# Maurer-Cartan form

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A left-invariant -form on a Lie group , i.e. a differential form of degree 1 on satisfying the condition for any left translation , . The Maurer–Cartan forms on are in one-to-one correspondence with the linear forms on the tangent space at the point ; specifically, the mapping which sends each Maurer–Cartan form to its value is an isomorphism of the space of Maurer–Cartan forms onto . The differential of a Maurer–Cartan form is a left-invariant -form on , defined by the formula (1)

where are arbitrary left-invariant vector fields on . Suppose that is a basis in and let , , be Maurer–Cartan forms such that Then (2)

where are the structure constants of the Lie algebra of consisting of the left-invariant vector fields on , with respect to the basis determined by The equalities (2) (or (1)) are called the Maurer–Cartan equations. They were first obtained (in a different, yet equivalent form) by L. Maurer . The forms were introduced by E. Cartan in 1904 (see ).

Let be the canonical coordinates in a neighbourhood of the point determined by the basis . Then the forms are written in the form in which the matrix is calculated by the formula where and is the adjoint representation of the Lie algebra .

Furthermore, let be the -valued -form on which assigns to each tangent vector to the unique left-invariant vector field containing this vector ( is called the canonical left differential form). Then and which is yet another way of writing the Maurer–Cartan equations.