In mathematics, the Jacobson–Morozov theorem is the assertion that

nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...

elements in a semi-simple

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...

can be extended to sl₂-triples. The theorem is named after , .

Statement

The statement of Jacobson–Morozov relies on the following preliminary notions: an sl₂-triple in a semi-simple Lie algebra

\mathfrak g

(throughout in this article, over a field of

characteristic zero In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...

) is a

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...

of Lie algebras

\mathfrak_2 \to \mathfrak g

. Equivalently, it is a triple

e, f, h

of elements in

\mathfrak g

satisfying the relations :

,e = 2e, \quad,f = -2f, \quad,f = h.

An element

x \in \mathfrak g

is called nilpotent, if the

endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a ...

: \mathfrak g \to \mathfrak g

(known as the

adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL( ...

) is a

nilpotent endomorphism In linear algebra, a nilpotent matrix is a square matrix ''N'' such that :N^k = 0\, for some positive integer k. The smallest such k is called the index of N, sometimes the degree of N. More generally, a nilpotent transformation is a linear transf ...

. It is an elementary fact that for any sl₂-triple

(e, f, h)

, ''e'' must be nilpotent. The Jacobson–Morozov theorem states that, conversely, any nilpotent non-zero element

e \in \mathfrak g

can be extended to an sl₂-triple. For

\mathfrak g = \mathfrak_n

, the sl₂-triples obtained in this way are made explicit in . The theorem can also be stated for

linear algebraic group In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n ...

s (again over a field ''k'' of characteristic zero): any morphism (of algebraic groups) from the

additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structure ...

G_a

to a

reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...

''H'' factors through the embedding :

G_a \to SL_2, x \mapsto \left ( \begin 1 & x \\ 0 & 1 \end \right ).

Furthermore, any two such factorizations :

SL_2 \to H

are conjugate by a ''k''-point of ''H''.

Generalization

A far-reaching generalization of the theorem as formulated above can be stated as follows: the inclusion of pro-reductive groups into all linear algebraic groups, where morphisms

G \to H

in both categories are taken up to conjugation by elements in

H(k)

, admits a

left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...

, the so-called pro-reductive envelope. This left adjoint sends the additive group

G_a

SL_2

(which happens to be semi-simple, as opposed to pro-reductive), thereby recovering the above form of Jacobson–Morozov. This generalized Jacobson–Morozov theorem was proven by by appealing to methods related to Tannakian categories and by by more geometric methods.

References

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