Lie Groups and Lie Algebras via Multi-Valued Topological Groups

1. Introduction

Lie theory establishes a remarkably strong correspondence between smooth groups and infinitesimal algebraic objects. For a Lie group G, the tangent space at the identity,

$$\mathfrak{g}=T_{e}G,$$

carries a canonical Lie bracket, and the resulting Lie algebra controls a substantial part of the local structure of G. The exponential map, invariant vector fields, the adjoint representation, and the Baker–Campbell–Hausdorff formula form the basic mechanisms connecting the global group with its infinitesimal structure; see, for example, [2,6,8].

A different direction begins when the product of two elements is allowed to have more than one possible value. Instead of a mapping

$$G\times G⟶G,$$

one considers a hyperoperation

$$★:G\times G⟶{\mathcal{P}}^{*}\left(G\right),$$

where ${\mathcal{P}}^{*}\left(G\right)$ is the set of all nonempty subsets of G. This point of view goes back to the hypergroup theory initiated by Marty [7]; systematic developments can be found in Corsini’s work and in Corsini–Leoreanu [3,4]. The multi-valued viewpoint has subsequently developed in several directions, including algebraic hypergroups, topological hypergroups, and measure-theoretic hypergroups; see also [1,5].

The purpose of this paper is to investigate the following question:

When does a smooth multi-valued topological group possess a Lie algebra, and how is the resulting Lie algebra related to the different branches of the multi-valued multiplication?

This question is not merely a routine extension of the ordinary theory. If

$$x★y=\{z_1,z_2\},$$

then there is no canonical product $xy$. Consequently, the expression

$$d{m}_{(e,e)}(X,Y)$$

has no immediate meaning because the multiplication m is not an ordinary function. Moreover, arbitrary set-valued continuity does not imply differentiability.

The main idea of this paper is therefore to distinguish three levels of structure.

(i)

The topological level, where the hyperoperation is continuous as a set-valued map.

(ii)

The smooth branch level, where locally the possible products can be represented by smooth maps.

(iii)

The infinitesimal-collapse level, where all local branches have the same first-order behavior at the identity.

At the second level each branch gives a local Lie group structure. At the third level the different local Lie structures have the same tangent Lie algebra.

The resulting theory can therefore be regarded as a bridge:

$$\mathrm{multi}\text{-}\mathrm{valued}\mathrm{topological}\mathrm{group}⟶\mathrm{smooth}\mathrm{branches}⟶\mathrm{local}\mathrm{Lie}\mathrm{groups}⟶\mathrm{Lie}\mathrm{algebras}.$$

The paper is organized as follows. Section 2 recalls the necessary facts concerning hyperoperations and set-valued topology. Section 3 introduces multi-valued topological groups. Section 4 develops smooth and locally branchable multi-valued groups. Section 5 constructs the associated local Lie groups. Section 6 establishes the infinitesimal Lie algebra. Section 7 studies quotient-induced multi-valued groups. Section 8 discusses the exponential map and a multi-valued Baker–Campbell–Hausdorff expansion. Section 9 contains examples and counterexamples. Section 10 discusses homomorphisms and representations. Section 11 studies the relation with ordinary Lie groups. Section 12 gives further structural results and open problems.

2. Preliminaries on Hyperoperations

Let X be a nonempty set. We write

$${\mathcal{P}}^{*}\left(X\right)=\{A\subseteq X:A\ne ⌀\}.$$

Definition 1.

A hyperoperation on X is a mapping

$$★:X\times X⟶{\mathcal{P}}^{*}\left(X\right).$$

For $x,y\in X$, the set $★(x,y)$ will also be denoted by

$$x★y.$$

Thus an ordinary binary operation is precisely a hyperoperation satisfying

$$|x★y|=1$$

for every $x,y\in X$.

For subsets $A,B\subseteq X$, define

$$A★B=\bigcup _{a\in A,b\in B}(a★b).$$

Definition 2.

A hyperoperation ★ on X is calledassociativeif

$$(x★y)★z=x★(y★z)$$

for all $x,y,z\in X$.

The concept is one of the fundamental starting points of hyperstructure theory [3,4,7].

Definition 3.

A hypergroupoid $(X,★)$ is called a multi-valued group if there exists an element $e\in X$ such that

$$e★x=x★e=\left\{x\right\}$$

for every $x\in X$, the hyperoperation is associative, and every $x\in X$ has an inverse $x^{-1}\in X$ satisfying

$$e\in x★x^{-1}\cap x^{-1}★x.$$

Remark 1.

There are several non-equivalent notions of hypergroup in the literature. Algebraic hypergroups in the sense of Marty and topological/measure-theoretic hypergroups should not be conflated. The present paper uses a set-valued algebraic operation together with an explicitly specified topology and smooth structure. This makes the terminology “multi-valued topological group” precise for the purposes of Lie theory.

3. Topological Multi-Valued Groups

Let X be a topological space. We equip ${\mathcal{P}}^{*}\left(X\right)$ with a hyperspace topology. For compact-valued operations the Vietoris topology is particularly natural.

If $\mathcal{U}$ is a family of open subsets of X, the basic Vietoris neighborhoods of a nonempty compact set K are of the form

$$\langle U_1,\dots ,U_n\rangle =\left\{K^{\prime }\in \mathcal{K}\left(X\right):K^{\prime }\subseteq U_1\cup \dots \cup U_n,K^{\prime }\cap U_i\ne ⌀\mathrm{for}\mathrm{all}i\right\},$$

where $\mathcal{K}\left(X\right)$ denotes the family of nonempty compact subsets of X.

Definition 4.

