Quantum group

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Quantum_group

In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their name, they do not themselves have a natural group structure, though they are in some sense 'close' to a group.

The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized by Vladimir Drinfeld and Michio Jimbo as a particular class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct" class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo.

In Drinfeld's approach, quantum groups arise as Hopf algebras depending on an auxiliary parameter q or h, which become universal enveloping algebras of a certain Lie algebra, frequently semisimple or affine, when q = 1 or h = 0. Closely related are certain dual objects, also Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimple algebraic group or a compact Lie group.

Intuitive meaning

The discovery of quantum groups was quite unexpected since it was known for a long time that compact groups and semisimple Lie algebras are "rigid" objects, in other words, they cannot be "deformed". One of the ideas behind quantum groups is that if we consider a structure that is in a sense equivalent but larger, namely a group algebra or a universal enveloping algebra, then a group or enveloping algebra can be "deformed", although the deformation will no longer remain a group or enveloping algebra. More precisely, deformation can be accomplished within the category of Hopf algebras that are not required to be either commutative or cocommutative. One can think of the deformed object as an algebra of functions on a "noncommutative space", in the spirit of the noncommutative geometry of Alain Connes. This intuition, however, came after particular classes of quantum groups had already proved their usefulness in the study of the quantum Yang–Baxter equation and quantum inverse scattering method developed by the Leningrad School (Ludwig Faddeev, Leon Takhtajan, Evgeny Sklyanin, Nicolai Reshetikhin and Vladimir Korepin) and related work by the Japanese School. The intuition behind the second, bicrossproduct, class of quantum groups was different and came from the search for self-dual objects as an approach to quantum gravity.

Drinfeld–Jimbo type quantum groups

One type of objects commonly called a "quantum group" appeared in the work of Vladimir Drinfeld and Michio Jimbo as a deformation of the universal enveloping algebra of a semisimple Lie algebra or, more generally, a Kac–Moody algebra, in the category of Hopf algebras. The resulting algebra has additional structure, making it into a quasitriangular Hopf algebra.

Let A = (a_ij) be the Cartan matrix of the Kac–Moody algebra, and let q ≠ 0, 1 be a complex number, then the quantum group, U_q(G), where G is the Lie algebra whose Cartan matrix is A, is defined as the unital associative algebra with generators k_λ (where λ is an element of the weight lattice, i.e. 2(λ, α_i)/(α_i, α_i) is an integer for all i), and e_i and f_i (for simple roots, α_i), subject to the following relations:

${\displaystyle \begin{array}{rl}{k}_0& =1\\ k_{\lambda }{k}_{\mu }& =k_{\lambda +\mu }\\ k_{\lambda }{e}_i{k}_{\lambda }^{-1}& =q^{(\lambda ,{\alpha }_i)}{e}_i\\ k_{\lambda }{f}_i{k}_{\lambda }^{-1}& =q^{-(\lambda ,{\alpha }_i)}{f}_i\\ \left[e_i,f_j\right]& ={\delta }_{ij}\frac{k_i-k_i^{-1}}{q_i-q_i^{-1}}& & k_i=k_{{\alpha }_i},q_i=q^{\frac{1}{2}({\alpha }_i,{\alpha }_i)}\end{array}}$

And for i ≠ j we have the q-Serre relations, which are deformations of the Serre relations:

${\displaystyle \begin{array}{rl}\sum _{n=0}^{1-a_{ij}}(-1{)}^n\frac{[1-a_{ij}{]}_{q_i}!}{[1-a_{ij}-n{]}_{q_i}![n{]}_{q_i}!}{e}_i^n{e}_j{e}_i^{1-a_{ij}-n}& =0\\ \sum _{n=0}^{1-a_{ij}}(-1{)}^n\frac{[1-a_{ij}{]}_{q_i}!}{[1-a_{ij}-n{]}_{q_i}![n{]}_{q_i}!}{f}_i^n{f}_j{f}_i^{1-a_{ij}-n}& =0\end{array}}$

where the q-factorial, the q-analog of the ordinary factorial, is defined recursively using q-number:

