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In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two
basis vector In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
s in the algebra as a linear combination. Given the structure constants, the resulting product is bilinear 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 Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, as the basis vectors indicate specific directions in physical space, or correspond to specific
particles In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...
. Recall that Lie algebras are algebras over a field, with the bilinear product being given by the
Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
or commutator.


Definition

Given a set of
basis vector In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
s \ for the underlying
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of the algebra, the structure constants or structure coefficients c_^ express the multiplication \cdot of pairs of vectors as a linear combination: :\mathbf_i \cdot \mathbf_j = \sum_ c_^ \mathbf_k. 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 In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
, on the algebra of the
indefinite orthogonal group In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature , where . It is also called the p ...
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 In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field (mathematics), field of scalar (mathematics), scalars (often, the real numbers or the complex numbers). ...
, i.e. are covariant under a
change of basis In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are consider ...
, 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 In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
; this is presented further down in the article, after some preliminary examples.


Example: Lie algebras

For a Lie algebra, the basis vectors are termed the generators of the algebra, and the product is given by the
Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
. That is, the algebra product \cdot is ''defined'' to be the Lie bracket: for two vectors A and B in the algebra, the product is A\cdot B\equiv ,B In particular, the algebra product \cdot ''must not'' be confused with a matrix product, and thus sometimes requires an alternate notation. There is no particular need to distinguish the upper and lower indices in this case; they can be written all up or all down. In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, it is common to use the notation T_i for the generators, and f_^ or f^ (ignoring the upper-lower distinction) for the structure constants. The Lie bracket of pairs of generators is a linear combination of generators from the set, i.e. : _a, T_b= \sum_ f_^ T_c. 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 In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
. For the basis vectors, it can be written as : _a,_[T_b,T_c_+_[T_b,_[T_c,_T_a.html" ;"title="_b,T_c.html" ;"title="_a, [T_b,T_c">_a, [T_b,T_c + [T_b, [T_c, T_a">_b,T_c.html" ;"title="_a, [T_b,T_c">_a, [T_b,T_c + [T_b, [T_c, T_a + [T_c, [T_a, T_b = 0 and this leads directly to a corresponding identity in terms of the structure constants: :f_^f_^ + f_^f_^ + f_^f_^ = 0. 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 In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
. The Killing form and the
Casimir invariant In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operato ...
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 X, Y of the Lie algebra, the structure of the Lie group near the identity element is given by :\exp(X)\exp(Y) \approx \exp(X + Y + \tfrac ,Y. Note the factor of 1/2. They also appear in explicit expressions for differentials, such as e^de^X; see Baker–Campbell–Hausdorff formula#Infinitesimal case for details.


Lie algebra examples


𝔰𝔲(2) and 𝔰𝔬(3)

The algebra \mathfrak(2) of the
special unitary group In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
SU(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
is three-dimensional, with generators given by the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
\sigma_i. The generators of the group SU(2) satisfy the commutation relations (where \varepsilon^ is the
Levi-Civita symbol In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the parity of a permutation, sign of a permutation of the n ...
): sigma_a, \sigma_b= 2 i \varepsilon^ \sigma_c where \sigma_1 = \begin 0 & 1 \\ 1 & 0 \end,~~ \sigma_2 = \begin 0 & -i \\ i & 0 \end,~~ \sigma_3 = \begin 1 & 0 \\ 0 & -1 \end In this case, the structure constants are f^ = 2 i \varepsilon^. Note that the constant 2''i'' can be absorbed into the definition of the basis vectors; thus, defining t_a = -i\sigma_a/2, one can equally well write _a, t_b= \varepsilon^ t_c Doing so emphasizes that the Lie algebra \mathfrak(2) of the Lie group SU(2) is isomorphic to the Lie algebra \mathfrak(3) of
SO(3) 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 \R^3 under the operation of composition. By definition, a rotation about the origin is a tr ...
. This brings the structure constants into line with those of the
rotation group SO(3) 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 \R^3 under the operation of composition. By definition, a rotation about the origin is a ...
. That is, the commutator for the
angular momentum operator In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum prob ...
s are then commonly written as _i, L_j= \varepsilon^ L_k where L_x = L_1 = \begin 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end,~~ L_y = L_2 = \begin 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end,~~ L_z = L_3 = \begin 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end are written so as to obey the right hand rule for rotations in 3-dimensional space. The difference of the factor of 2''i'' 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 representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group or Lie algebra whose highest weig ...
s of \mathfrak(2), 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. In the case of three dimensions, there is only one three-dimensional representation, the
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
, which is a real representation; more precisely, it is the same as its
dual representation In mathematics, if is a group and is a linear representation of it on the vector space , then the dual representation is defined over the dual vector space as follows: : is the transpose of , that is, = for all . The dual representation ...
, shown above. That is, one has that the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
is minus itself: L_k^T = -L_k. 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.


