Unipotent

Unipotent is an adjective used in the context of linear algebraic groups and related representation theory. In a linear algebraic group G defined over a field k, an element g in GL(V) is unipotent if g − I is nilpotent; equivalently, all eigenvalues of g are equal to 1. In terms of Jordan form, a unipotent matrix is similar to a Jordan decomposition with 1s on the diagonal. For a subgroup of GL(V), the same criterion applies to its elements when realized in the defining representation.

Examples: In GL_n(k), unipotent elements are precisely those matrices with 1s on the diagonal; the subgroup UT_n(k) of unitriangular matrices is a classic unipotent subgroup. Unipotent subgroups occur as unipotent radicals of parabolic subgroups, and in reductive groups many unipotent elements lie in Borel subgroups.

Properties: If char(k) = 0, a unipotent element has infinite order unless it is the identity. In characteristic p > 0, unipotent elements have order a power of p. Over algebraically closed fields, the set of unipotent elements is closed and stable under conjugation; each unipotent element is contained in some Borel subgroup. In Lie theory, unipotent elements can be written as exp(N) for nilpotent N in the Lie algebra on appropriate domains, linking them to nilpotent orbits and the Jacobson–Morozov theory.

Structure: The unipotent radical Ru(G) of a connected solvable linear algebraic group G is the largest connected normal unipotent subgroup; in a reductive group, Ru(G) is trivial. The distribution and geometry of unipotent elements are central to the Bruhat decomposition and the representation theory of G.

Finite groups of Lie type: when defined over a finite field, unipotent elements are those that become unipotent in the algebraic group over the algebraic closure; their orders are powers of the defining characteristic p and they form a major component of conjugacy class structure.