Lie-Semigroup Methods in Algebra and Dynamics

• Lie‐semigroup‐based methods are algebraic constructions that use associative semigroup laws to generate closure in Lie brackets and dynamical models.
• They extend Lie algebras and groups via S‐expansion, yielding new models in phylogenetics, network dynamics, and even constructing Maxwell and Virasoro extensions.
• These methods also underpin analytic splitting techniques for PDEs and control systems on Lie groups, ensuring stability and improved numerical performance.

Lie-semigroup-based methods are constructions in which a semigroup law controls a Lie-theoretic, dynamical, or stochastic object. In the cited literature, this role is played by finite semigroups that determine Lie brackets or rate-matrix bases, by semigroups of maps that encode network architecture, by semigroups of annuli that substitute for a non-existent complex Lie group, by semigroups attached to reachable sets of control systems on Lie groups, and by analytic semigroups generated by elliptic operators in splitting schemes (Artebani et al., 2016, Astudillo et al., 2010, Sumner et al., 2017, Rink et al., 2012, Henriques et al., 2024, Ayala et al., 2016, Dang et al., 2024). The common structural theme is closure: associativity, semigroup invariance, or semigroup generation supplies multiplicative closure, closure under commutators, or stable composition laws that remain internal to the model.

1. Algebraic premise and closure mechanism

A finite semigroup of degree $k$ is a set $S=\{a_1,\dots,a_k\}$ equipped with an associative binary operation $m:S\times S\to S$, $m(a_i,a_j)=a_i a_j$, so that $(ab)c=a(bc)$ for all $a,b,c\in S$. Two semigroups of the same size are isomorphic if there is a bijection preserving the product, and anti-isomorphic if the product is reversed under the bijection (Sumner et al., 2017). In the S-expansion literature, the semigroup is typically finite, discrete, and abelian, with product encoded by selectors $K_{ij}{}^k\in\{0,1\}$ defined by $\lambda_i\lambda_j=\lambda_k$ (Artebani et al., 2016, Astudillo et al., 2010).

The basic closure mechanism appears in several equivalent guises. For finite semigroups used in phylogenetics, one passes from the multiplication table $a_i a_j=a_{m(i,j)}$ to $0\!-\!1$ multiplication matrices $S=\{a_1,\dots,a_k\}$0 satisfying $S=\{a_1,\dots,a_k\}$1, and then to rate-matrix generators $S=\{a_1,\dots,a_k\}$2 (Sumner et al., 2017). For S-expansions of Lie algebras, one combines the semigroup selectors with the structure constants $S=\{a_1,\dots,a_k\}$3 of a Lie algebra $S=\{a_1,\dots,a_k\}$4 to obtain new generators $S=\{a_1,\dots,a_k\}$5 and bracket

$S=\{a_1,\dots,a_k\}$6

Associativity of the semigroup and the Jacobi identity in $S=\{a_1,\dots,a_k\}$7 imply that the expanded bracket again defines a Lie algebra (Artebani et al., 2016, Astudillo et al., 2010).

An analogous phenomenon occurs in coupled cell networks. A homogeneous network architecture is specified by maps $S=\{a_1,\dots,a_k\}$8 on the cell set, and $S=\{a_1,\dots,a_k\}$9 is called a semigroup when it is closed under composition. In that case one can construct linear maps $m:S\times S\to S$0 satisfying $m:S\times S\to S$1, and the associated network vector fields $m:S\times S\to S$2 form a Lie subalgebra under the usual Lie bracket of vector fields (Rink et al., 2012). This suggests that the semigroup law acts as a universal bookkeeping device for internal composition.

