Unipotency

Unipotency is a property used in linear algebra, group theory, and algebraic geometry to describe a close relationship to the identity transformation. In a linear representation, an element g of GL(V) is unipotent if g − I is a nilpotent operator. Equivalently, all eigenvalues of g are equal to 1, and in a suitable basis g is a Jordan matrix with ones on the diagonal. In GL_n over a field, these conditions are equivalent; a typical form is g = I + N with N nilpotent.

Examples include the strictly upper triangular matrices with ones on the diagonal, which form a canonical unipotent subgroup of GL_n. More generally, the upper triangular matrices with ones on the diagonal constitute a standard unipotent subgroup.

In the context of algebraic groups, a unipotent element of a linear algebraic group G is one that becomes unipotent in every rational finite-dimensional representation; equivalently, g lies in a connected unipotent algebraic subgroup of G. Unipotent elements generate unipotent subgroups, and the unipotent radical of a parabolic subgroup is the maximal connected unipotent normal subgroup.

In Lie theory, over fields of characteristic zero, unipotent elements are closely related to nilpotent elements of the Lie algebra via the exponential map. In positive characteristic, the correspondence is more delicate and requires care with the exponential map.

Unipotent conjugacy classes in reductive groups are classified by combinatorial data, such as partitions for GL_n, and provide important structural information about representations and geometric structure.