${\displaystyle \begin{array}{rl}{[0]}_{q_i}!& =1\\ {[n]}_{q_i}!& =\prod _{m=1}^n[m{]}_{q_i},& & [m{]}_{q_i}=\frac{q_i^m-q_i^{-m}}{q_i-q_i^{-1}}\end{array}}$

In the limit as q → 1, these relations approach the relations for the universal enveloping algebra U(G), where

${\displaystyle k_{\lambda }\to 1,\frac{k_{\lambda }-k_{-\lambda }}{q-q^{-1}}\to t_{\lambda }}$

and t_λ is the element of the Cartan subalgebra satisfying (t_λ, h) = λ(h) for all h in the Cartan subalgebra.

There are various coassociative coproducts under which these algebras are Hopf algebras, for example,

${\displaystyle \begin{array}{lll}{\mathrm{\Delta }}_1(k_{\lambda })=k_{\lambda }\otimes k_{\lambda }& {\mathrm{\Delta }}_1(e_i)=1\otimes e_i+e_i\otimes k_i& {\mathrm{\Delta }}_1(f_i)=k_i^{-1}\otimes f_i+f_i\otimes 1\\ {\mathrm{\Delta }}_2(k_{\lambda })=k_{\lambda }\otimes k_{\lambda }& {\mathrm{\Delta }}_2(e_i)=k_i^{-1}\otimes e_i+e_i\otimes 1& {\mathrm{\Delta }}_2(f_i)=1\otimes f_i+f_i\otimes k_i\\ {\mathrm{\Delta }}_3(k_{\lambda })=k_{\lambda }\otimes k_{\lambda }& {\mathrm{\Delta }}_3(e_i)=k_i^{-\frac{1}{2}}\otimes e_i+e_i\otimes k_i^{\frac{1}{2}}& {\mathrm{\Delta }}_3(f_i)=k_i^{-\frac{1}{2}}\otimes f_i+f_i\otimes k_i^{\frac{1}{2}}\end{array}}$

where the set of generators has been extended, if required, to include k_λ for λ which is expressible as the sum of an element of the weight lattice and half an element of the root lattice.

In addition, any Hopf algebra leads to another with reversed coproduct T o Δ, where T is given by T(x ⊗ y) = y ⊗ x, giving three more possible versions.

The counit on U_q(A) is the same for all these coproducts: ε(k_λ) = 1, ε(e_i) = ε(f_i) = 0, and the respective antipodes for the above coproducts are given by

${\displaystyle \begin{array}{lll}{S}_1(k_{\lambda })=k_{-\lambda }& S_1(e_i)=-e_i{k}_i^{-1}& S_1(f_i)=-k_i{f}_i\\ S_2(k_{\lambda })=k_{-\lambda }& S_2(e_i)=-k_i{e}_i& S_2(f_i)=-f_i{k}_i^{-1}\\ S_3(k_{\lambda })=k_{-\lambda }& S_3(e_i)=-q_i{e}_i& S_3(f_i)=-q_i^{-1}{f}_i\end{array}}$

Alternatively, the quantum group U_q(G) can be regarded as an algebra over the field C(q), the field of all rational functions of an indeterminate q over C.

Similarly, the quantum group U_q(G) can be regarded as an algebra over the field Q(q), the field of all rational functions of an indeterminate q over Q (see below in the section on quantum groups at q = 0). The center of quantum group can be described by quantum determinant.

Representation theory

Just as there are many different types of representations for Kac–Moody algebras and their universal enveloping algebras, so there are many different types of representation for quantum groups.

As is the case for all Hopf algebras, U_q(G) has an adjoint representation on itself as a module, with the action being given by

${\displaystyle {\mathrm{A}\mathrm{d}}_x\cdot y=\sum _{(x)}{x}_{(1)}yS(x_{(2)}),}$

where

${\displaystyle \mathrm{\Delta }(x)=\sum _{(x)}{x}_{(1)}\otimes x_{(2)}.}$

Case 1: q is not a root of unity

One important type of representation is a weight representation, and the corresponding module is called a weight module. A weight module is a module with a basis of weight vectors. A weight vector is a nonzero vector v such that k_λ · v = d_λv for all λ, where d_λ are complex numbers for all weights λ such that

${\displaystyle d_0=1,}$

${\displaystyle d_{\lambda }{d}_{\mu }=d_{\lambda +\mu },}$ for all weights λ and μ.