2. S-expansion of Lie algebras and Lie groups

The S-expansion procedure starts from a Lie algebra $m:S\times S\to S$3 with basis $m:S\times S\to S$4 and a finite abelian semigroup $m:S\times S\to S$5. The expanded vector space is $m:S\times S\to S$6 or $m:S\times S\to S$7, with generators $m:S\times S\to S$8, and the semigroup product deforms the original Lie bracket through the selectors $m:S\times S\to S$9 (Artebani et al., 2016, Astudillo et al., 2010). When $m(a_i,a_j)=a_i a_j$0 contains a zero element $m(a_i,a_j)=a_i a_j$1, one may impose the ideal condition $m(a_i,a_j)=a_i a_j$2; the resulting quotient is the $m(a_i,a_j)=a_i a_j$3-reduced algebra (Artebani et al., 2016). When $m(a_i,a_j)=a_i a_j$4 admits a subspace decomposition compatible with the bracket and $m(a_i,a_j)=a_i a_j$5 admits a resonant partition $m(a_i,a_j)=a_i a_j$6 satisfying $m(a_i,a_j)=a_i a_j$7, the direct sum $m(a_i,a_j)=a_i a_j$8 is a resonant subalgebra (Artebani et al., 2016).

The method extends to the group manifold. If an element of the original Lie group is parametrized by $m(a_i,a_j)=a_i a_j$9, then the coordinates are expanded by an S-map $(ab)c=a(bc)$0, producing $(ab)c=a(bc)$1. The left-invariant Maurer–Cartan forms expand in parallel, $(ab)c=a(bc)$2, and satisfy the expanded Maurer–Cartan equations with the same selectors $(ab)c=a(bc)$3 (Astudillo et al., 2010). In the geometric analysis of the procedure, the Killing–Cartan form is deformed by the semigroup selectors, so that lengths and scalar products on the group manifold are rescaled by the semigroup data (Artebani et al., 2016).

Several explicit constructions are standard. The Maxwell algebra is obtained from $(ab)c=a(bc)$4 using a finite abelian semigroup with zero and a resonant partition; after $(ab)c=a(bc)$5-reduction, the generators $(ab)c=a(bc)$6, $(ab)c=a(bc)$7, and $(ab)c=a(bc)$8 satisfy $(ab)c=a(bc)$9 (Concha et al., 2014). The same formalism yields the minimal Maxwell superalgebra $a,b,c\in S$0, the $a,b,c\in S$1-extended Maxwell superalgebra $a,b,c\in S$2, and new minimal Maxwell superalgebras $a,b,c\in S$3 from $a,b,c\in S$4 (Concha et al., 2014). In three-dimensional non-relativistic supergravity, the method is applied to a supersymmetric extension of the Nappi–Witten algebra, with semigroups $a,b,c\in S$5 and $a,b,c\in S$6, to generate generalized extended Bargmann and generalized extended Newton–Hooke families together with their invariant tensors and Chern–Simons actions (Concha et al., 2020).

The method also has an infinite-dimensional branch. Taking $a,b,c\in S$7 under addition and realizing $a,b,c\in S$8 as the Fourier mode $a,b,c\in S$9, one obtains generators $K_{ij}{}^k\in\{0,1\}$0 with bracket $K_{ij}{}^k\in\{0,1\}$1, which is precisely the loop algebra $K_{ij}{}^k\in\{0,1\}$2. The same construction with bases of spherical harmonics or Wigner $K_{ij}{}^k\in\{0,1\}$3-functions produces $K_{ij}{}^k\in\{0,1\}$4 and $K_{ij}{}^k\in\{0,1\}$5 (Astudillo et al., 2010). A further structural result is that if $K_{ij}{}^k\in\{0,1\}$6 is simple and $K_{ij}{}^k\in\{0,1\}$7 has $K_{ij}{}^k\in\{0,1\}$8 elements, then the S-expanded algebra is non-simple; in the stated faithful setting it decomposes as a direct sum of $K_{ij}{}^k\in\{0,1\}$9 copies of $\lambda_i\lambda_j=\lambda_k$0 (Artebani et al., 2016).