Let X be a topological space and let

$$★:X\times X⟶\mathcal{K}\left(X\right)$$

be a compact-valued hyperoperation. We call $(X,★)$ a topological multi-valued group if:

(i)

$(X,★)$ is a multi-valued group;

(ii)

the map

$$★:X\times X⟶\mathcal{K}\left(X\right)$$

is continuous for the Vietoris topology;

(iii)

the inverse map $x\mapsto x^{-1}$ is continuous.

Proposition 1.

Every topological group is a topological multi-valued group.

Proof. 

Let G be a topological group with multiplication $m:G\times G\to G$. Define

$$x★y=\left\{xy\right\}.$$

The singleton embedding

$$G⟶\mathcal{K}\left(G\right),x⟼\left\{x\right\},$$

is continuous. Hence the ordinary multiplication induces a continuous compact-valued map. All group axioms are inherited immediately.    □

Remark 2.

The converse is false in general. A continuous set-valued multiplication may have genuinely multi-valued fibers, and there is no reason for it to admit a continuous single-valued selection. This is one of the main differences between ordinary topological groups and multi-valued topological groups.

4. Smooth Multi-Valued Topological Groups

Let M be a smooth manifold. A set-valued map

$$F:M⟶\mathcal{K}\left(M\right)$$

is not, by itself, differentiable. We therefore introduce a local branch concept.

Definition 5.

Let G be a smooth manifold equipped with a multi-valued operation ★. A map

$$s:U\times U⟶G,$$

where U is an open subset of G, is called a local branch of ★ if

$$s(x,y)\in x★y$$

for all $x,y\in U$.

Definition 6.

A multi-valued topological group $(G,★)$ is called smoothly locally branchable if for every $(x,y)\in G\times G$ there exist neighborhoods U of x, V of y, finitely many smooth maps

$$s_1,\dots ,s_k:U\times V⟶G$$

such that

$$u★v=\{s_1(u,v),\dots ,s_k(u,v)\}$$

for every $(u,v)\in U\times V$.

This definition captures finite local ambiguity. The case of infinitely many branches requires additional hypotheses and is discussed later.

Definition 7.

A smoothly locally branchable multi-valued group $(G,★)$ is called identity-branchable if there exists a neighborhood U of e and a smooth branch

$$m:U\times U⟶G$$

such that

$$m(e,x)=m(x,e)=x$$

whenever both sides are defined.

The following observation is fundamental.

Lemma 1.

Let $(G,★)$ be identity-branchable and let m be an identity branch. If

$$m\left(m\right(x,y),z)=m(x,m(y,z\left)\right)$$

whenever all expressions are defined, then m defines a local group operation on a sufficiently small neighborhood of e.

Proof. 

Associativity is assumed. The identity condition follows from

$$m(e,x)=m(x,e)=x.$$

By continuity, after shrinking the neighborhood if necessary, $m(x,y)$ remains in the neighborhood. If the inverse operation has a smooth local branch j, then

$$m(x,j(x\left)\right)=m\left(j\right(x),x)=e.$$

Thus all local group axioms hold.    □

Definition 8.

An identity branch m is called a Lie branch if it is smooth, associative, admits a smooth local inverse, and satisfies the local identity axiom.

Theorem 1.

Let $(G,★)$ be a smoothly locally branchable multi-valued topological group. Suppose that an identity branch m is a Lie branch. Then, after restricting to a sufficiently small neighborhood U of e, $(U,m)$ is a local Lie group.

Proof. 

The manifold structure of G gives a smooth structure on U. The branch m is smooth by assumption, associative, and has identity e. The local inverse is smooth. Therefore $(U,m)$ satisfies the definition of a local Lie group.    □

This elementary theorem is the first bridge between multi-valued groups and Lie groups.

5. The Lie Algebra of a Multi-Valued Group

Let m be a Lie branch on a neighborhood U of e. The usual construction of invariant vector fields can now be performed with respect to m.

For $x\in U$, define the left translation

$$L_x\left(y\right)=m(x,y).$$

Similarly,

$$R_x\left(y\right)=m(y,x).$$

Proposition 2.

For a Lie branch m, the differential

$$d{L}_x\left(e\right):T_{e}G⟶T_{x}G$$

is an isomorphism for x sufficiently close to e.

Proof. 

Since $L_x$ is a local diffeomorphism with local inverse $L_{x^{-1}}$, the inverse function theorem gives that $d{L}_x\left(e\right)$ is an isomorphism.    □

Let $\mathfrak{g}=T_{e}G$. For $X\in \mathfrak{g}$, define the left invariant vector field

$$X^L\left(x\right)=d{L}_x\left(e\right)X.$$

Theorem 2.

The vector space

$$\mathfrak{g}=T_{e}G$$

associated with a Lie branch carries a natural Lie algebra structure defined by

$$[X,Y]=[X^L,Y^L]\left(e\right).$$

Proof. 

The proof is the standard proof for a local Lie group. Since left translations are local diffeomorphisms, left invariant vector fields are closed under the ordinary vector-field commutator:

$$[X^L,Y^L]$$

is again left invariant. Hence there exists a unique element $[X,Y]\in T_{e}G$ such that

$$[X^L,Y^L]\left(e\right)=[X,Y].$$

Bilinearity and antisymmetry follow from the corresponding properties of vector-field commutators. The Jacobi identity follows from the Jacobi identity for vector fields:

$$[X^L,[Y^L,Z^L]]+[Y^L,[Z^L,X^L]]+[Z^L,[X^L,Y^L]]=0.$$

Evaluating at e proves

$$[X,[Y,Z\left]\right]+[Y,[Z,X\left]\right]+[Z,[X,Y\left]\right]=0.$$

Thus $\mathfrak{g}$ is a Lie algebra.    □

The important issue is whether different branches give different Lie algebras. We next formulate a condition that prevents this.

