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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?

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    $\begingroup$ Intuitively speaking, any element of a monoid infinitesimally close to the identity should be invertible (e.g. the group of units of any Banach algebra is open), so the obvious analogue of a Lie algebra here should still be a Lie algebra. $\endgroup$ Dec 28, 2013 at 3:05
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    $\begingroup$ So that prompts the question: is there some analogue of the exponential map in this context? Also: does this mean that the tangent spaces at other monoid elements are "different" to the algebra? I guess that this is technically manifested as the tangent bundle not necessarily being trivial, as it is in a lie group? $\endgroup$
    – Benjamin
    Dec 28, 2013 at 4:08
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    $\begingroup$ @QiaochuYuan, Actually there are many Lie algebras. To every idempotent e of a semigroup S there is the monoid eSe with identity e. In the Lie or algebraic settings eSe will again be Lie or algebraic. It has its own group of units and that group has a Lie algebra. My guess is there may be a lie algebroid lurking about. $\endgroup$ Dec 28, 2013 at 16:52
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    $\begingroup$ One example of Lie monoid is that of invertible real matrices with positive entries. It's open in $GL_n$ so should have the same Lie algebra, and the exponential map cannot be defined everywhere. $\endgroup$
    – YCor
    Jun 5, 2014 at 21:18
  • $\begingroup$ There is a theory of smooth Moufang loops. Loops are groups without the nonassociative law and a Moufang loop is a loop satisfying the Moufang identities, a weakened associative law. These have an analogue of a Lie algebra called Malcev algebras (Wikipedia even calls them Lie-Moufang algebras). And likewise with the analogue to Lie algebras, it turns that the Moufang loop structure can be recovered from the Malcev structure. $\endgroup$ Feb 13, 2021 at 23:58

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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.

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  • $\begingroup$ Thanks for that. Are there any key words or good resources I should know? $\endgroup$
    – Benjamin
    Dec 28, 2013 at 0:14
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    $\begingroup$ @Benjamin: What Putcha and Renner do is mostly in the spirit of linear algebraic groups rather than Lie groups, so I'm unsure how a close analogy with Lie groups would fit here. Maybe it's partly a question of what characteristic the underlying field has. $\endgroup$ Dec 28, 2013 at 18:06
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    $\begingroup$ I'm actually only interested in the case of matrices as I have a specific application in mind. Work analogous to linear algebraic groups would be ideal. $\endgroup$
    – Benjamin
    Dec 28, 2013 at 19:57
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    $\begingroup$ @YCor, usually people study the irreducible case. $\endgroup$ Mar 8 at 11:16
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    $\begingroup$ The irreducible component of the unit of course is not helpful if the unit is isolated as you suggest. Also connected and irreducible are different for monoids. I think people do look at connected but not irreducible monoids but the theory is not as good. $\endgroup$ Mar 8 at 11:28
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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.

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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.

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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'.

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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).

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    $\begingroup$ This isn't an answer to the question. It describes a notion of Lie algebra (as an algebra over the Lie operad; in particular, this definition is by no means due to Aguiar and Mahajan), not a notion of an analogue of Lie groups but where invertibility is dropped. $\endgroup$ Jan 2, 2016 at 7:17

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