Lie groups vs Lie monoids

Question

Does there exist a well developed theory of a class of objects which might rightfully be called Lie monoids? By this I mean with axioms similar to those of Lie groups, but with the axiomatic existence of inverses dropped. If so what is the analogous structure to lie algebras if one exists?

Answer 1

There is a well developed theory of algebraic monoids, due principally to Putcha and Renner. I think Lie semigroups is less well developed but there is work by Hoffmann, Lawson and the thesis of Langlands was on this subject.

Answer 2

See the following:

• MR1317811 Hilgert, Joachim; Neeb, Karl-Hermann Lie semigroups and their applications. Lecture Notes in Mathematics, 1552. Springer-Verlag, Berlin, 1993. xii+315 pp. ISBN: 3-540-56954-5 (Reviewer: Gestur Ólafsson)

• MR1179336 Neeb, Karl-Hermann On the foundations of Lie semigroups. J. Reine Angew. Math. 431 (1992), 165–189. (Reviewer: Jimmie D. Lawson)

• MR1235759 Mittenhuber, Dirk; Neeb, Karl-Hermann Remarks on our paper: "On the exponential function of an invariant Lie semigroup'' [Sem. Sophus Lie 2 (1992), no. 1, 21–30; MR1188629 (93j:22007)]. Sem. Sophus Lie 3 (1993), no. 1, 119–120.

Moreover, completions of infinite dimensional Lie groups (like diffeomorphism groups) with respect to right invariant Riemannian metrics tend to be semigroups, for an easy example see 4.8 of here.

Answer 3

Anders Kock mentions Lie monoids and some of their properties in his book on synthetic differential geometry. Basically, in SDG, a Lie monoid is a microlinear monoid object. It is easy to show that the tangent space at the identity of a Lie monoid is an $R$-Lie algebra, defined in the same way as for a Lie group, and isomorphic to the left- or right-invariant vector fields on the monoid (depending on your bracket convention). We also still have a Lie functor, taking Lie monoids to their Lie algebras, and Lie monoid homomorphisms to Lie algebra homomorphisms.

For example, for any microlinear space $M$ the mapping space $M^M$ is a Lie monoid, with the composition of maps as the multiplication. Then the Lie algebra of $M^M$ is the space $\frak{X}$$(M)$ of vector fields on $M$. Since each infinitesimal transformation is invertible, it follows that this is also the Lie algebra of $\text{Diff}(M)$.

One thing to note is that since we do not have an inversion map on a Lie monoid, then I do not see how would have a canonical isomorphism of Lie algebras between left- and right-invariant vector fields on the monoid. Usually the isomorphism would be given by pushing forward a left- or right-invariant vector field by the inversion map.

Answer 4

The basic theory is elaborated in Hilgert, Hofmann & Lawson's Lie Groups, Convex Semicones and Semigroups.

The analagous structure to a Lie algebra is what they call a Lie wedge. This being a wedge (a closed convex cone) $W$ that is invariant under the ad action. That is:

$e^{ad(v)}.W = W$ for all $v \in HW$.

Here $HW$ is the edge of the wedge and defined as $W \cap -W$. Lawson also defines subtangent vectors and these form a wedge and the tangent wedge to any local semigroup is a Lie wedge. It turns out that the converse is true but takes quite a bit of work. Two easy special cases are for strictly positive wedges and split wedges which are always Lie wedges.

Hilgert & Neeb's Lie Semigroups and applications focuses on a certain class of Lie semigroups, those that are a closed subsemigroup of a Lie group generated by the images of all one parameter semigroups lying in the ambient subsemigroup.

They choose this definition, they say:

"to avoid the complications arising from the semigroup analogues of dense winds on tori".

Presumably irrational tori, though I think I prefer their more colourful language.

You may also be interested in a different weakening of a Lie group. Loops are groups where the associative law has been weakened. The analogue to a Lie algebra for a Lie-Moufang loop is a Lie-Moufang or Mal'cev algebra. This also is an anti-symmetric non-associative algebra satisfying:

$(uv)(uw) = ((uv)w)u + ((vw)u)u + ((wu)u)v$

Not quite as symmetrical as the Jacobi law, but there you go. Both $\mathbb{O}^{\times}$ and $S(\mathbb{O})$, the set of invertible and unit octonions respectively, form Moufang loops, so they turn up naturally. Notably octonions turn up in classifying supergravities coupled to super Yang-Mills and also in defining the Rosenfield projective planes which give a systematic geometric method in constructing the exceptional Lie groups upto a fully determined interpretation of 'systematic'.

Answer 5

In "Monoidal functors, species and Hopf algebras", by M. Aguiar and S. Mahajan (CRM Monograph Series, 29), they defined a Lie monoid (definition 1.25) in the following sense:

Let $(\mathsf{C}, \bullet, \beta)$ be a $\mathbb{K}$-linear symmetric monoidal category (possibly without unit). A Lie monoid in $(\mathsf{C}, \bullet, \beta)$ is a pair $(L, \gamma)$, where $$\gamma: L \bullet L \to L$$ satisfies $$\gamma+ \gamma \circ \beta_{L,L}=0 \qquad \text{and} \qquad \gamma \circ (\gamma \bullet \text{id})\circ (\text{id}+\xi + \xi^2)=0,$$ where $\xi$ denotes the composite $$L \bullet L \bullet L\xrightarrow{\ \text{id}\,\bullet\,\beta_{L,L} \ }L\bullet L \bullet L \xrightarrow{\ \beta_{L,L}\,\bullet \,\text{id}\ }L \bullet L \bullet L.$$

Here "$\mathbb{K}$-linear" means that each set $\text{Hom}(A, B)$ of morphism in $\mathsf{C}$ is a $\mathbb{K}$-module and composition of arrows $$\text{Hom}(A,B)\times \text{Hom}(B,C)\to \text{Hom}(A,C)$$ is $\mathbb{K}$-bilinear. The application $\beta$ is a natural isomorphism where for each $A, B \in \text{Obj}(\mathsf{C})$, the isomorphism $$\beta_{A,B}: A \bullet B \to B \bullet A$$ satisfies $\beta_{A,B}\circ \beta_{B,A}=\text{id}$ and two axioms of compatibility with $\bullet$ (see definition 1.2).