6. Infinitesimal Collapse

Suppose that

$$x★y=\{m_1(x,y),\dots ,m_k(x,y)\}$$

near $(e,e)$, where each $m_i$ is smooth.

Definition 9.

The multi-valued group $(G,★)$ satisfies the first-order infinitesimal-collapse condition at e if

$$d{m}_{i_{(e,e)}}=d{m}_{j_{(e,e)}}$$

for every pair of local branches $m_i,m_j$.

This says that the multi-valued ambiguity disappears to first order.

Theorem 3

(Branch independence). Let $(G,★)$ be smoothly locally branchable. Assume that all identity branches are Lie branches and that the first-order infinitesimal-collapse condition holds at e. Then all identity branches induce the same tangent Lie algebra on $T_{e}G$.

Proof. 

Let $m_1$ and $m_2$ be two identity branches. Their differentials at $(e,e)$ agree by hypothesis:

$$d{m}_{1_{(e,e)}}=d{m}_{2_{(e,e)}}.$$

In local coordinates centered at e, the multiplication maps have the expansions

$$m_i(x,y)=x+y+B_i(x,y)+{O(\parallel (x,y)\parallel }^3),$$

where the bilinear term $B_i$ determines the infinitesimal commutator and hence the Lie bracket. Equality of the first-order differentials together with the compatibility of the local associative structures implies equality of the induced invariant first-order commutator. Therefore

$${[X,Y]}_{m_1}={[X,Y]}_{m_2}$$

for all $X,Y\in T_{e}G$. Hence the identity map of $T_{e}G$ is a Lie algebra isomorphism between the two tangent Lie algebras.    □

Remark 3.

For a stronger and more transparent sufficient condition one may require that

$$m_i(x,y)-m_j(x,y)={O(\parallel (x,y)-(e,e)\parallel }^2)$$

in a common coordinate chart. Then all branches agree to first order.

7. Local Coordinate Form and the Lie Bracket

Choose coordinates around e identifying a neighborhood of e with a neighborhood of $0\in {\mathbb{R}}^n$. A smooth branch can be written as

$$m(x,y)=x+y+B(x,y)+O\left(3\right),$$

where B is a smooth function with

$$B(0,0)=0.$$

For a local Lie group, the antisymmetric part of the quadratic term determines the Lie bracket.

Theorem 4.

Let m be a smooth Lie branch with local expression

$$m(x,y)=x+y+{\displaystyle \frac{1}{2}}B(x,y)+O\left(3\right),$$

where B is bilinear. Then the Lie bracket is represented by

$$[X,Y]=B(X,Y)-B(Y,X).$$

Proof. 

Consider the commutator curve

$$c\left(t\right)=exp\left(tX\right)exp\left(tY\right)exp(-tX)exp(-tY).$$

The standard local calculation gives

$$c\left(t\right)=e+t^2[X,Y]+O\left(t^3\right).$$

Using the coordinate expansion of m, the order-$t^2$ contribution is precisely

$$B(X,Y)-B(Y,X).$$

Therefore the tangent vector of the commutator is the stated bracket.    □

Corollary 1.

If two local branches $m_1,m_2$ have the same antisymmetric quadratic term, then they induce the same Lie bracket on $T_{e}G$.

Proof. 

Apply the theorem to each branch.    □

8. Multi-Valued Exponential Maps

Let $\mathfrak{g}$ be the Lie algebra associated with a Lie branch.

The ordinary exponential map is obtained from the one-parameter subgroup equation

$${\gamma }_X^{\prime }\left(t\right)=X^L\left({\gamma }_X\left(t\right)\right),{\gamma }_X\left(0\right)=e.$$

We define the multi-valued exponential by allowing all compatible branches.

Definition 10.

Let $(G,★)$ be a smoothly branchable multi-valued group. The multi-valued exponential set of $X\in \mathfrak{g}$ is

$${Exp}_{★}\left(X\right)=\left\{\gamma \left(1\right):{\gamma }^{\prime }\left(t\right)=X_{\alpha \left(t\right)}^L\left(\gamma \left(t\right)\right),\gamma \left(0\right)=e\right\},$$

where α ranges over admissible local branch choices.

If all branches coincide, this set is a singleton.

Proposition 3.

Suppose that all local branches have the same left invariant vector field associated with X. Then

$${Exp}_{★}\left(X\right)=\{exp\left(X\right)\}.$$

Proof. 

Every admissible integral curve solves the same ordinary differential equation with the same initial condition. Uniqueness of solutions therefore implies that all admissible curves coincide.    □

Remark 4.

Thus genuine multi-valuedness may survive globally even when the infinitesimal exponential is single-valued.

9. A Multi-Valued Baker–Campbell–Hausdorff Principle

For an ordinary Lie group, the Baker–Campbell–Hausdorff formula describes the local product in exponential coordinates [2,6]:

$$log(expXexpY)=X+Y+{\displaystyle \frac{1}{2}}[X,Y]+{\displaystyle \frac{1}{12}}[X,[X,Y]]+{\displaystyle \frac{1}{12}}[Y,[Y,X]]+\dots .$$

For a branch-compatible multi-valued group, every smooth branch $m_{\alpha }$ has a corresponding logarithmic product.

Definition 11.

Let $\mathfrak{g}$ be the tangent Lie algebra of a smoothly branchable multi-valued group. For every local branch $m_{\alpha }$ define

$${BCH}_{\alpha }(X,Y)=log\left(m_{\alpha }(expX,expY)\right)$$

whenever the logarithm is defined.

Theorem 5.

Suppose the multi-valued group is branch-compatible and all branches have the same tangent Lie algebra $\mathfrak{g}$. Then each local branch satisfies

$${BCH}_{\alpha }(X,Y)=X+Y+{\displaystyle \frac{1}{2}}[X,Y]+{\displaystyle \frac{1}{12}}[X,[X,Y]]+{\displaystyle \frac{1}{12}}[Y,[Y,X]]+O\left(4\right).$$

In particular, the terms of degree at most three are independent of the branch.

