Magma contains one of the most complete computational toolkits anywhere for Lie theory: Coxeter groups, root systems, Lie algebras, quantum groups, and groups of Lie type. This is a survey, not a deep dive — the goal is to give you a mental map of the roughly eleven chapters that make up this part of the Handbook, so you know which chapter to reach for when a Lie-theoretic question comes up. Each topic here easily fills a chapter on its own; treat this as a table of contents with commentary.
The Big Picture
Lie theory in Magma is built around a chain of increasingly rich descriptions of the same underlying combinatorics. At the bottom sit Coxeter matrices and Cartan matrices — small integer matrices encoding braid relations between generators. These feed into root systems and root data, which record the actual root vectors and their pairings. Root data in turn classify Coxeter groups, reflection groups, Lie algebras, and groups of Lie type. On top of all this sit representation theory and the quantum group deformation. The introductory chapter of the Handbook (lie-introduction) lists exactly these eight areas as one connected family, and the rest of the chapters flesh each one out.
Coxeter Matrices, Cartan Matrices, and Naming
A Coxeter system is a group generated by involutions $s_1,\dots,s_n$ subject only to braid relations $s_is_js_i\cdots = s_js_is_j\cdots$ of length $m_{ij}$. The coxeter-description chapter (Cartan.text) is where Magma stores and converts between the various combinatorial encodings of this data: Coxeter matrices, Coxeter graphs, (generalized) Cartan matrices, and Dynkin digraphs, together with Cartan's naming system (things like "A2 B3 G2") that classifies finite and affine Coxeter groups. Almost everything downstream — root systems, Weyl groups, Lie algebra types — is ultimately specified via one of these names or matrices.
> C := CartanMatrix("G2");
> IsCoxeterMatrix(CoxeterMatrix(C));
true
Root Systems and Root Data
Root systems (root-systems) describe the reflections of a finite Coxeter group combinatorially, as vectors in a real vector space with a bilinear pairing. Root data (root-data) are the refinement needed once you want to talk about Lie algebras or algebraic groups: they track not just the roots but an ambient lattice (weights, coroots, an isogeny type), which is exactly the extra information a Lie algebra or a group of Lie type needs. The rule of thumb given in the Handbook is simple: if you're working with Coxeter or reflection groups, a root system is enough; if you're working with Lie algebras or groups of Lie type, use a root datum.
> R := RootDatum("E8");
> Rank(R), NumPosRoots(R);
Coxeter Groups and Reflection Groups
Once you have the combinatorial data, GrpCox.text and GrpRfl.text describe how to realize the abstract group. Magma offers a Coxeter group in three concrete guises: as a finitely presented group (category GrpFPCox), as a permutation group, and as a reflection group acting on a vector space — all Coxeter-isomorphic representations of the same group, chosen depending on whether you need presentations, orbit/stabilizer computation, or linear-algebraic reflection data. Reflection groups generalize this further: a reflection group is any finite-dimensional linear group generated by reflections (diagonalizable maps fixing a hyperplane) over an arbitrary field, not just a real one. Every Coxeter group is a real reflection group, but complex reflection groups (the Shephard–Todd classification) are handled here too, going beyond what Coxeter theory alone covers.
Lie Algebras and Kac–Moody Lie Algebras
AlgLie.text covers finite-dimensional Lie algebras, Magma's most heavily developed algebraic structure in this area. A Lie algebra is stored as a structure-constant algebra, a matrix algebra, or a finitely presented algebra, and the classical simple types (A$_n$ through G$_2$) can be built directly from a root datum or a Cartan name rather than by hand-specifying structure constants.
> L := LieAlgebra("F4", RationalField());
> Dimension(L);
AlgLieKM.text extends this to the infinite-dimensional case: a generalized Cartan matrix (dropping the positivity restrictions of an ordinary Cartan matrix) generates a Kac–Moody Lie algebra via the Chevalley–Serre relations. These split into finite-dimensional/reductive, affine (polynomial growth), and indefinite (exponential growth) types; only the first two are currently supported computationally.
Groups of Lie Type
GrpLie.text assembles a group of Lie type from a root datum plus a ring, using the Steinberg presentation with root elements $x_r(t)$ and a torus. Depending on the ring you plug in, the same construction yields a reductive Lie group (ring = $\mathbb{R}$ or $\mathbb{C}$), a reductive algebraic group (ring = an algebraically closed field), or a finite group of Lie type (ring = a finite field) — e.g. the classical groups of Lie type such as untwisted and twisted Chevalley groups. Because the generating set is parametrised by field elements, these groups don't fit the finitely-presented-group category and get their own type, GrpLie.
> G := GroupOfLieType("A2", GF(5));
Quantum Groups
AlgQEA.text describes quantized enveloping algebras (quantum groups), the one-parameter deformation $U_v(\mathfrak{g})$ of the universal enveloping algebra of a semisimple Lie algebra, built from the same Cartan matrix data as the classical algebra. These have their own category AlgQUE, built on a general PBW-basis algebra framework, and specialize back to the classical enveloping algebra as the deformation parameter $v \to 1$.
Representations
Finally, LieReps.text (Representations of Lie Groups and Algebras) handles highest-weight representations of semisimple Lie algebras and connected reductive algebraic groups — the class that captures all finite-dimensional representations over the complex numbers. Rather than building the (potentially huge) representation itself, Magma works combinatorially with decomposition multisets of dominant weights, letting you compute dimensions, full weight multisets, and decompositions of symmetric powers, tensor products, and restrictions/inductions purely combinatorially.
Finding Your Way
As a quick map: describe combinatorics with Cartan/Coxeter matrices, upgrade to root systems for Coxeter/reflection groups or root data for Lie algebras/groups of Lie type, build the group or algebra itself, then move to LieReps for representation questions or AlgQEA if you need the quantum deformation. Each chapter's own introduction is worth reading in full once you know which one you need.