associated unipotent groups. For example, on , the central series are the matrix groups , , , and given some induced examples of unipotent groups. Gan The additive group is a unipotent group through the embedding Notice the matrix multiplication gives hence this is a group embedding. More generally, there is an embedding from the map Using scheme theory, is given by the functor where Kernel of the Frobenius Consider the functor on the subcategory , there is the subfunctor where so it is given by the kernel of the Frobenius endomorphism. Classification of unipotent groups over characteristic 0 Over characteristic 0 there is a nice classification of unipotent algebraic groups with respect to nilpotent Lie algebras. Recall that a nilpotent Lie algebra is a subalgebra of some such that the iterated adjoint action eventually terminates to the zero-map. In terms of matrices, this means it is a subalgebra of , the matrices with for . Then, there is an equivalence of categories of finite-dimensional nilpotent Lie algebras and unipotent algebraic groups.page 261 This can be constructed using the Baker–Campbell–Hausdorff series , where given a finite-dimensional nilpotent Lie algebra, the map gives a Unipotent algebraic group structure on . In the other direction the exponential map takes any nilpotent square matrix to a unipotent matrix. Moreover, if U is a commutative unipotent group, the exponential map induces an isomorphism from the Lie algebra of U to U itself. Remarks Unipotent groups over an algebraically closed field of any given dimension can in principle be classified, but in practice the complexity of the classification increases very rapidly with the dimension, so people tend to give up somewhere around dimension 6. Unipotent radical The unipotent radical of an algebraic group G is the set of unipotent elements in the radical of G. It is a connected unipotent normal subgroup of G, and contains all other such subgroups. A group is called reductive if its unipotent radical is trivial. If G is reductive then its radical is a torus. Decomposition of algebraic groups Algebraic groups can be decomposed into unipotent groups, multiplicative groups, and abelian varieties, but the statement of how they decompose depends upon the characteristic of their base field. Characteristic 0 Over characteristic 0 there is a nice decomposition theorem of an algebraic group relating its structure to the structure of a linear algebraic group and an Abelian variety. There is