Proof. 

Each branch defines a local Lie group with tangent Lie algebra $\mathfrak{g}$. The BCH formula for that local Lie group gives

$${BCH}_{\alpha }(X,Y)=X+Y+{\displaystyle \frac{1}{2}}[X,Y]+{\displaystyle \frac{1}{12}}[X,[X,Y]]+{\displaystyle \frac{1}{12}}[Y,[Y,X]]+O\left(4\right).$$

Branch independence of the Lie bracket implies that all displayed terms coincide.    □

Remark 5.

This theorem gives a precise meaning to the statement that multi-valuedness may be a higher-order or global phenomenon while the infinitesimal geometry remains classical.

10. Quotient-Induced Multi-Valued Groups

We now construct an important class of examples.

Let G be an ordinary group and let $N◃G$ be a normal subgroup. Consider the quotient map

$$q:G⟶G/N.$$

Suppose that $S\subseteq G$ is a transversal for N, so every $g\in G$ has a unique representation

$$g=sn,s\in S,n\in N.$$

If one forgets the choice of representative, multiplication becomes naturally multi-valued.

For $x,y\in G/N$, define

$$x★y=\{q\left(s_x{t}_{y}n\right):n\in N\},$$

where $s_x,s_y$ are representatives.

A cleaner construction uses a quotient by a family of possible representatives.

Definition 12.

Let G be a group and let $\mathcal{A}\left(x\right)$ be a nonempty family of representatives of an equivalence class x. Define

$$x★y=\{ab:a\in \mathcal{A}(x),b\in \mathcal{A}(y\left)\right\}/\sim ,$$

where ∼ identifies elements belonging to the same prescribed equivalence class.

The most useful smooth version is obtained from a covering group.

Theorem 6.

Let $\tilde{G}$ be a Lie group and let $\mathsf{\Gamma }\subseteq Z\left(\tilde{G}\right)$ be a discrete central subgroup. Put

$$G=\tilde{G}/\mathsf{\Gamma }.$$

Suppose a local section

$$s:U⟶\tilde{G}$$

of the quotient map exists on a neighborhood U of the identity. Then the quotient construction induces a local multi-valued product whose every local branch is a Lie group multiplication and whose tangent Lie algebra is canonically

$$Lie\left(G\right)\cong Lie\left(\tilde{G}\right).$$

Proof. 

Since $\mathsf{\Gamma }$ is discrete and central, the quotient map

$$q:\tilde{G}\to G$$

is a local diffeomorphism. Choose a local section s near the identity. For $x,y$ sufficiently close to e, every possible lift of x has the form

$$\gamma s\left(x\right),\gamma \in \mathsf{\Gamma }.$$

Likewise for y. Their products are

$${\gamma }_{1}s\left(x\right){\gamma }_{2}s\left(y\right)={\gamma }_1{\gamma }_{2}s\left(x\right)s\left(y\right),$$

because $\mathsf{\Gamma }$ is central. Hence all branches differ only by discrete central factors. Their differentials at the identity therefore agree. Since q is a local diffeomorphism,

$$d{q}_e:T_e\tilde{G}\to T_{e}G$$

is an isomorphism. Compatibility of left invariant vector fields then identifies the Lie algebras.    □

11. Example: Ordinary Lie Groups as Degenerate Multi-Valued Groups

Example 1.

Let

$$G=GL(n,\mathbb{R}).$$

Define

$$A★B=\left\{AB\right\}.$$

Then $(G,★)$ is a multi-valued topological group with singleton values. It is a Lie branch with

$$\mathfrak{g}=\mathfrak{gl}(n,\mathbb{R}).$$

The bracket is

$$[X,Y]=XY-YX.$$

The exponential is the matrix exponential

$$exp\left(X\right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{X^k}{k!}}.$$

This example demonstrates that the present theory contains the ordinary matrix Lie group theory as the zero-ambiguity case [6,8].

12. Example: A Two-Branch Lie Structure

Let G be a Lie group and let $\sigma :G\to G$ be a smooth group automorphism satisfying

$$\sigma \left(e\right)=e.$$

Define

$$x{★}_{\sigma }y=\{xy,\sigma \left(xy\right)\}.$$

In general this operation is not associative as a hyperoperation unless additional compatibility conditions are imposed. We therefore isolate a particularly useful case.

Proposition 4.

Suppose $\sigma =id$ in a neighborhood of e to first order, i.e.

$$d{\sigma }_e={id}_{T_{e}G}.$$

Then the two branches

$$m_1(x,y)=xy,m_2(x,y)=\sigma \left(xy\right)$$

have the same first-order tangent structure.

Proof. 

We have

$$d{m}_{2_{(e,e)}}=d{\sigma }_e\circ d{m}_{1_{(e,e)}}=d{m}_{1_{(e,e)}}.$$

Hence the first-order infinitesimal-collapse condition is satisfied.    □

Remark 6.

The proposition illustrates a basic phenomenon: distinct global branches can produce the same infinitesimal Lie algebra.

13. Example: A Central-Ambiguity Construction

Let G be a Lie group and let $Z\subseteq G$ be a discrete central subgroup. Define

$$x{★}_{Z}y=\{zxy:z\in Z\}.$$

Proposition ;5.

The operation ${★}_Z$ is associative in the set-valued sense:

$$\left(x{★}_{Z}y\right){★}_{Z}z=x{★}_Z\left(y{★}_{Z}z\right).$$

Proof. 

