}_n$.

Connections [Master]

• Vector space 01.01.03 — the underlying linear structure on which the bracket lives.

• Invariant polynomial on a Lie algebra 03.06.05 — adjoint-invariant polynomials on $\mathfrak{g}$ are the input to Chern-Weil theory.

• Chern-Weil homomorphism 03.06.06 — turns invariant polynomials into characteristic classes via curvature.

• Yang-Mills action 03.07.05 — gauge potentials are $\mathrm{ad}P$-valued forms; field strength is the curvature 2-form valued in $\mathfrak{g}$.

• Spin group 03.09.03 — the Lie algebra $\mathfrak{spin}(n)$ is isomorphic to $\mathfrak{so}(n)$, and rotations live as exponentials of skew-symmetric matrices.

• Central extension of a Lie algebra 03.11.01 — modifies the bracket by a 2-cocycle, producing the Heisenberg, Virasoro, and affine Kac-Moody algebras.

• Virasoro algebra 03.11.03 — the central extension of the Witt algebra; the symmetry algebra of two-dimensional CFT.

Historical & philosophical context [Master]

Sophus Lie introduced what we now call Lie groups in the 1870s as continuous symmetry groups of differential equations. His infinitesimal method linearised these groups at the identity and studied the resulting bracket structure. Killing classified the simple complex Lie algebras in 1888–1890; Cartan corrected and completed the classification in his 1894 thesis, fixing the structural mistakes Killing had made and identifying the five exceptional algebras ^{[Humphreys §1]}.

The modern axiomatic definition — vector space with a bilinear, antisymmetric, Jacobi-respecting bracket — was crystallised by Hermann Weyl and others in the 1920s, separating the algebraic content from the original analytic motivation. Bourbaki's treatment (1960s) made the abstraction definitive ^{[Serre Part I]}.

In physics, Lie algebras describe infinitesimal symmetries of every kind: gauge symmetries (Yang-Mills theory uses $\mathfrak{su}(2),\mathfrak{su}(3),\mathfrak{u}(1)$), spacetime symmetries (Poincaré algebra), and conformal symmetries (Virasoro algebra in two dimensions). Noether's theorem translates each Lie-algebra generator of a symmetry into a conserved current of the corresponding field theory.

Bibliography [Master]

• Humphreys, J. E., Introduction to Lie Algebras and Representation Theory, Springer GTM 9, 1972.
• Fulton, W. & Harris, J., Representation Theory: A First Course, Springer GTM 129, 1991.
• Serre, J.-P., Lie Algebras and Lie Groups, Springer Lecture Notes in Mathematics, 1965.
• Bourbaki, N., Lie Groups and Lie Algebras, Chapters 1–3, Springer, 1989.
• Knapp, A. W., Lie Groups Beyond an Introduction, Birkhäuser, 2nd ed., 2002.

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Wave 2 Phase 2.3 unit #1. Foundation for invariant polynomial, Chern-Weil, Yang-Mills, and the central-extension / Virasoro chain.