A weight module is called integrable if the actions of e_i and f_i are locally nilpotent (i.e. for any vector v in the module, there exists a positive integer k, possibly dependent on v, such that $e_i^k.v=f_i^k.v=0$ for all i). In the case of integrable modules, the complex numbers d_λ associated with a weight vector satisfy ${\displaystyle d_{\lambda }=c_{\lambda }{q}^{(\lambda ,\nu )}}$, where ν is an element of the weight lattice, and c_λ are complex numbers such that

• ${\displaystyle c_0=1,}$
• ${\displaystyle c_{\lambda }{c}_{\mu }=c_{\lambda +\mu },}$ for all weights λ and μ,
• $c_{2{\alpha }_i}=1$ for all i.

Of special interest are highest-weight representations, and the corresponding highest weight modules. A highest weight module is a module generated by a weight vector v, subject to k_λ · v = d_λv for all weights μ, and e_i · v = 0 for all i. Similarly, a quantum group can have a lowest weight representation and lowest weight module, i.e. a module generated by a weight vector v, subject to k_λ · v = d_λv for all weights λ, and f_i · v = 0 for all i.

Define a vector v to have weight ν if ${\displaystyle k_{\lambda }\cdot v=q^{(\lambda ,\nu )}v}$ for all λ in the weight lattice.

If G is a Kac–Moody algebra, then in any irreducible highest weight representation of U_q(G), with highest weight ν, the multiplicities of the weights are equal to their multiplicities in an irreducible representation of U(G) with equal highest weight. If the highest weight is dominant and integral (a weight μ is dominant and integral if μ satisfies the condition that $2(\mu ,{\alpha }_i)/({\alpha }_i,{\alpha }_i)$ is a non-negative integer for all i), then the weight spectrum of the irreducible representation is invariant under the Weyl group for G, and the representation is integrable.

Conversely, if a highest weight module is integrable, then its highest weight vector v satisfies ${\displaystyle k_{\lambda }\cdot v=c_{\lambda }{q}^{(\lambda ,\nu )}v}$, where c_λ · v = d_λv are complex numbers such that

• ${\displaystyle c_0=1,}$
• ${\displaystyle c_{\lambda }{c}_{\mu }=c_{\lambda +\mu },}$ for all weights λ and μ,
• $c_{2{\alpha }_i}=1$ for all i,

and ν is dominant and integral.

As is the case for all Hopf algebras, the tensor product of two modules is another module. For an element x of U_q(G), and for vectors v and w in the respective modules, x ⋅ (v ⊗ w) = Δ(x) ⋅ (v ⊗ w), so that ${\displaystyle k_{\lambda }\cdot (v\otimes w)=k_{\lambda }\cdot v\otimes k_{\lambda }.w}$, and in the case of coproduct Δ₁, ${\displaystyle e_i\cdot (v\otimes w)=k_i\cdot v\otimes e_i\cdot w+e_i\cdot v\otimes w}$ and ${\displaystyle f_i\cdot (v\otimes w)=v\otimes f_i\cdot w+f_i\cdot v\otimes k_i^{-1}\cdot w.}$

The integrable highest weight module described above is a tensor product of a one-dimensional module (on which k_λ = c_λ for all λ, and e_i = f_i = 0 for all i) and a highest weight module generated by a nonzero vector v₀, subject to ${\displaystyle k_{\lambda }\cdot v_0=q^{(\lambda ,\nu )}{v}_0}$ for all weights λ, and ${\displaystyle e_i\cdot v_0=0}$ for all i.

In the specific case where G is a finite-dimensional Lie algebra (as a special case of a Kac–Moody algebra), then the irreducible representations with dominant integral highest weights are also finite-dimensional.

In the case of a tensor product of highest weight modules, its decomposition into submodules is the same as for the tensor product of the corresponding modules of the Kac–Moody algebra (the highest weights are the same, as are their multiplicities).