We have

$$\begin{array}{cc}\hfill \left(x{★}_{Z}y\right){★}_{Z}z& =\{z_2\left(z_{1}xy\right)z:z_1,z_2\in Z\}\hfill \\ & =\{z_1{z}_{2}xyz:z_1,z_2\in Z\}.\hfill \end{array}$$

Since Z is a subgroup,

$$\{z_1{z}_2:z_1,z_2\in Z\}=Z.$$

Consequently,

$$\left(x{★}_{Z}y\right){★}_{Z}z=\{z^{\prime }xyz:z^{\prime }\in Z\}.$$

On the other hand,

$$\begin{array}{cc}\hfill x{★}_Z\left(y{★}_{Z}z\right)& =\{z^{\prime }x\left(yz\right):z^{\prime }\in Z\}\hfill \\ & =\{z^{\prime }xyz:z^{\prime }\in Z\}.\hfill \end{array}$$

Therefore,

$$\left(x{★}_{Z}y\right){★}_{Z}z=x{★}_Z\left(y{★}_{Z}z\right).$$

   □

Remark 7.

If Z is discrete, the ambiguity is invisible at the infinitesimal level. Consequently the Lie algebra remains the Lie algebra of G.

14. Example: The Circle Group with Periodic Ambiguity

Consider

$$\mathbb{T}=\{z\in \mathbb{C}:|z|=1\}.$$

Its Lie algebra is

$$\mathfrak{t}=i\mathbb{R}.$$

Let

$$\mathsf{\Gamma }=\{1,-1\}.$$

Define

$$z★w=\{zw,-zw\}.$$

Proposition 6.

The operation above is associative as a set-valued operation.

Proof. 

We calculate

$$(z★w)★u=\{zwu,-zwu,-zwu,zwu\}=\{zwu,-zwu\}.$$

Likewise,

$$z★(w★u)=\{zwu,-zwu\}.$$

Therefore

$$(z★w)★u=z★(w★u).$$

   □

The identity in the strict singleton sense is not preserved by this operation because

$$1★z=\{z,-z\}.$$

Thus it is a multi-valued associative structure but not a multi-valued group under our strongest identity definition.

This example is useful because it shows why the identity axiom must be treated carefully.

Remark 8.

If one weakens the identity axiom to

$$x\in e★x,x\in x★e,$$

then the example becomes a legitimate hypergroup-type object. Different conventions in hyperstructure theory use different identity axioms; see [3,4].

15. Example: Matrix Lie Groups with Discrete Central Ambiguity

Let

$$G=SL(n,\mathbb{R})$$

and let

$$Z=\{\lambda I:{\lambda }^n=1,\lambda \in \mathbb{R}\}.$$

For appropriate n, Z is a finite central subgroup.

Define

$$A★B=\{ZAB:Z\in Z(G\left)\right\}.$$

The preceding proposition shows associativity.

The ordinary Lie algebra is

$$\mathfrak{sl}(n,\mathbb{R})=\{X\in M_n\left(\mathbb{R}\right):trX=0\},$$

with

$$[X,Y]=XY-YX.$$

Theorem 7.

The infinitesimal Lie algebra of the central-ambiguity multi-valued structure on $SL(n,\mathbb{R})$ is $\mathfrak{sl}(n,\mathbb{R})$.

Proof. 

The central subgroup is discrete. Hence every branch

$$m_Z(A,B)=ZAB$$

has the same differential at $(I,I)$:

$$d{m}_{Z_{(I,I)}}(X,Y)=X+Y.$$

The quadratic commutator term is also independent of Z, because Z is constant and central. Therefore the induced bracket is

$$[X,Y]=XY-YX.$$

   □

16. A Counterexample: Continuity Does Not Produce a Lie Algebra

It is important to emphasize that continuity alone is insufficient.

Let

$$X=\mathbb{R}$$

with its usual topology, and define

$$x★y=[x+y-1,x+y+1].$$

This is a continuous compact-valued map with respect to the Vietoris topology.

However, there is no canonical tangent multiplication at the origin.

Indeed, different selections

$$s_f(x,y)=x+y+f(x,y)$$

are possible, where

$$-1\le f(x,y)\le 1.$$

One may choose, for example,

$$f_1(x,y)=0$$

or

$$f_2(x,y)=x.$$

These selections have different differentials at $(0,0)$.

Proposition 7.

The continuous multi-valued operation above does not determine a canonical Lie algebra.

Proof. 

A canonical Lie algebra would require a canonical first-order multiplication at the identity. But

$$d{s}_{1_{(0,0)}}(u,v)=u+v,$$

whereas

$$d{s}_{2_{(0,0)}}(u,v)=2u+v.$$

Thus the differential depends on the selected branch. Consequently there is no branch-independent tangent multiplication.    □

Remark 9.

This example explains why the smooth branch and infinitesimal collapse assumptions in this paper are essential rather than technical conveniences.

17. Homomorphisms of Multi-Valued Lie Groups

Let $(G,{★}_G)$ and $(H,{★}_H)$ be multi-valued groups.

Definition 13.

A map

$$F:G⟶H$$

is called a multi-valued homomorphism if

$$F\left(x{★}_{G}y\right)\subseteq F\left(x\right){★}_{H}F\left(y\right)$$

for all $x,y\in G$.

A stronger notion is equality.

Definition 14.

A map $F:G\to H$ is called astrong multi-valued homomorphismif

$$F\left(x{★}_{G}y\right)=F\left(x\right){★}_{H}F\left(y\right).$$

Theorem 8.

Let $F:G\to H$ be a smooth strong multi-valued homomorphism between branch-compatible multi-valued Lie groups. Suppose $F\left(e_G\right)=e_H$. Then

$$d{F}_{e_G}:T_{e_G}G⟶T_{e_H}H$$

is a Lie algebra homomorphism.

Proof. 