𝔰𝔲(3)

A less trivial example is given by
SU(3) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the specia ...
: Its generators, ''T'', in the defining representation, are: :T^a = \frac.\, where \lambda \,, the
Gell-Mann matrices 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 t ...
, are the SU(3) analog of the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
for SU(2): : These obey the relations :\left ^a, T^b \right= i f^ T^c \, : \ = \frac\delta^ + d^ T^c. \, The structure constants are totally antisymmetric. They are given by: :f^ = 1 \, :f^ = -f^ = f^ = f^ = f^ = -f^ = \frac \, :f^ = f^ = \frac, \, and all other f^ not related to these by permuting indices are zero. The ''d'' take the values: :d^ = d^ = d^ = -d^ = \frac \, :d^ = d^ = d^ = d^ = -\frac \, :d^ = d^ = -d^ = d^ = d^ = d^ = -d^ = -d^ = \frac. \,


𝔰𝔲(N)

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 N^-1 generators of 𝔰𝔲(N), based on a generalisation of the Pauli matrices and of the Gell-Mann matrices (using the bra-ket notation). There are N(N-1)/2 symmetric matrices, :\hat_=\frac(, m\rangle\langle n, +, n\rangle\langle m, ), N(N-1)/2 anti-symmetric matrices, :\hat_=-i\frac(, m\rangle\langle n, -, n\rangle\langle m, ), and N-1 diagonal matrices, :\hat_=\frac\Big(\sum_^, l\rangle\langle l, +(1-n), n\rangle\langle n, )\Big). To differenciate those matrices we define the following indices: :\alpha_=n^2+2(m-n)-1, :\beta_=n^2+2(m-n), :\gamma_=n^2-1, with the condition 1\leq m. All the non-zero totally anti-symmetric structure constants are :f^=f^=f^=\frac, :f^=\frac, :f^=-\sqrt,~f^=\sqrt, :f^=\sqrt,~m. All the non-zero totally symmetric structure constants are :d^=d^=d^=\frac, :d^=-\frac, :d^=d^=-\sqrt, :d^=d^=\sqrt,~m, :d^=d^=\frac, :d^=d^=\sqrt,~n, :d^=\sqrt,~k, :d^=(2-n)\sqrt. For more details on the derivation see and.


Examples from other algebras


Hall polynomials

The Hall polynomials are the structure constants of the
Hall algebra In mathematics, the Hall algebra is an associative algebra with a basis corresponding to isomorphism classes of finite abelian p-group, ''p''-groups. It was first discussed by but forgotten until it was rediscovered by , both of whom published no ...
.


Hopf algebras

In addition to the product, the
coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduc ...
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.


Applications

*A Lie group is abelian exactly when all structure constants are 0. *A Lie group is
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
exactly when its structure constants are real. *The structure constants are completely anti-symmetric in all indices if and only if the Lie algebra is a direct sum of
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
compact Lie algebra In the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algebr ...
s. *A
nilpotent Lie group In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with . Intuit ...
admits a lattice if and only if its Lie algebra admits a basis with rational structure constants: this is Malcev's criterion. Not all nilpotent Lie groups admit lattices; for more details, see also Raghunathan. *In quantum chromodynamics, the symbol G^a_ \, represents the gauge covariant gluon field strength tensor, analogous to the electromagnetic field strength tensor, ''F''μν, in
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
. It is given by: G^a_ = \partial_\mu \mathcal^a_\nu - \partial_\nu \mathcal^a_\mu + g f^ \mathcal^b_\mu \mathcal^c_\nu \,, where ''fabc'' are the structure constants of SU(3). Note that the rules to push-up or pull-down the ''a'', ''b'', or ''c'' indexes are ''trivial'', (+,... +), so that ''fabc'' = ''fabc'' = ''f'' whereas for the ''μ'' or ''ν'' indexes one has the non-trivial ''relativistic'' rules, corresponding e.g. to the
metric signature In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and ...
(+ − − −).


Choosing a basis for a Lie algebra

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 In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
. 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 In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...
. Given a Lie algebra \mathfrak, the Cartan subalgebra \mathfrak\subset\mathfrak is the maximal Abelian subalgebra. By definition, it consists of those elements that commute with one-another. An
orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
basis can be freely chosen on \mathfrak; write this basis as H_1,\cdots, H_r with :\langle H_i,H_j\rangle=\delta_ where \langle \cdot,\cdot\rangle is the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on the vector space. The dimension r of this subalgebra is called the rank of the algebra. In the
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
, the matrices \mathrm(H_i) are mutually commuting, and can be simultaneously diagonalized. The matrices \mathrm(H_i) have (simultaneous)
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s; those with a non-zero eigenvalue \alpha are conventionally denoted by E_\alpha. Together with the H_i these span the entire vector space \mathfrak. The commutation relations are then : _i,H_j0 \quad \mbox \quad _i, E_\alpha\alpha_i E_\alpha The eigenvectors E_\alpha are determined only up to overall scale; one conventional normalization is to set :\langle E_\alpha,E_\rangle=1 This allows the remaining commutation relations to be written as : _\alpha,E_\alpha_i H_i and : _\alpha,E_\betaN_E_ with this last subject to the condition that the roots (defined below) \alpha,\beta sum to a non-zero value: \alpha+\beta\ne 0. The E_\alpha are sometimes called ladder operators, as they have this property of raising/lowering the value of \beta. For a given \alpha, there are as many \alpha_i as there are H_i and so one may define the vector \alpha=\alpha_iH_i, this vector is termed a
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
of the algebra. The roots of Lie algebras appear in regular structures (for example, in
simple Lie algebra In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan. A direct sum of s ...
s, the roots can have only two different lengths); see
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
for details. The structure constants N_ have the property that they are non-zero only when \alpha+\beta are a root. In addition, they are antisymmetric: :N_=-N_ and can always be chosen such that :N_=-N_ They also obey cocycle conditions: :N_=N_=N_ whenever \alpha+\beta+\gamma=0, and also that :N_N_ + N_N_ + N_N_ = 0 whenever \alpha+\beta+\gamma+\delta=0.


References

{{reflist Lie algebras