3. Semigroup-derived stochastic and network dynamics

In phylogenetics, a finite semigroup of degree $\lambda_i\lambda_j=\lambda_k$1 gives rise to a continuous-time Markov chain on $\lambda_i\lambda_j=\lambda_k$2 states. From the multiplication matrices $\lambda_i\lambda_j=\lambda_k$3, one defines $\lambda_i\lambda_j=\lambda_k$4, forms the real span $\lambda_i\lambda_j=\lambda_k$5, and then takes $\lambda_i\lambda_j=\lambda_k$6, where $\lambda_i\lambda_j=\lambda_k$7 is the set of rate-matrices with non-negative off-diagonals and zero column-sum (Sumner et al., 2017). Every rate-matrix in the semigroup-based model has the form

$\lambda_i\lambda_j=\lambda_k$8

Because $\lambda_i\lambda_j=\lambda_k$9, one has $a_i a_j=a_{m(i,j)}$0, so $a_i a_j=a_{m(i,j)}$1 is closed under commutators and is therefore a matrix Lie algebra (Sumner et al., 2017). By the stated equivalence, multiplicative closure is equivalent to the ambient space being a linear subspace closed under commutators; hence every semigroup-based model is a Lie-Markov model (Sumner et al., 2017).

The phylogenetic significance is that the product of substitution matrices taken from the model again lies in the model. If $a_i a_j=a_{m(i,j)}$2 is a finite group, the regular-representation construction recovers the usual group-based model, and for $a_i a_j=a_{m(i,j)}$3 abelian one recovers the classical Fourier-diagonalizable group-based models (Sumner et al., 2017). The semigroup construction is broader because semigroups do not require inverses, anti-isomorphic semigroups can yield inequivalent models, and some semigroup-based Lie-Markov models do not come from any group (Sumner et al., 2017). Enumeration for small state spaces makes this explicit. For $a_i a_j=a_{m(i,j)}$4, the five non-isomorphic semigroups yield, up to state permutations, three distinct Lie-Markov models: the absorbing-state model, the 2-state general Markov (equal-input) model, and the binary symmetric group-based model $a_i a_j=a_{m(i,j)}$5 (Sumner et al., 2017). For $a_i a_j=a_{m(i,j)}$6, among the $a_i a_j=a_{m(i,j)}$7 non-isomorphic semigroups, exactly two irreducible models remain after discarding reducible or absorbing-state cases: the equal-input model and the cyclic-group $a_i a_j=a_{m(i,j)}$8 group-based model (Sumner et al., 2017). For $a_i a_j=a_{m(i,j)}$9, among the $0\!-\!1$0 non-isomorphic semigroups, the irreducible non-absorbing output reduces to four non-isomorphic models: F81, Kimura 3ST, Model 3.3b, and a new 4-state model (Sumner et al., 2017).

A parallel use of semigroup closure appears in coupled cell networks. Given a semigroup architecture $0\!-\!1$1, the network vector fields satisfy

$0\!-\!1$2

where the symbolic bracket on $0\!-\!1$3 is

$0\!-\!1$4

Thus the infinitesimal generators form a Lie algebra, and near a dynamical equilibrium the local normal form of a semigroup network is a semigroup network itself (Rink et al., 2012). Networks without the semigroup property may have normal forms with a more general network architecture, although the normal forms preserve the same symmetries and synchronous solutions as the original network (Rink et al., 2012).

4. Infinite-dimensional integration by semigroups

A distinctive infinite-dimensional use of the method occurs for the complexification of vector fields on the circle. The Lie algebra of vector fields on $0\!-\!1$5 integrates to the Lie group $0\!-\!1$6, but there is no Lie group whose Lie algebra is the complexification of vector fields on $0\!-\!1$7. The semigroup $0\!-\!1$8 of parametrised annuli serves as a substitute for that non-existent group (Henriques et al., 2024). Its elements are genus-zero Riemann surfaces with two boundary circles parametrized by $0\!-\!1$9, and the enlargement considered in the cited work allows the annuli to be partially thin, meaning that the two boundary circles may touch along an arbitrary closed subset (Henriques et al., 2024). The semigroup product is conformal welding: $S=\{a_1,\dots,a_k\}$00

The infinitesimal data are paths in the cone of inward-pointing complexified vector fields

$S=\{a_1,\dots,a_k\}$01

For a framing $S=\{a_1,\dots,a_k\}$02, one sets $S=\{a_1,\dots,a_k\}$03 and writes

$S=\{a_1,\dots,a_k\}$04

The surjectivity theorem states that every $S=\{a_1,\dots,a_k\}$05 arises in this way from a smooth path with sitting instants near $S=\{a_1,\dots,a_k\}$06 (Henriques et al., 2024). This is a semigroup-level integration statement: the time-ordered exponential is surjective onto the annulus semigroup.