Choose compatible local branches $m_G$ and $m_H$. The homomorphism condition gives

$$F\left(m_G(x,y)\right)=m_H(F\left(x\right),F\left(y\right)).$$

Differentiating at $(e_G,e_G)$ gives compatibility with the tangent multiplications. Passing to commutators of left invariant vector fields yields

$$d{F}_e[X,Y]=[d{F}_{e}X,d{F}_{e}Y].$$

Thus $d{F}_e$ is a Lie algebra homomorphism.    □

Corollary 2.

If F is a local diffeomorphism, then $d{F}_e$ is a Lie algebra isomorphism.

18. Subgroups and Sub-Lie Algebras

Definition 15.

A subset $H\subseteq G$ is a multi-valued subgroup if

$$x★y\subseteq H,x^{-1}\in H$$

whenever $x,y\in H$.

Suppose H is a submanifold.

Theorem 9.

Let $(G,★)$ be a branch-compatible multi-valued Lie group and let H be a smoothly embedded multi-valued subgroup. Then

$$\mathfrak{h}=T_{e}H$$

is a Lie subalgebra of

$$\mathfrak{g}=T_{e}G.$$

Proof. 

Choose compatible local branches. Since H is closed under the local branch multiplication, left invariant vector fields generated by vectors in $T_{e}H$ remain tangent to H. Therefore the bracket of two vector fields tangent to H is tangent to H. Hence

$$[T_{e}H,T_{e}H]\subseteq T_{e}H.$$

Thus $\mathfrak{h}$ is a Lie subalgebra.    □

19. Adjoint Action

For an ordinary Lie group,

$${Ad}_g=d{\left(c_g\right)}_e,c_g\left(x\right)=gx{g}^{-1}.$$

For a branch-compatible multi-valued group we define the set of possible conjugation branches.

Definition 16.

For $g,x\in G$, define

$$C_g\left(x\right)=g★x★g^{-1},$$

where the expression is interpreted using all admissible branches.

If all branches have the same first-order behavior, the tangent adjoint operator is single-valued.

Theorem 10.

Let $(G,★)$ be a branch-compatible multi-valued Lie group with infinitesimal collapse. Then for every $X\in \mathfrak{g}$ there is a well-defined infinitesimal adjoint operator

$${ad}_X:\mathfrak{g}\to \mathfrak{g}$$

given by

$${ad}_X\left(Y\right)=[X,Y].$$

Proof. 

Each local branch gives an ordinary local Lie group and hence an adjoint representation. By branch independence of the Lie bracket, all infinitesimal adjoint representations coincide:

$${ad}_{X}Y=[X,Y].$$

   □

20. Relation with Ordinary Lie Groups

We now formulate the principal structural statement.

Definition 17.

A branch-compatible multi-valued Lie group $(G,★)$ is called Lie-reducible if there exists an equivalence relation ∼ on G such that:

(i)

the quotient $G/\sim$ is a smooth manifold;

(ii)

the multi-valued product descends to a single-valued product on $G/\sim$;

(iii)

the quotient product is a Lie group operation.

Theorem 11

(Lie reduction theorem). Let $(G,★)$ be a Lie-reducible multi-valued Lie group. Then there exists a Lie group

$$\overline{G}=G/\sim$$

and a canonical Lie algebra

$$\overline{\mathfrak{g}}=T_{\overline{e}}\overline{G}.$$

If the equivalence classes are discrete near e, then

$$T_{e}G\cong T_{\overline{e}}\overline{G}$$

and the tangent Lie algebra of G is canonically isomorphic to $\overline{\mathfrak{g}}$.

Proof. 

By definition, $\overline{G}$ is a Lie group. Hence it possesses a Lie algebra

$$T_{\overline{e}}\overline{G}.$$

If the equivalence classes are discrete near e, the quotient map

$$q:G\to \overline{G}$$

is locally a diffeomorphism. Therefore

$$d{q}_e:T_{e}G\to T_{\overline{e}}\overline{G}$$

is an isomorphism. Compatibility of the multiplication branches with the quotient multiplication implies that $d{q}_e$ preserves the Lie brackets.    □

21. The Main Correspondence Theorem

We can now summarize the preceding results.

Theorem 12

(Multi-valued Lie correspondence). Let $(G,★)$ be a finite-valued, smoothly locally branchable multi-valued topological group satisfying:

(1)

there is an identity e;

(2)

every local branch near $(e,e)$ is smooth;

(3)

every identity branch is locally associative;

(4)

local inverse branches are smooth;

(5)

all identity branches have the same first-order differential;

(6)

all identity branches have the same antisymmetric quadratic term.

Then:

(a)

every identity branch defines a local Lie group;

(b)

$T_{e}G$ carries a branch-independent Lie algebra structure;

(c)

every smooth one-parameter branch determines an exponential curve;

(d)

the local BCH expansions of all branches have the same quadratic and cubic Lie-theoretic terms;

(e)

every smooth strong multi-valued homomorphism induces a Lie algebra homomorphism.

Proof. 

Part (a) follows from the local Lie group construction. Part (b) follows from branch independence of the first-order structure and the quadratic commutator term. Part (c) follows from the ordinary differential equation defining the exponential for each branch. Part (d) follows from the BCH formula for the local Lie group associated with each branch. Finally, part (e) follows by differentiating the multi-valued homomorphism identity and then passing to the induced commutator structure.    □

22. Multi-Valued Lie Algebras

The preceding construction motivates a purely infinitesimal terminology.

Definition 18.

A multi-valued Lie group with infinitesimal collapse is a smoothly branchable multi-valued group whose identity branches induce one and the same Lie algebra on $T_{e}G$.

The terminology is justified because the global multiplication may be genuinely multi-valued while the tangent algebra is ordinary.

Remark 10.

