Structure constants
Because the product operation in the algebra is bilinear, by linearity knowing the product of basis vectors allows to compute the product of any elements (just like a matrix allows to compute the action of the linear operator on any vector by providing the action of the operator on basis vectors).
Given the structure constants, the resulting product is obtained by bilinearity and can be uniquely extended to all vectors in the vector space, thus uniquely determining the product for the algebra.
Thus, they are frequently used when discussing Lie algebras in physics, as the basis vectors indicate specific directions in physical space, or correspond to specific particles (recall that Lie algebras are algebras over a field, with the bilinear product being given by the Lie bracket, usually defined via the commutator).
for the underlying vector space of the algebra, the product operation is uniquely defined by the products of basis vectors: The structure constants or structure coefficients
The upper and lower indices are frequently not distinguished, unless the algebra is endowed with some other structure that would require this (for example, a pseudo-Riemannian metric, on the algebra of the indefinite orthogonal group so(p,q)).
That is, structure constants are often written with all-upper, or all-lower indexes.
The distinction between upper and lower is then a convention, reminding the reader that lower indices behave like the components of a dual vector, i.e. are covariant under a change of basis, while upper indices are contravariant.
For Lie algebras, one frequently used convention for the basis is in terms of the ladder operators defined by the Cartan subalgebra; this is presented further down in the article, after some preliminary examples.
For a Lie algebra, the basis vectors are termed the generators of the algebra, and the product rather called the Lie bracket (often the Lie bracket is an additional product operation beyond the already existing product, thus necessitating a separate name).
For the basis vectors, it can be written as and this leads directly to a corresponding identity in terms of the structure constants: The above, and the remainder of this article, make use of the Einstein summation convention for repeated indexes.
The structure constants play a role in Lie algebra representations, and in fact, give exactly the matrix elements of the adjoint representation.
The structure constants often make an appearance in the approximation to the Baker–Campbell–Hausdorff formula for the product of two elements of a Lie group.
of the special unitary group SU(2) is three-dimensional, with generators given by the Pauli matrices
The generators of the group SU(2) satisfy the commutation relations (where
Note that the constant 2i can be absorbed into the definition of the basis vectors; thus, defining
This brings the structure constants into line with those of the rotation group SO(3).
That is, the commutator for the angular momentum operators are then commonly written as
are written so as to obey the right hand rule for rotations in 3-dimensional space.
The difference of the factor of 2i between these two sets of structure constants can be infuriating, as it involves some subtlety.
Thus, for example, the two-dimensional complex vector space can be given a real structure.
A less trivial example is given by SU(3):[2] Its generators, T, in the defining representation, are: where
, the Gell-Mann matrices, are the SU(3) analog of the Pauli matrices for SU(2): These obey the relations The structure constants are totally antisymmetric.
The d take the values: For the general case of 𝔰𝔲(N), there exists closed formula to obtain the structure constant, without having to compute commutation and anti-commutation relations between the generators.
All the non-zero totally anti-symmetric structure constants are All the non-zero totally symmetric structure constants are For more details on the derivation see [3] and.
In addition to the product, the coproduct and the antipode of a Hopf algebra can be expressed in terms of structure constants.
The connecting axiom, which defines a consistency condition on the Hopf algebra, can be expressed as a relation between these various structure constants.
One conventional approach to providing a basis for a Lie algebra is by means of the so-called "ladder operators" appearing as eigenvectors of the Cartan subalgebra.
The construction of this basis, using conventional notation, is quickly sketched here.
are determined only up to overall scale; one conventional normalization is to set This allows the remaining commutation relations to be written as and with this last subject to the condition that the roots (defined below)
In addition, they are antisymmetric: and can always be chosen such that They also obey cocycle conditions:[7] whenever