Case 2: q is a root of unity

Quasitriangularity

Case 1: q is not a root of unity

Strictly, the quantum group U_q(G) is not quasitriangular, but it can be thought of as being "nearly quasitriangular" in that there exists an infinite formal sum which plays the role of an R-matrix. This infinite formal sum is expressible in terms of generators e_i and f_i, and Cartan generators t_λ, where k_λ is formally identified with q^t_λ. The infinite formal sum is the product of two factors:

$q^{\eta \sum _j{t}_{{\lambda }_j}\otimes t_{{\mu }_j}}$

and an infinite formal sum, where λ_j is a basis for the dual space to the Cartan subalgebra, and μ_j is the dual basis, and η = ±1.

The formal infinite sum which plays the part of the R-matrix has a well-defined action on the tensor product of two irreducible highest weight modules, and also on the tensor product of two lowest weight modules. Specifically, if v has weight α and w has weight β, then

${\displaystyle q^{\eta \sum _j{t}_{{\lambda }_j}\otimes t_{{\mu }_j}}\cdot (v\otimes w)=q^{\eta (\alpha ,\beta )}v\otimes w,}$

and the fact that the modules are both highest weight modules or both lowest weight modules reduces the action of the other factor on v ⊗ W to a finite sum.

Specifically, if V is a highest weight module, then the formal infinite sum, R, has a well-defined, and invertible, action on V ⊗ V, and this value of R (as an element of End(V ⊗ V)) satisfies the Yang–Baxter equation, and therefore allows us to determine a representation of the braid group, and to define quasi-invariants for knots, links and braids.

Case 2: q is a root of unity

Quantum groups at q = 0

Main article: Crystal base

Masaki Kashiwara has researched the limiting behaviour of quantum groups as q → 0, and found a particularly well behaved base called a crystal base.

Description and classification by root-systems and Dynkin diagrams

There has been considerable progress in describing finite quotients of quantum groups such as the above U_q(g) for qⁿ = 1; one usually considers the class of pointed Hopf algebras, meaning that all subcoideals are 1-dimensional and thus there sum form a group called coradical:

• In 2002 H.-J. Schneider and N. Andruskiewitsch finished their classification of pointed Hopf algebras with an abelian co-radical group (excluding primes 2, 3, 5, 7), especially as the above finite quotients of U_q(g) decompose into E′s (Borel part), dual F′s and K′s (Cartan algebra) just like ordinary Semisimple Lie algebras:

${\left(\mathfrak{B}(V)\otimes k[{\mathbf{Z}}^n]\otimes \mathfrak{B}(V^{\ast })\right)}^{\sigma }$

Here, as in the classical theory V is a braided vector space of dimension n spanned by the E′s, and σ (a so-called cocylce twist) creates the nontrivial linking between E′s and F′s. Note that in contrast to classical theory, more than two linked components may appear. The role of the quantum Borel algebra is taken by a Nichols algebra $\mathfrak{B}(V)$ of the braided vectorspace.

generalized Dynkin diagram for a pointed Hopf algebra linking four A3 copies

• A crucial ingredient was I. Heckenberger's classification of finite Nichols algebras for abelian groups in terms of generalized Dynkin diagrams. When small primes are present, some exotic examples, such as a triangle, occur (see also the Figure of a rank 3 Dankin diagram).

A rank 3 Dynkin diagram associated to a finite-dimensional Nichols algebra

• Meanwhile, Schneider and Heckenberger have generally proven the existence of an arithmetic root system also in the nonabelian case, generating a PBW basis as proven by Kharcheko in the abelian case (without the assumption on finite dimension). This can be used on specific cases U_q(g) and explains e.g. the numerical coincidence between certain coideal subalgebras of these quantum groups and the order of the Weyl group of the Lie algebra g.

Compact matrix quantum groups

Main article: Compact quantum group

S. L. Woronowicz introduced compact matrix quantum groups. Compact matrix quantum groups are abstract structures on which the "continuous functions" on the structure are given by elements of a C*-algebra. The geometry of a compact matrix quantum group is a special case of a noncommutative geometry.