The same framework carries the Virasoro extension. The classical Virasoro cocycle is lifted to a closed, left-invariant $S=\{a_1,\dots,a_k\}$07-form on the thick parameter space of annuli, and the quotient construction

$S=\{a_1,\dots,a_k\}$08

produces a central extension integrating the universal central extension of the Lie algebra of vector fields on $S=\{a_1,\dots,a_k\}$09 (Henriques et al., 2024). A plausible implication is that semigroup objects can replace missing complex Lie groups when time-ordered exponentials remain available but genuine group integration fails.

5. Semigroups in Lie groups, flag manifolds, and control

For connected real semisimple Lie groups with finite center, semigroups arise directly inside the group. If $S=\{a_1,\dots,a_k\}$10, the semigroup it generates is $S=\{a_1,\dots,a_k\}$11, consisting of all finite products of elements of $S=\{a_1,\dots,a_k\}$12 (Santos et al., 2015). In the cited topological generation theorem, $S=\{a_1,\dots,a_k\}$13 is equivalent to $S=\{a_1,\dots,a_k\}$14 generating a Zariski-dense subgroup of $S=\{a_1,\dots,a_k\}$15; the proof uses Abels’s result on interior points together with a flag-manifold obstruction showing that any proper closed semigroup with nonempty interior must preserve a contractible subset in some minimal flag manifold (Santos et al., 2015). When $S=\{a_1,\dots,a_k\}$16 contains a connected semisimple subgroup $S=\{a_1,\dots,a_k\}$17 and every closed $S=\{a_1,\dots,a_k\}$18-orbit in the relevant minimal flags is non-contractible, the topological criterion forces $S=\{a_1,\dots,a_k\}$19 (Santos et al., 2015). Explicit applications are given for split real forms with root-$S=\{a_1,\dots,a_k\}$20 subgroups, irreducible representations of $S=\{a_1,\dots,a_k\}$21 inside $S=\{a_1,\dots,a_k\}$22, and complex semisimple subgroups containing a regular real one-parameter group (Santos et al., 2015).

In linear control systems on connected Lie groups, the accessible set from the identity is generally not a semigroup. For a system

$S=\{a_1,\dots,a_k\}$23

with linear drift $S=\{a_1,\dots,a_k\}$24, flow $S=\{a_1,\dots,a_k\}$25, and right-invariant control fields $S=\{a_1,\dots,a_k\}$26, the reachable set $S=\{a_1,\dots,a_k\}$27 satisfies $S=\{a_1,\dots,a_k\}$28, which obstructs ordinary product closure (Ayala et al., 2016). The associated semigroup is

$S=\{a_1,\dots,a_k\}$29

It is the largest $S=\{a_1,\dots,a_k\}$30-invariant subset of $S=\{a_1,\dots,a_k\}$31, it contains the identity, it is connected, and it is closed under products (Ayala et al., 2016). The central criterion is that the system is controllable, meaning $S=\{a_1,\dots,a_k\}$32, if and only if $S=\{a_1,\dots,a_k\}$33; in particular, controllability is equivalent to $S=\{a_1,\dots,a_k\}$34 (Ayala et al., 2016). In the noncompact semisimple case with finite center and open reachable set, controllability is equivalent to $S=\{a_1,\dots,a_k\}$35 (Ayala et al., 2016).

