Group versus Lie ring

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Topologists, algebraists and geometers know of the close relation and correspondence between a Lie group and its Lie algebra. The close relation between these can be studied in a more general context: how are the theory of groups and the theory of Lie rings related? We explore this question from many different angles.

Definitions

Group

Lie ring

As varieties of algebras

Both are varieties of algebras

As described above, a group is a set with three binary operations, satisfying some universal identities. Thus, groups form a variety of algebras. In a similar way, a Lie ring is described by four binary operations (the group operations, and the Lie bracket) satisfying certain universal identities. Thus, Lie rings form a variety of algebras.

An interesting question is: how similar are these varieties? We describe some similarities and differences.

Both varieties are ideal-determined

Further information: variety of groups is ideal-determined, variety of Lie rings is ideal-determined

For groups, every normal subgroup is the kernel of a homomorphism, and a surjective homomorphism is completely determined by its kernel, which is a normal subgroup. Thus, there is a bijection between the normal subgroups of a group, and the surjective homomorphisms emanating from the group (which can in turn be identified with congruences on the group).

For Lie rings, every ideal is the kernel of a homomorphism, and a surjective homomorphism of Lie rings is completely determined by its kernel, which is an ideal. Thus, there is a bijection between the ideals of a Lie ring, and the surjective homomorphisms emanating from it (which can in turn be identified with congruences on the Lie ring).

In the language of universal algebra, the variety of groups, as well as the variety of Lie rings, are ideal-determined varieties. An ideal is a subset that is closed under certain kinds of terms called ideal terms.

Being ideal-determined does not strongly distinguish these varieties from many others. For instance, the variety of commutative unital rings, the variety of unital rings, the variety of algebras over a field, are all ideal-determined.

In both varieties, ideals are subalgebras

Further information: ideals are subalgebras in the variety of groups, ideals are subalgebras in the variety of Lie rings

For groups, ideals are the same as normal subgroups, which are certainly subgroups. For Lie rings, any ideal is a Lie subring.

This means, for instance, that we can talk of an ideal of an ideal. We can talk of a normal subgroup of a normal subgroup, or an ideal of an ideal in a Lie ring.

Ideals are not subalgebras in most of the other commonly encountered varieties, such as the variety of commutative unital rings (where no proper ideal is a unital subring) or the variety of unital rings.

In groups, characteristic subalgebras are ideals, but not in Lie rings

Further information: characteristic subalgebras are ideals in the variety of groups

Any characteristic subgroup of a group is necessarily a normal subgroup. Thus, in the variety of groups, characteristic subalgebras are normal. This is important because it tells us that every subgroup-defining function yields a normal subgroup, and hence we can study the associated quotient.

The equivalent fails to hold for Lie rings. A characteristic Lie subring of a Lie ring need not be an ideal. Nonetheless, it is true in many of the situations we are interested in (see below regarding the relation between automorphisms and derivations). In practice, most of the ways we obtain characteristic subrings in a Lie ring, yield ideals: for instance, the center of a Lie ring. Nonetheless, we need to prove in each case that we do end up with an ideal.

Automorphisms and derivations

For groups, only automorphisms

For a group, there is no notion of derivation, and only the notion of automorphism makes sense. The automorphisms of a group themselves form a group, called the automorphism group.

Moreover, the group elements themselves give automorphisms by their conjugation action. This gives a homomorphism from the group to its automorphism group, and the elements in the image are the inner automorphisms. The inner automorphism group forms a normal subgroup inside the automorphism group.

For Lie rings, both automorphisms and derivations

For a Lie ring, we have two relevant notions: the automorphisms of the Lie ring, that form a group, and the derivations of the Lie ring, that themselves form a Lie ring.

Moreover, parallel to the way that a group acts on itself by conjugation, elements of a Lie ring themselves give rise to derivations, by their left adjoint action. Derivations arising in this manner are termed inner derivations, and the inner derivations form an ideal inside the Lie ring of all derivations.

For a Lie ring, both the automorphisms and the derivations matter. The derivations create a theory parallel to the theory of automorphisms for a group, but automorphisms are important as well. Thus, we have a mix of notions, like:

Relation between automorphisms and derivations for Lie groups

In the geometric situation of a Lie group and its Lie algebra, the (smooth) automorphisms and derivations are linked by the exponential map: given any derivation, we can exponentiate it to obtain an automorphism, and given any automorphism, we can differentiate it to obtain a derivation. There are actually three things we're studying:

  1. The smooth automorphisms of the Lie group
  2. The automorphisms of the Lie algebra
  3. The derivations of the Lie algebra

The relations are as follows:

  1. (1) to (2): There is a natural (injective, if the Lie group is connected, and surjective if it is simply connected) homomorphism from the automorphisms of the Lie group to the automorphisms of the Lie algebra, obtained by studying the effect on the tangent space at the identity.
  2. (3) to (2): Given any derivation of the Lie algebra, the exponential of that derivation gives an automorphism of the Lie algebra.
  3. Composite map from (3) to (1): Starting with a derivation of the Lie algebra, we can exponentiate it to an automorphism of the Lie algebra, and then, under suitable conditions, we might find an automorphism of the Lie group corresponding to it. In the particular case where we start with an inner derivation by an element x of the Lie algebra, the automorphism of the Lie group that we end up with, is the inner automorphism by the exponential of the element x.

We can thus talk of inner automorphism of a Lie algebra as an automorphism that comes about by exponentiating an inner derivation. Then, it turns out that:

  1. The ideals are the same as the subalgebras invariant under inner automorphisms
  2. The derivation-invariant ideals are the same as the characteristic subalgebras

All these break down for arbitrary Lie rings because there is in general no way to relate the derivations and automorphisms.

Direct and semidirect products

Direct products

We can define direct products of groups, both as internal direct product and external direct product.

We can define direct product of Lie rings in an internal and external sense.

Semidirect products

Interestingly, there are notions of semidirect product, both for groups, and for Lie rings. In the case of groups, a semidirect product is based on a map from one group, to the automorphism group of another. In the case of Lie rings, a semidirect product is based on a map from one Lie ring, to the Lie ring of derivations of the other. (We again see derivations playing the role parallel to automorphisms).

Analogous ideals, quotients, ascending and descending series

Given a notion in groups, it is not clear how one could rigorously define the analogous notion in Lie rings. A reasonable definition of analogousness should be such that in the specific cases of a correspondence between Lie groups and their Lie algebras, the two notions do match up, and that the definitions in general look reasonably similar. Some examples of analogies are described.

Center, lower and upper central series

Commutator, derived series

Induced notions of nilpotent, solvable, perfect