The continuous complex-valued functions on a compact Hausdorff topological space form a commutative C*-algebra. By the Gelfand theorem, a commutative C*-algebra is isomorphic to the C*-algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C*-algebra up to homeomorphism.

For a compact topological group, G, there exists a C*-algebra homomorphism Δ: C(G) → C(G) ⊗ C(G) (where C(G) ⊗ C(G) is the C*-algebra tensor product - the completion of the algebraic tensor product of C(G) and C(G)), such that Δ(f)(x, y) = f(xy) for all f ∈ C(G), and for all x, y ∈ G (where (f ⊗ g)(x, y) = f(x)g(y) for all f, g ∈ C(G) and all x, y ∈ G). There also exists a linear multiplicative mapping κ: C(G) → C(G), such that κ(f)(x) = f(x⁻¹) for all f ∈ C(G) and all x ∈ G. Strictly, this does not make C(G) a Hopf algebra, unless G is finite. On the other hand, a finite-dimensional representation of G can be used to generate a *-subalgebra of C(G) which is also a Hopf *-algebra. Specifically, if $g\mapsto (u_{ij}(g){)}_{i,j}$ is an n-dimensional representation of G, then for all i, j u_ij ∈ C(G) and

$\mathrm{\Delta }(u_{ij})=\sum _k{u}_{ik}\otimes u_{kj}.$

It follows that the *-algebra generated by u_ij for all i, j and κ(u_ij) for all i, j is a Hopf *-algebra: the counit is determined by ε(u_ij) = δ_ij for all i, j (where δ_ij is the Kronecker delta), the antipode is κ, and the unit is given by

$1=\sum _k{u}_{1k}\kappa (u_{k1})=\sum _k\kappa (u_{1k})u_{k1}.$

General definition

As a generalization, a compact matrix quantum group is defined as a pair (C, u), where C is a C*-algebra and $u=(u_{ij}{)}_{i,j=1,\dots ,n}$ is a matrix with entries in C such that

• The *-subalgebra, C₀, of C, which is generated by the matrix elements of u, is dense in C;

• There exists a C*-algebra homomorphism called the comultiplication Δ: C → C ⊗ C (where C ⊗ C is the C*-algebra tensor product - the completion of the algebraic tensor product of C and C) such that for all i, j we have:

$\mathrm{\Delta }(u_{ij})=\sum _k{u}_{ik}\otimes u_{kj}$

• There exists a linear antimultiplicative map κ: C₀ → C₀ (the coinverse) such that κ(κ(v*)*) = v for all v ∈ C₀ and

$\sum _k\kappa (u_{ik})u_{kj}=\sum _k{u}_{ik}\kappa (u_{kj})={\delta }_{ij}I,$

where I is the identity element of C. Since κ is antimultiplicative, then κ(vw) = κ(w) κ(v) for all v, w in C₀.

As a consequence of continuity, the comultiplication on C is coassociative.

In general, C is not a bialgebra, and C₀ is a Hopf *-algebra.

Informally, C can be regarded as the *-algebra of continuous complex-valued functions over the compact matrix quantum group, and u can be regarded as a finite-dimensional representation of the compact matrix quantum group.

Representations

A representation of the compact matrix quantum group is given by a corepresentation of the Hopf *-algebra (a corepresentation of a counital coassociative coalgebra A is a square matrix $v=(v_{ij}{)}_{i,j=1,\dots ,n}$ with entries in A (so v belongs to M(n, A)) such that

$\mathrm{\Delta }(v_{ij})=\sum _{k=1}^n{v}_{ik}\otimes v_{kj}$

for all i, j and ε(v_ij) = δ_ij for all i, j). Furthermore, a representation v, is called unitary if the matrix for v is unitary (or equivalently, if κ(v_ij) = v*_ij for all i, j).