A related invariant for proper semigroups $S=\{a_1,\dots,a_k\}$36 with nonempty interior is the flag type $S=\{a_1,\dots,a_k\}$37, defined through the invariant control set $S=\{a_1,\dots,a_k\}$38 on the maximal flag manifold $S=\{a_1,\dots,a_k\}$39. It is the unique minimal $S=\{a_1,\dots,a_k\}$40 such that $S=\{a_1,\dots,a_k\}$41, and equivalently the unique maximal $S=\{a_1,\dots,a_k\}$42 for which the $S=\{a_1,\dots,a_k\}$43-control set in the partial flag manifold $S=\{a_1,\dots,a_k\}$44 is contractible (Silva et al., 13 Jan 2025). The cocycle method reconstructs $S=\{a_1,\dots,a_k\}$45 from lower bounds of the $S=\{a_1,\dots,a_k\}$46-invariant cocycles $S=\{a_1,\dots,a_k\}$47: for $S=\{a_1,\dots,a_k\}$48, if $S=\{a_1,\dots,a_k\}$49, then there exists $S=\{a_1,\dots,a_k\}$50 such that $S=\{a_1,\dots,a_k\}$51 for all $S=\{a_1,\dots,a_k\}$52; if $S=\{a_1,\dots,a_k\}$53, then $S=\{a_1,\dots,a_k\}$54 (Silva et al., 13 Jan 2025). This links algebraic information about parabolics, roots, and weights to semigroup dynamics on flag manifolds.

6. Analytic-semigroup Lie splitting and conceptual boundaries

The cited literature also contains a distinct analytic-semigroup usage of Lie-semigroup-based ideas. For convection–diffusion problems

$S=\{a_1,\dots,a_k\}$55

rewritten abstractly as $S=\{a_1,\dots,a_k\}$56, the relevant semigroup is the analytic semigroup generated by the strongly elliptic operator $S=\{a_1,\dots,a_k\}$57 on $S=\{a_1,\dots,a_k\}$58, satisfying $S=\{a_1,\dots,a_k\}$59 and $S=\{a_1,\dots,a_k\}$60 (Dang et al., 2024). The classical first-order Lie splitting composes a diffusion step with a convection step, but when $S=\{a_1,\dots,a_k\}$61 and $S=\{a_1,\dots,a_k\}$62 is unbounded, the operator $S=\{a_1,\dots,a_k\}$63 may blow up on $S=\{a_1,\dots,a_k\}$64, and the numerical tests in the cited work show unbounded growth of $S=\{a_1,\dots,a_k\}$65 (Dang et al., 2024).

The adapted Lie splitting method circumvents this by decomposing the singular field $S=\{a_1,\dots,a_k\}$66, defining $S=\{a_1,\dots,a_k\}$67, and introducing the modified operators

$S=\{a_1,\dots,a_k\}$68

One step of size $S=\{a_1,\dots,a_k\}$69 is

$S=\{a_1,\dots,a_k\}$70

Under Assumptions 4.1–4.2 and for $S=\{a_1,\dots,a_k\}$71 sufficiently small, the global error satisfies

$S=\{a_1,\dots,a_k\}$72

and the local consistency error is $S=\{a_1,\dots,a_k\}$73 (Dang et al., 2024). Numerically, in the two-dimensional tests with $S=\{a_1,\dots,a_k\}$74, the classical Lie method becomes unstable as $S=\{a_1,\dots,a_k\}$75 decreases, whereas the adapted Lie method is stable, converges with slope $S=\{a_1,\dots,a_k\}$76, and has error approximately $S=\{a_1,\dots,a_k\}$77 smaller for the same $S=\{a_1,\dots,a_k\}$78; in the three-dimensional test, classical splitting blows up for large $S=\{a_1,\dots,a_k\}$79, while adapted splitting remains stable and more accurate (Dang et al., 2024).

Taken together, these works delimit the scope of the term. In one branch, semigroup multiplication is an algebraic input that generates Lie brackets, resonant subalgebras, Lie-Markov models, or network normal forms (Artebani et al., 2016, Sumner et al., 2017, Rink et al., 2012). In another, semigroups act as geometric replacements for missing Lie groups or as invariant objects attached to semigroup actions on flags and reachable sets in Lie groups (Henriques et al., 2024, Santos et al., 2015, Ayala et al., 2016, Silva et al., 13 Jan 2025). In the analytic branch, the central object is not a finite algebraic semigroup but the analytic semigroup of an operator, used to control Lie splitting in PDEs (Dang et al., 2024). A common misconception is that these constructions require group structure; the cited results show instead that associativity alone can be sufficient, and in several cases the absence of inverses is precisely what enlarges the available class of models (Sumner et al., 2017).