Cartan's lemma

In mathematics, Cartan's lemma refers to a number of results named after either Élie Cartan or his son Henri Cartan:

• In exterior algebra:^[1] Suppose that v₁, ..., v_p are linearly independent elements of a vector space V and w₁, ..., w_p are such that

$v_1\wedge w_1 + \cdots + v_p\wedge w_p = 0$

in ΛV. Then there are scalars h_ij = h_ji such that

$w_i = \sum_{j=1}^p h_{ij}v_j.$

• In several complex variables:^[2] Let a₁ < a₂ < a₃ < a₄ and b₁ < b₂ and define rectangles in the complex plane C by

$\begin{align} K_1 &= \{ z_1=x_1+iy_1 | a_2 < x_1 < a_3, b_1 < y_1 < b_2\} \\ K_1' &= \{ z_1=x_1+iy_1 | a_1 < x_1 < a_3, b_1 < y_1 < b_2\} \\ K_1'' &= \{ z_1=x_1+iy_1 | a_2 < x_1 < a_4, b_1 < y_1 < b_2\} \end{align}$

so that $K_1 = K_1'\cap K_1''$. Let K₂, ..., K_n be simply connected domains in C and let

$\begin{align} K &= K_1\times K_2\times\cdots \times K_n\\ K' &= K_1'\times K_2\times\cdots \times K_n\\ K'' &= K_1''\times K_2\times\cdots \times K_n \end{align}$

so that again $K = K'\cap K''$. Suppose that F(z) is a complex analytic matrix-valued function on a rectangle K in Cⁿ such that F(z) is an invertible matrix for each z in K. Then there exist analytic functions $F'\,$ in $K'\,$ and $F''\,$ in $K''\,$ such that

$F(z) = F'(z)F''(z)\,$

in K.

• In potential theory, a result that estimates the Hausdorff measure of the set on which a logarithmic Newtonian potential is small. See Cartan's lemma (potential theory).

References

1. • Sternberg, S. (1983). Lectures on Differential Geometry ((2nd ed.) ed.). New York: Chelsea Publishing Co. p. 18. ISBN 0-8218-1385-4. OCLC 43032711.
2. Robert C. Gunning and Hugo Rossi (1965). Analytic Functions of Several Complex Variables. Prentice-Hall. p. 199.