Example

An example of a compact matrix quantum group is SU_μ(2), where the parameter μ is a positive real number. So SU_μ(2) = (C(SU_μ(2)), u), where C(SU_μ(2)) is the C*-algebra generated by α and γ, subject to

$\gamma {\gamma }^{\ast }={\gamma }^{\ast }\gamma ,$

$\alpha \gamma =\mu \gamma \alpha ,$

$\alpha {\gamma }^{\ast }=\mu {\gamma }^{\ast }\alpha ,$

$\alpha {\alpha }^{\ast }+\mu {\gamma }^{\ast }\gamma ={\alpha }^{\ast }\alpha +{\mu }^{-1}{\gamma }^{\ast }\gamma =I,$

and

$u=\left(\begin{array}{cc}\alpha & \gamma \\ -{\gamma }^{\ast }& {\alpha }^{\ast }\end{array}\right),$

so that the comultiplication is determined by ∆(α) = α ⊗ α − γ ⊗ γ*, ∆(γ) = α ⊗ γ + γ ⊗ α*, and the coinverse is determined by κ(α) = α*, κ(γ) = −μ⁻¹γ, κ(γ*) = −μγ*, κ(α*) = α. Note that u is a representation, but not a unitary representation. u is equivalent to the unitary representation

$v=\left(\begin{array}{cc}\alpha & \sqrt{\mu }\gamma \\ -\frac{1}{\sqrt{\mu }}{\gamma }^{\ast }& {\alpha }^{\ast }\end{array}\right).$

Equivalently, SU_μ(2) = (C(SU_μ(2)), w), where C(SU_μ(2)) is the C*-algebra generated by α and β, subject to

$\beta {\beta }^{\ast }={\beta }^{\ast }\beta ,$

$\alpha \beta =\mu \beta \alpha ,$

$\alpha {\beta }^{\ast }=\mu {\beta }^{\ast }\alpha ,$

$\alpha {\alpha }^{\ast }+{\mu }^2{\beta }^{\ast }\beta ={\alpha }^{\ast }\alpha +{\beta }^{\ast }\beta =I,$

and

$w=\left(\begin{array}{cc}\alpha & \mu \beta \\ -{\beta }^{\ast }& {\alpha }^{\ast }\end{array}\right),$

so that the comultiplication is determined by ∆(α) = α ⊗ α − μβ ⊗ β*, Δ(β) = α ⊗ β + β ⊗ α*, and the coinverse is determined by κ(α) = α*, κ(β) = −μ⁻¹β, κ(β*) = −μβ*, κ(α*) = α. Note that w is a unitary representation. The realizations can be identified by equating $\gamma =\sqrt{\mu }\beta$.

When μ = 1, then SU_μ(2) is equal to the algebra C(SU(2)) of functions on the concrete compact group SU(2).

Bicrossproduct quantum groups

Whereas compact matrix pseudogroups are typically versions of Drinfeld-Jimbo quantum groups in a dual function algebra formulation, with additional structure, the bicrossproduct ones are a distinct second family of quantum groups of increasing importance as deformations of solvable rather than semisimple Lie groups. They are associated to Lie splittings of Lie algebras or local factorisations of Lie groups and can be viewed as the cross product or Mackey quantisation of one of the factors acting on the other for the algebra and a similar story for the coproduct Δ with the second factor acting back on the first.

The very simplest nontrivial example corresponds to two copies of R locally acting on each other and results in a quantum group (given here in an algebraic form) with generators p, K, K⁻¹, say, and coproduct

$[p,K]=hK(K-1)$

$\mathrm{\Delta }p=p\otimes K+1\otimes p$

$\mathrm{\Delta }K=K\otimes K$

where h is the deformation parameter.

This quantum group was linked to a toy model of Planck scale physics implementing Born reciprocity when viewed as a deformation of the Heisenberg algebra of quantum mechanics. Also, starting with any compact real form of a semisimple Lie algebra g its complexification as a real Lie algebra of twice the dimension splits into g and a certain solvable Lie algebra (the Iwasawa decomposition), and this provides a canonical bicrossproduct quantum group associated to g. For su(2) one obtains a quantum group deformation of the Euclidean group E(3) of motions in 3 dimensions.

Hilbert's axioms

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Hilbert's_axioms

Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie (tr. The Foundations of Geometry) as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff.

The axioms

Hilbert's axiom system is constructed with six primitive notions: three primitive terms:

• point;
• line;
• plane;

and three primitive relations:

• Betweenness, a ternary relation linking points;
• Lies on (Containment), three binary relations, one linking points and straight lines, one linking points and planes, and one linking straight lines and planes;
• Congruence, two binary relations, one linking line segments and one linking angles, each denoted by an infix ≅.

