Structure constants

In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting product is bilinear and can be uniquely extended to all vectors in the vector space, thus uniquely determining the product for the algebra. In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting product is bilinear and can be uniquely extended to all vectors in the vector space, thus uniquely determining the product for the algebra. Structure constants are used whenever an explicit form for the algebra must be given. 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 or commutator. Given a set of basis vectors { e i } {displaystyle {mathbf {e} _{i}}} for the underlying vector space of the algebra, the structure constants or structure coefficients c i j k {displaystyle c_{ij}^{;k}} express the multiplication ⋅ {displaystyle cdot } of pairs of vectors as a linear combination: 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 a Lie algebra, the basis vectors are termed the generators of the algebra, and the product is given by the Lie bracket. That is, the algebra product ⋅ {displaystyle cdot } is defined to be the Lie bracket: for two vectors A {displaystyle A} and B {displaystyle B} in the algebra, the product is A ⋅ B ≡ [ A , B ] . {displaystyle Acdot Bequiv .} In particular, the algebra product ⋅ {displaystyle cdot } must not be confused with a matrix product, and thus sometimes requires an alternate notation. There is no particular need to distinguish the upper and lower indices in this case; they can be written all up or all down. In physics, it is common to use the notation T i {displaystyle T_{i}} for the generators, and f a b c {displaystyle f_{ab}^{;;c}} or f a b c {displaystyle f^{abc}} (ignoring the upper-lower distinction) for the structure constants. The Lie bracket of pairs of generators is a linear combination of generators from the set, i.e. By linear extension, the structure constants completely determine the Lie brackets of all elements of the Lie algebra. All Lie algebras satisfy the Jacobi identity. For the basis vectors, it can be written as