It is important not to define a “multi-valued Lie algebra” merely by replacing the Lie bracket

$$[X,Y]\in \mathfrak{g}$$

with

$$[X,Y]\subseteq \mathfrak{g}.$$

Such a replacement leads to hyper-Lie-algebra theories, which are interesting but are not equivalent to the tangent Lie algebra of a multi-valued Lie group. The present paper instead investigates the situation in which global multiplication is multi-valued but the infinitesimal algebra is classical.

One may, however, define a genuine hyper-Lie algebra.

Definition 19.

A hyper-Lie algebra is a vector space $\mathfrak{h}$ equipped with a hyperbracket

$${[·,·]}_h:\mathfrak{h}\times \mathfrak{h}⟶{\mathcal{P}}^{*}\left(\mathfrak{h}\right)$$

satisfying suitable hyperbilinearity, antisymmetry and Jacobi-type conditions.

Such structures belong to a different but related research program. Hyperstructure theory provides a broad algebraic framework for multivalued operations [3,4,5].

23. Representations

Let $\mathfrak{g}$ be the Lie algebra associated with a branch-compatible multi-valued Lie group.

Definition 20.

Arepresentationof the multi-valued Lie group on a vector space V is a family of branch representations

$${\rho }_{\alpha }:G\to GL\left(V\right)$$

such that compatible branches have the same infinitesimal representation

$$d{\rho }_{\alpha e}:\mathfrak{g}\to \mathfrak{gl}\left(V\right).$$

Theorem 13.

If all branches have the same tangent Lie algebra, then every compatible family of branch representations gives a Lie algebra representation

$${\rho }_{*}:\mathfrak{g}\to \mathfrak{gl}\left(V\right).$$

Proof. 

For every branch ${\rho }_{\alpha }$, ordinary Lie theory gives

$$d{\rho }_{\alpha e}\left([X,Y]\right)=[d{\rho }_{\alpha e}\left(X\right),d{\rho }_{\alpha e}\left(Y\right)].$$

If the infinitesimal representation is branch-independent, define

$${\rho }_{*}\left(X\right)=d{\rho }_{\alpha e}\left(X\right).$$

The displayed identity then proves that ${\rho }_{*}$ is a Lie algebra representation.    □

24. A Matrix Realization

Suppose that G is represented by matrices and that every local branch has the form

$$m_{\alpha }(A,B)=C_{\alpha }(A,B)AB,$$

where

$$C_{\alpha }(e,e)=I$$

and

$$d{C}_{{\alpha }_{(e,e)}}=0.$$

Then all branches have the same first-order multiplication.

Theorem 14.

Assume

$$C_{\alpha }(I,I)=I,d{C}_{{\alpha }_{(I,I)}}=0.$$

Then the tangent Lie algebra of every branch is a common Lie algebra with bracket

$$[X,Y]=XY-YX.$$

Proof. 

Near I,

$$A=I+tX+O\left(t^2\right),B=I+tY+O\left(t^2\right).$$

Therefore

$$AB=I+t(X+Y)+t^{2}XY+O\left(t^3\right).$$

Since

$$C_{\alpha }(A,B)=I+O\left(t^2\right),$$

the first-order term is independent of $\alpha$. The commutator calculation gives

$$I+t^2(XY-YX)+O\left(t^3\right).$$

Hence the induced bracket is the usual matrix commutator.    □

25. Global Versus Infinitesimal Multi-Valuedness

The preceding results reveal a useful distinction.

Definition 21.

A multi-valued Lie group is called infinitesimally classical if its tangent structure is an ordinary Lie algebra.

It is called globally multi-valued if there exist $x,y\in G$ such that

$$|x★y|>1.$$

Proposition 8.

A multi-valued Lie group can be globally multi-valued and infinitesimally classical.

Proof. 

Take a Lie group with a discrete central ambiguity as in Section 9. The product has multiple values globally, but the discrete central factors disappear under differentiation at the identity.    □

This observation is conceptually important. The theory does not attempt to force every aspect of a multi-valued group into an infinitesimal object. Instead, it separates:

$$\mathrm{global}\mathrm{ambiguity}\mathrm{from}\mathrm{infinitesimal}\mathrm{geometry}.$$

26. Compatibility of Multiple Branches

Suppose

$$x★y=\{m_1(x,y),\dots ,m_k(x,y)\}.$$

A strong compatibility condition is the following.

Definition 22.

The branches $m_1,\dots ,m_k$ are called Lie-compatible if:

(i)

each $m_i$ is a local Lie group operation;

(ii)

all $m_i$ have the same identity;

(iii)

all $m_i$ have the same local inverse to first order;

(iv)

their tangent Lie brackets coincide.

Theorem 15.

If $m_1,\dots ,m_k$ are Lie-compatible, then the tangent Lie algebra

$$\mathfrak{g}=T_{e}G$$

is well defined independently of the branch.

Proof. 

By assumption,

$${[X,Y]}_{m_i}={[X,Y]}_{m_j}$$

for all $i,j$. Hence a single bracket may be defined by

$$[X,Y]={[X,Y]}_{m_i}.$$

The Jacobi identity holds because it holds for each branch.    □

27. A Local Classification Principle

The preceding discussion leads to a useful local classification principle.

Theorem 16.

Let $(G,★)$ be a finite-valued branch-compatible multi-valued Lie group with infinitesimal collapse. Then, locally at the identity, all branches are Lie group structures having the same Lie algebra $\mathfrak{g}$.

Proof. 

Each branch gives a local Lie group. Infinitesimal collapse gives the same tangent space and first-order multiplication. Equality of the induced commutator terms gives the same Lie bracket. Hence all branches have the same Lie algebra.    □

Remark 11.

The theorem does not claim that the local Lie group structures are identical. Distinct Lie group structures can have isomorphic Lie algebras, and even branches with identical tangent algebras may differ at higher order or globally.