Line segments, angles, and triangles may each be defined in terms of points and straight lines, using the relations of betweenness and containment. All points, straight lines, and planes in the following axioms are distinct unless otherwise stated.

I. Incidence

1. For every two points A and B there exists a line a that contains them both. We write AB = a or BA = a. Instead of "contains", we may also employ other forms of expression; for example, we may say "A lies upon a", "A is a point of a", "a goes through A and through B", "a joins A to B", etc. If A lies upon a and at the same time upon another line b, we make use also of the expression: "The lines a and b have the point A in common", etc.
2. For every two points there exists no more than one line that contains them both; consequently, if AB = a and AC = a, where B ≠ C, then also BC = a.
3. There exist at least two points on a line. There exist at least three points that do not lie on the same line.
4. For every three points A, B, C not situated on the same line there exists a plane α that contains all of them. For every plane there exists a point which lies on it. We write ABC = α. We employ also the expressions: "A, B, C lie in α"; "A, B, C are points of α", etc.
5. For every three points A, B, C which do not lie in the same line, there exists no more than one plane that contains them all.
6. If two points A, B of a line a lie in a plane α, then every point of a lies in α. In this case we say: "The line a lies in the plane α", etc.
7. If two planes α, β have a point A in common, then they have at least a second point B in common.
8. There exist at least four points not lying in a plane.

II. Order

1. If a point B lies between points A and C, B is also between C and A, and there exists a line containing the distinct points A, B, C.
2. If A and C are two points, then there exists at least one point B on the line AC such that C lies between A and B.
3. Of any three points situated on a line, there is no more than one which lies between the other two.
4. Pasch's Axiom: Let A, B, C be three points not lying in the same line and let a be a line lying in the plane ABC and not passing through any of the points A, B, C. Then, if the line a passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC.

III. Congruence

1. If A, B are two points on a line a, and if A′ is a point upon the same or another line a′, then, upon a given side of A′ on the straight line a′, we can always find a point B′ so that the segment AB is congruent to the segment A′B′. We indicate this relation by writing AB ≅ A′B′. Every segment is congruent to itself; that is, we always have AB ≅ AB.
   We can state the above axiom briefly by saying that every segment can be laid off upon a given side of a given point of a given straight line in at least one way.
2. If a segment AB is congruent to the segment A′B′ and also to the segment A″B″, then the segment A′B′ is congruent to the segment A″B″; that is, if AB ≅ A′B′ and AB ≅ A″B″, then A′B′ ≅ A″B″.
3. Let AB and BC be two segments of a line a which have no points in common aside from the point B, and, furthermore, let A′B′ and B′C′ be two segments of the same or of another line a′ having, likewise, no point other than B′ in common. Then, if AB ≅ A′B′ and BC ≅ B′C′, we have AC ≅ A′C′.
4. Let an angle ∠ (h,k) be given in the plane α and let a line a′ be given in a plane α′. Suppose also that, in the plane α′, a definite side of the straight line a′ be assigned. Denote by h′ a ray of the straight line a′ emanating from a point O′ of this line. Then in the plane α′ there is one and only one ray k′ such that the angle ∠ (h, k), or ∠ (k, h), is congruent to the angle ∠ (h′, k′) and at the same time all interior points of the angle ∠ (h′, k′) lie upon the given side of a′. We express this relation by means of the notation ∠ (h, k) ≅ ∠ (h′, k′).
5. If the angle ∠ (h, k) is congruent to the angle ∠ (h′, k′) and to the angle ∠ (h″, k″), then the angle ∠ (h′, k′) is congruent to the angle ∠ (h″, k″); that is to say, if ∠ (h, k) ≅ ∠ (h′, k′) and ∠ (h, k) ≅ ∠ (h″, k″), then ∠ (h′, k′) ≅ ∠ (h″, k″).
6. If, in the two triangles ABC and A′B′C′ the congruences AB ≅ A′B′, AC ≅ A′C′, ∠BAC ≅ ∠B′A′C′ hold, then the congruence ∠ABC ≅ ∠A′B′C′ holds (and, by a change of notation, it follows that ∠ACB ≅ ∠A′C′B′ also holds).

