In mathematics, the structure constants or structure coefficients of an algebra over a field are the coefficients of the basis expansion (into linear combination of basis vectors) of the products of basis vectors. Because the product operation in the algebra is bilinear, by linearity knowing the product of basis vectors allows to compute the product of any elements (just like a matrix allows to compute the action of the linear operator on any vector by providing the action of the operator on basis vectors). Therefore, the structure constants can be used to specify the product operation of the algebra (just like a matrix defines a linear operator). Given the structure constants, the resulting product is obtained by bilinearity and can be uniquely extended to all vectors in the vector space, thus uniquely determining the product for the algebra.
Structure constants are used whenever an explicit form for the algebra must be given. Thus, they are frequently used when discussing Lie algebras in physics, as the basis vectors indicate specific directions in physical space, or correspond to specific particles (recall that Lie algebras are algebras over a field, with the bilinear product being given by the Lie bracket, usually defined via the commutator).
Given a set of basis vectors for the underlying vector space of the algebra, the product operation is uniquely defined by the products of basis vectors:
The structure constants or structure coefficients are just the coefficients of in the same basis:
Otherwise said they are the coefficients that express as linear combination of the basis vectors .
The upper and lower indices are frequently not distinguished, unless the algebra is endowed with some other structure that would require this (for example, a pseudo-Riemannian metric, on the algebra of the indefinite orthogonal group so(p,q)). That is, structure constants are often written with all-upper, or all-lower indexes. The distinction between upper and lower is then a convention, reminding the reader that lower indices behave like the components of a dual vector, i.e. are covariant under a change of basis, while upper indices are contravariant.
The structure constants obviously depend on the chosen basis. For Lie algebras, one frequently used convention for the basis is in terms of the ladder operators defined by the Cartan subalgebra; this is presented further down in the article, after some preliminary examples.
For a Lie algebra, the basis vectors are termed the generators of the algebra, and the product rather called the Lie bracket (often the Lie bracket is an additional product operation beyond the already existing product, thus necessitating a separate name). For two vectors and in the algebra, the Lie bracket is denoted .
Again, there is no particular need to distinguish the upper and lower indices; they can be written all up or all down. In physics, it is common to use the notation for the generators, and or (ignoring the upper-lower distinction) for the structure constants. The linear expansion of the Lie bracket of pairs of generators then looks like
Again, by linear extension, the structure constants completely determine the Lie brackets of all elements of the Lie algebra.
All Lie algebras satisfy the Jacobi identity. For the basis vectors, it can be written as
and this leads directly to a corresponding identity in terms of the structure constants:
The above, and the remainder of this article, make use of the Einstein summation convention for repeated indexes.
The structure constants play a role in Lie algebra representations, and in fact, give exactly the matrix elements of the adjoint representation. The Killing form and the Casimir invariant also have a particularly simple form, when written in terms of the structure constants.
The structure constants often make an appearance in the approximation to the Baker–Campbell–Hausdorff formula for the product of two elements of a Lie group. For small elements of the Lie algebra, the structure of the Lie group near the identity element is given by
Note the factor of 1/2. They also appear in explicit expressions for differentials, such as ; see Baker–Campbell–Hausdorff formula#Infinitesimal case for details.
The algebra of the special unitary group SU(2) is three-dimensional, with generators given by the Pauli matrices . The generators of the group SU(2) satisfy the commutation relations (where is the Levi-Civita symbol): where
In this case, the structure constants are . Note that the constant 2i can be absorbed into the definition of the basis vectors; thus, defining , one can equally well write
Doing so emphasizes that the Lie algebra of the Lie group SU(2) is isomorphic to the Lie algebra of SO(3). This brings the structure constants into line with those of the rotation group SO(3). That is, the commutator for the angular momentum operators are then commonly written as where are written so as to obey the right hand rule for rotations in 3-dimensional space.
The difference of the factor of 2i between these two sets of structure constants can be infuriating, as it involves some subtlety. Thus, for example, the two-dimensional complex vector space can be given a real structure. This leads to two inequivalent two-dimensional fundamental representations of , which are isomorphic, but are complex conjugate representations; both, however, are considered to be real representations, precisely because they act on a space with a real structure. [1] In the case of three dimensions, there is only one three-dimensional representation, the adjoint representation, which is a real representation; more precisely, it is the same as its dual representation, shown above. That is, one has that the transpose is minus itself:
In any case, the Lie groups are considered to be real, precisely because it is possible to write the structure constants so that they are purely real.
A less trivial example is given by SU(3): [2]
Its generators, T, in the defining representation, are:
where , the Gell-Mann matrices, are the SU(3) analog of the Pauli matrices for SU(2):
These obey the relations
The structure constants are totally antisymmetric. They are given by:
and all other not related to these by permuting indices are zero.
