sl(2) triples and weight strings

An sl(2) triple consists of e,h,f satisfying [h,e]=2e, [h,f]=−2f, and [e,f]=h. These are represented by the usual off-diagonal matrix units and diag(1,−1) in sl(2,C). In any representation, h measures weights while e raises them by two and f lowers them by two. The number two belongs to the chosen relation; replacing h or the operators arbitrarily changes the formulas. A quantum angular-momentum basis often introduces different scale factors and normalized vectors, which should not be mixed silently with this algebraic convention.

The Jacobson–Morozov theorem states that a nonzero adjoint-nilpotent element of a finite-dimensional complex semisimple Lie algebra can be included as e in such a triple. This is a structural theorem, not something established merely by finding one nilpotent matrix in a representation. Once a triple is available, restricting a finite-dimensional representation to it gives sl(2) modules and their strings. This reduction is one reason rank-one calculations control root geometry and larger highest-weight constructions.

In a finite-dimensional irreducible sl(2,C) module, choose a nonzero highest vector v₀ with ev₀=0 and hv₀=mv₀. Then m is a nonnegative integer, the module has dimension m+1, and vⱼ=fʲv₀ for j from zero through m is a basis. Here these vectors are not divided by factorials and not normalized by an inner product. The raising coefficient is j(m−j+1), while lowering simply takes vⱼ to vⱼ₊₁ until the final zero. This convention makes direct commutator checks straightforward.

The coefficient formula follows by commuting e through a power of f and using [h,f]=−2f. Finite dimensionality forces the string to stop, and the first vanishing coefficient identifies the nonnegative integer m. If one instead permits an infinite-dimensional highest-weight module, a general complex highest weight is possible and repeated lowering need not terminate. It is therefore wrong to read dimension m+1 from a highest-weight symbol before checking the module category and the quotient being considered.

Every finite-dimensional complex sl(2) module is a direct sum of these irreducible strings. Several strings can overlap at a weight, so a weight's multiplicity need not be one in a reducible module. A highest eigenvalue and a list of distinct weights do not by themselves recover the entire module unless multiplicities and decomposition information are supplied. In a semisimple algebra of higher rank, the sl(2) subalgebra associated with a root imposes corresponding integral-string constraints on restricted weights.

The compact real form su(2) and its unitary representations give a compatible geometric picture after complexification, but an algebraic weight is not a probability. For the standard two-dimensional module, h has weights 1 and −1. A normalized superposition of its two basis vectors has amplitudes and measurement probabilities, while the representation's weight labels remain fixed. A continuous change of a state vector does not create a new finite-dimensional irreducible highest weight. This distinction keeps the qubit visualization useful without replacing the algebraic classification by a state-space picture.