IV. Parallels

1. Euclid's Axiom: Let a be any line and A a point not on it. Then there is at most one line in the plane, determined by a and A, that passes through A and does not intersect a.

V. Continuity

1. Axiom of Archimedes: If AB and CD are any segments then there exists a number n such that n segments CD constructed contiguously from A, along the ray from A through B, will pass beyond the point B.
2. Axiom of line completeness: An extension (An extended line from a line that already exists, usually used in geometry) of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-III and from V-1 is impossible.

Hilbert's discarded axiom

Hilbert (1899) included a 21st axiom that read as follows:

II.4. Any four points A, B, C, D of a line can always be labeled so that B shall lie between A and C and also between A and D, and, furthermore, that C shall lie between A and D and also between B and D.

E.H. Moore and R.L. Moore independently proved that this axiom is redundant, and the former published this result in an article appearing in the Transactions of the American Mathematical Society in 1902.

Before this, the axiom now listed as II.4. was numbered II.5.

Editions and translations of Grundlagen der Geometrie

The original monograph, based on his own lectures, was organized and written by Hilbert for a memorial address given in 1899. This was quickly followed by a French translation, in which Hilbert added V.2, the Completeness Axiom. An English translation, authorized by Hilbert, was made by E.J. Townsend and copyrighted in 1902. This translation incorporated the changes made in the French translation and so is considered to be a translation of the 2nd edition. Hilbert continued to make changes in the text and several editions appeared in German. The 7th edition was the last to appear in Hilbert's lifetime. In the Preface of this edition Hilbert wrote:

"The present Seventh Edition of my book Foundations of Geometry brings considerable improvements and additions to the previous edition, partly from my subsequent lectures on this subject and partly from improvements made in the meantime by other writers. The main text of the book has been revised accordingly."

New editions followed the 7th, but the main text was essentially not revised. The modifications in these editions occur in the appendices and in supplements. The changes in the text were large when compared to the original and a new English translation was commissioned by Open Court Publishers, who had published the Townsend translation. So, the 2nd English Edition was translated by Leo Unger from the 10th German edition in 1971. This translation incorporates several revisions and enlargements of the later German editions by Paul Bernays.

The Unger translation differs from the Townsend translation with respect to the axioms in the following ways:

• Old axiom II.4 is renamed as Theorem 5 and moved.
• Old axiom II.5 (Pasch's Axiom) is renumbered as II.4.
• V.2, the Axiom of Line Completeness, replaced:

Axiom of completeness. To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid.

• The old axiom V.2 is now Theorem 32.

The last two modifications are due to P. Bernays.

Other changes of note are:

• The term straight line used by Townsend has been replaced by line throughout.
• The Axioms of Incidence were called Axioms of Connection by Townsend.

Application

These axioms axiomatize Euclidean solid geometry. Removing five axioms mentioning "plane" in an essential way, namely I.4–8, and modifying III.4 and IV.1 to omit mention of planes, yields an axiomatization of Euclidean plane geometry.

Hilbert's axioms, unlike Tarski's axioms, do not constitute a first-order theory because the axioms V.1–2 cannot be expressed in first-order logic.

The value of Hilbert's Grundlagen was more methodological than substantive or pedagogical. Other major contributions to the axiomatics of geometry were those of Moritz Pasch, Mario Pieri, Oswald Veblen, Edward Vermilye Huntington, Gilbert Robinson, and Henry George Forder. The value of the Grundlagen is its pioneering approach to metamathematical questions, including the use of models to prove axioms independent; and the need to prove the consistency and completeness of an axiom system.

Mathematics in the twentieth century evolved into a network of axiomatic formal systems. This was, in considerable part, influenced by the example Hilbert set in the Grundlagen. A 2003 effort (Meikle and Fleuriot) to formalize the Grundlagen with a computer, though, found that some of Hilbert's proofs appear to rely on diagrams and geometric intuition, and as such revealed some potential ambiguities and omissions in his definitions.