The d take the values:
For the general case of 𝔰𝔲(N), there exists closed formula to obtain the structure constant, without having to compute commutation and anti-commutation relations between the generators. We first define the generators of 𝔰𝔲(N), based on a generalisation of the Pauli matrices and of the Gell-Mann matrices (using the bra-ket notation where is the matrix unit). There are symmetric matrices,
anti-symmetric matrices,
and diagonal matrices,
To differenciate those matrices we define the following indices:
with the condition .
All the non-zero totally anti-symmetric structure constants are
All the non-zero totally symmetric structure constants are
The Hall polynomials are the structure constants of the Hall algebra.
In addition to the product, the coproduct and the antipode of a Hopf algebra can be expressed in terms of structure constants. The connecting axiom, which defines a consistency condition on the Hopf algebra, can be expressed as a relation between these various structure constants.
One conventional approach to providing a basis for a Lie algebra is by means of the so-called "ladder operators" appearing as eigenvectors of the Cartan subalgebra. The construction of this basis, using conventional notation, is quickly sketched here. An alternative construction (the Serre construction) can be found in the article semisimple Lie algebra.
Given a Lie algebra , the Cartan subalgebra is the maximal Abelian subalgebra. By definition, it consists of those elements that commute with one-another. An orthonormal basis can be freely chosen on ; write this basis as with
where is the inner product on the vector space. The dimension of this subalgebra is called the rank of the algebra. In the adjoint representation, the matrices are mutually commuting, and can be simultaneously diagonalized. The matrices have (simultaneous) eigenvectors; those with a non-zero eigenvalue are conventionally denoted by . Together with the these span the entire vector space . The commutation relations are then
The eigenvectors are determined only up to overall scale; one conventional normalization is to set
This allows the remaining commutation relations to be written as
and
with this last subject to the condition that the roots (defined below) sum to a non-zero value: . The are sometimes called ladder operators, as they have this property of raising/lowering the value of .
For a given , there are as many as there are and so one may define the vector , this vector is termed a root of the algebra. The roots of Lie algebras appear in regular structures (for example, in simple Lie algebras, the roots can have only two different lengths); see root system for details.
The structure constants have the property that they are non-zero only when are a root. In addition, they are antisymmetric:
and can always be chosen such that
They also obey cocycle conditions: [7]
whenever , and also that
whenever .
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition.
In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
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, compact matrix quantum groups, 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 Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics. They span the Lie algebra of the SU(3) group in the defining representation.
In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also interact with matrix multiplication.
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written concisely, and allows for the quantization of the electromagnetic field by the Lagrangian formulation described below.
In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.
The principle of detailed balance can be used in kinetic systems which are decomposed into elementary processes. It states that at equilibrium, each elementary process is in equilibrium with its reverse process.
The Newman–Penrose (NP) formalism is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members often asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.
The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanical theory of angular momentum. The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The letter D stands for Darstellung, which means "representation" in German.
In mathematics, the Bianchi classification provides a list of all real 3-dimensional Lie algebras. The classification contains 11 classes, 9 of which contain a single Lie algebra and two of which contain a continuum-sized family of Lie algebras. The classification is important in geometry and physics, because the associated Lie groups serve as symmetry groups of 3-dimensional Riemannian manifolds. It is named for Luigi Bianchi, who worked it out in 1898.
There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.
In continuum mechanics, a compatible deformation tensor field in a body is that unique tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. Compatibility is the study of the conditions under which such a displacement field can be guaranteed. Compatibility conditions are particular cases of integrability conditions and were first derived for linear elasticity by Barré de Saint-Venant in 1864 and proved rigorously by Beltrami in 1886.
In mathematics, the Kodaira–Spencer map, introduced by Kunihiko Kodaira and Donald C. Spencer, is a map associated to a deformation of a scheme or complex manifold X, taking a tangent space of a point of the deformation space to the first cohomology group of the sheaf of vector fields on X.
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be called the absolute differential calculus, developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early twentieth century.
In theoretical particle physics, the gluon field strength tensor is a second order tensor field characterizing the gluon interaction between quarks.
In physics, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the three-dimensional quantity in classical mechanics.
In mathematical physics, nonlinear realization of a Lie group G possessing a Cartan subgroup H is a particular induced representation of G. In fact, it is a representation of a Lie algebra of G in a neighborhood of its origin. A nonlinear realization, when restricted to the subgroup H reduces to a linear representation.
This is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory. Because of the lack of other options, the glossary also includes some generalizations such as quantum group.