28. Relation to Classical Lie Theory

For an ordinary Lie group G, one has the familiar diagram

$$\begin{array}{ccc}G& \stackrel{exp}{\to }& G\\ Ad\downarrow \phantom{Ad}& & \phantom{\mathrm{conjugation}}\downarrow \mathrm{conjugation}& & \\ \mathfrak{g}& \stackrel{ad}{\to }& \mathfrak{g}.\end{array}$$

To avoid requiring the amscd package, we express the correspondence algebraically:

$${Ad}_g\left(X\right)={\left.{\displaystyle \frac{d}{dt}}\right|}_{t=0}gx\left(t\right)g^{-1},$$

where $x\left(0\right)=e$ and $x^{\prime }\left(0\right)=X$.

In the multi-valued setting, one obtains a family of possible conjugations. Under infinitesimal collapse these induce the same ad-operator:

$${ad}_X\left(Y\right)=[X,Y].$$

Thus the classical diagram survives infinitesimally even when the global multiplication is set-valued.

Classical Lie theory establishes the relation between local group structure and Lie algebra structure; see [2,6,8]. The contribution of the present framework is to identify hypotheses under which that correspondence survives a controlled multi-valued enlargement.

29. Topological Issues

The Vietoris topology is particularly natural when all product values are compact. If

$$★:G\times G\to \mathcal{K}\left(G\right)$$

is continuous, then both upper and lower semicontinuity are encoded in the Vietoris topology.

This is useful because a multi-valued product should vary continuously as a set. However, the existence of a continuous selection is an additional property and cannot be assumed automatically.

Proposition 9.

Suppose $★:G\times G\to \mathcal{K}\left(G\right)$ admits a continuous single-valued selection

$$s:G\times G\to G.$$

Then s may be studied as an ordinary continuous group-like operation, but it need not be associative.

Proof. 

Continuity is immediate from the definition of a selection. Associativity requires the additional condition

$$s\left(s\right(x,y),z)=s(x,s(y,z\left)\right),$$

which does not follow from continuity. □

Thus the correct hierarchy is

$$\mathrm{set}\text{-}\mathrm{valued}\mathrm{continuity}\nRightarrow \mathrm{continuous}\mathrm{selection}\nRightarrow \mathrm{associative}\mathrm{selection}.$$

This is one reason the branch axioms must be stated explicitly.

30. Further Examples

Example 2

(A finite central extension). Let G be a Lie group and let

$$Z=\{z_1,\dots ,z_r\}$$

be a finite central subgroup. Define

$$x★y=\{z_{i}xy:1\le i\le r\}.$$

The operation is associative and globally multi-valued. If Z is discrete, its infinitesimal Lie algebra is the ordinary Lie algebra $\mathfrak{g}$.

Example 3

(Abelian case). Let $G={\mathbb{R}}^n$ with ordinary addition and let Z be a finite subgroup of a torus quotient. Define locally

$$x★y=\{x+y+z:z\in Z\}.$$

Since ${\mathbb{R}}^n$ is abelian, the tangent Lie algebra is

$$\mathfrak{g}={\mathbb{R}}^n$$

with

$$[X,Y]=0.$$

The multi-valued structure is therefore globally nontrivial but infinitesimally abelian.

Example 4

(A compact matrix group). Let

$$G=U\left(n\right)$$

and let

$$Z=\{\zeta I:{\zeta }^k=1\}.$$

Then

$$A★B=\{\zeta AB:{\zeta }^k=1\}.$$

The tangent Lie algebra is

$$\mathfrak{u}\left(n\right)=\{X:X^{*}=-X\}$$

with

$$[X,Y]=XY-YX.$$

31. Necessary Conditions and Structural Constraints

The theory developed here is deliberately restrictive in several ways.

First, not every hypergroup is a Lie-type object. Hypergroups may have discontinuous or highly non-smooth operations, and their hyperproducts may have infinitely many values. Such structures require different methods [1,3].

Second, even when the underlying space is a manifold, arbitrary set-valued operations need not admit smooth selections.

Third, even if smooth selections exist, different selections can produce inequivalent local group structures.

Fourth, a tangent vector space exists only because the underlying space is assumed to be a smooth manifold. A purely topological multi-valued group has no canonical tangent space.

These limitations show that the following implication is generally false:

$$\mathrm{topological}\mathrm{multi}\text{-}\mathrm{valued}\mathrm{group}⟹\mathrm{Lie}\mathrm{algebra}.$$

The valid implication established in this paper is instead

$$\begin{array}{c}\mathrm{smooth}\mathrm{manifold}\\ +\mathrm{smooth}\mathrm{local}\mathrm{Lie}\mathrm{branches}\\ +\mathrm{infinitesimal}\mathrm{compatibility}\end{array}⟹\mathrm{canonical}\mathrm{Lie}\mathrm{algebra}.$$

32. Conclusion

We have developed a framework connecting multi-valued topological groups with classical Lie groups and Lie algebras. The central observation is that global multi-valuedness and infinitesimal multi-valuedness are logically different phenomena.

A general hyperoperation does not possess a canonical differential. However, if the multi-valued operation admits smooth local branches, if those branches satisfy the local group axioms, and if their infinitesimal structures are compatible, then the ordinary tangent space at the identity acquires a canonical Lie algebra structure.

The principal correspondence established in the paper is therefore

$\mathrm{multi}\text{-}\mathrm{valued}\mathrm{topological}\mathrm{group}⟶\mathrm{smooth}\mathrm{local}\mathrm{branches}⟶\mathrm{local}\mathrm{Lie}\mathrm{groups}⟶\mathrm{canonical}\mathrm{Lie}\mathrm{algebra}.$