Lie algebra representations

A Lie algebra is a vector space equipped with an alternating bilinear bracket satisfying the Jacobi identity. Matrices supply an example with bracket AB−BA. A representation is a linear map from the abstract Lie algebra to endomorphisms of V that sends the abstract bracket to this operator commutator. Linearity matters as much as the bracket law: mapping each named generator to a matrix without extending consistently to linear combinations does not define a representation. One checks generator relations only when the stated presentation really generates all relations needed.

Commuting generators describe mutually compatible infinitesimal directions, whereas nonzero brackets explain why applying two small transformations in opposite orders differs. The Jacobi identity ensures those interactions fit together. It does not mean every nested commutator vanishes. For example, a diagonal matrix can act on an off-diagonal matrix by rescaling its commutator. These algebraic relations constrain spectra and connect eigenvectors into ladders. Their power comes from exact relations rather than a visual resemblance between arrows in a diagram and imagined quantum particles.

The complex Lie algebra sl₂ consists of two-by-two complex matrices of trace zero. A convenient basis has a diagonal element H and raising and lowering elements E and F. Their commutators satisfy [H,E]=2E, [H,F]=−2F, and [E,F]=H. Because these elements form a basis and the bracket is bilinear, checking those relations verifies the standard defining representation. These are algebra generators acting on a two-dimensional space, not three mutually independent spatial directions with the same geometric interpretation.

If a vector v has H-eigenvalue m, the bracket relation gives H(Ev)=(m+2)Ev whenever Ev is nonzero. Likewise F lowers the eigenvalue by two. In the defining representation, the two weights are 1 and −1; trying to raise beyond the upper vector gives zero. In general finite-dimensional irreducible complex sl₂ representations, a nonnegative integer highest weight determines the familiar finite ladder. Infinite-dimensional modules admit different behavior, so finiteness and irreducibility are substantive assumptions in this classification.

Differentiating a smooth finite-dimensional representation of a Lie group gives a Lie algebra representation. Conversely, in the finite-dimensional setting a Lie algebra representation integrates to the connected simply connected Lie group with that algebra. Descending to another group with the same algebra may fail if a covering kernel acts nontrivially. Spinorial representations illustrate this distinction between rotation-related groups. Thus the commutation relations do not erase global topology. A statement about a representation of an algebra should not quietly become a statement about every group sharing that algebra.

Infinite-dimensional representations introduce another boundary. Operators may be unbounded, products may have different domains, and formal commutators need a common invariant domain before they become operator identities. Exponentiation to unitary groups requires analytic hypotheses beyond multiplying symbols. The two-level lab offers a finite complex state with mixing and phase controls, useful for observing that basis probabilities need not determine a state. It is not an sl₂ module classifier or an integration theorem; keeping those distinctions explicit prevents a familiar picture from replacing the actual mathematics.