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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a Lie algebra (pronounced "Lee") is a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
\mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, \ (x, y) \mapsto
, y The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
/math>, that satisfies the
Jacobi identity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. The vector space \mathfrak g together with this operation is a
non-associative algebra A non-associative algebra (or distributive algebra) is an algebra over a field where the binary operation, binary multiplication operation is not assumed to be associative operation, associative. That is, an algebraic structure ''A'' is a non-ass ...
, meaning that the Lie bracket is not necessarily
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. Lie algebras are closely related to
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s, which are
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
s that are also
smooth manifolds In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an ...
: any Lie group gives rise to a Lie algebra, which is its
tangent space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (
Lie's third theoremIn the mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
). This correspondence allows one to study the structure and
classification Classification is a process related to categorization Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience Experience refers to conscious , an English Paracels ...
of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
and particle physics. An elementary example is the space of three dimensional vectors \mathfrak=\mathbb^3 with the bracket operation defined by the
cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

cross product
,yx\times y. This is skew-symmetric since x\times y = -y\times x, and instead of associativity it satisfies the Jacobi identity: : x\times(y\times z) \ =\ (x\times y)\times z \ +\ y\times(x\times z). This is the Lie algebra of the Lie group of rotations of space, and each vector v\in\R^3 may be pictured as an infinitesimal rotation around the axis v, with velocity equal to the magnitude of v. The Lie bracket is a measure of the non-commutativity between two rotations: since a rotation commutes with itself, we have the alternating property ,xx\times x = 0.


History

Lie algebras were introduced to study the concept of
infinitesimal transformation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s by
Marius Sophus Lie
Marius Sophus Lie
in the 1870s, and independently discovered by
Wilhelm Killing Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of ...
in the 1880s. The name ''Lie algebra'' was given by
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens o ...

Hermann Weyl
in the 1930s; in older texts, the term ''infinitesimal group'' is used.


Definitions


Definition of a Lie algebra

A Lie algebra is a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
\,\mathfrak over some
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
F together with a
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
,\cdot\,,\cdot\, \mathfrak\times\mathfrak\to\mathfrak called the Lie bracket satisfying the following axioms: *
Bilinearity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
, :: x + b y, z= a
, z The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
+ b
, z The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
:: , a x + b y= a
, x The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
+ b
, y The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
:for all scalars a, b in F and all elements x, ''y'', ''z'' in \mathfrak. *
Alternativity In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
, :: ,x0\ :for all x in \mathfrak. * The
Jacobi identity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, :: ,[y,z_+_[y,[z,x.html"_;"title=",z.html"_;"title=",[y,z">,[y,z_+_[y,[z,x">,z.html"_;"title=",[y,z">,[y,z_+_[y,[z,x_+_[z,[x,y.html" ;"title=",z">,[y,z_+_[y,[z,x.html" ;"title=",z.html" ;"title=",[y,z">,[y,z + [y,[z,x">,z.html" ;"title=",[y,z">,[y,z + [y,[z,x + [z,[x,y">,z">,[y,z_+_[y,[z,x.html" ;"title=",z.html" ;"title=",[y,z">,[y,z + [y,[z,x">,z.html" ;"title=",[y,z">,[y,z + [y,[z,x + [z,[x,y = 0 \ :for all x, ''y'', ''z'' in \mathfrak. Using bilinearity to expand the Lie bracket [x+y,x+y] and using alternativity shows that ,y+ [y,x]=0\ for all elements x, ''y'' in \mathfrak, showing that bilinearity and alternativity together imply *
Anticommutativity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, :: ,y= - ,x\ :for all elements x, ''y'' in \mathfrak. If the field's
characteristic Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
is not 2 then anticommutativity implies alternativity, since it implies ,x- ,x It is customary to denote a Lie algebra by a lower-case
fraktur Fraktur () is a calligraphic hand of the Latin alphabet and any of several blackletter typefaces derived from this hand. The blackletter lines are broken up; that is, their forms contain many angles when compared to the curves of the Antiqua ...

fraktur
letter such as \mathfrak. If a Lie algebra is associated with a
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, then the algebra is denoted by the fraktur version of the group: for example the Lie algebra of SU(''n'') is \mathfrak(n).


Generators and dimension

Elements of a Lie algebra \mathfrak are said to generate it if the smallest subalgebra containing these elements is \mathfrak itself. The ''dimension'' of a Lie algebra is its dimension as a vector space over ''F''. The cardinality of a minimal generating set of a Lie algebra is always less than or equal to its dimension. See the classification of low-dimensional real Lie algebras for other small examples.


Subalgebras, ideals and homomorphisms

The Lie bracket is not required to be
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, meaning that x,yz] need not equal ,[y,z._However,_it_is_flexible_algebra, flexible.html" ;"title=",z.html" ;"title=",[y,z">,[y,z. However, it is flexible algebra, flexible">,z.html" ;"title=",[y,z">,[y,z. However, it is flexible algebra, flexible. Nonetheless, much of the terminology of associative ring (mathematics), rings and associative algebra, algebras is commonly applied to Lie algebras. A ''Lie subalgebra'' is a subspace \mathfrak \subseteq \mathfrak which is closed under the Lie bracket. An ''ideal'' \mathfrak i\subseteq\mathfrak is a subalgebra satisfying the stronger condition: : mathfrak,\mathfrak isubseteq \mathfrak i. A Lie algebra ''homomorphism'' is a linear map compatible with the respective Lie brackets: : \phi: \mathfrak\to\mathfrak, \quad \phi( ,y= phi(x),\phi(y)\ \text\ x,y \in \mathfrak g. As for associative rings, ideals are precisely the kernels of homomorphisms; given a Lie algebra \mathfrak and an ideal \mathfrak i in it, one constructs the ''factor algebra'' or ''quotient algebra'' \mathfrak/\mathfrak i, and the
first isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between Quotient (universal algebra), quotients, homomorphisms, and subobjects. Vers ...

first isomorphism theorem
holds for Lie algebras. Since the Lie bracket is a kind of infinitesimal
commutator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of the corresponding Lie group, we say that two elements x,y\in\mathfrak g ''commute'' if their bracket vanishes: ,y0. The
centralizer In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
subalgebra of a subset S\subset \mathfrak is the set of elements commuting with ''S'': that is, \mathfrak_(S) = \. The centralizer of \mathfrak itself is the ''center'' \mathfrak(\mathfrak). Similarly, for a subspace ''S'', the
normalizer In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
subalgebra of ''S'' is \mathfrak_(S) = \. Equivalently, if S is a Lie subalgebra, \mathfrak_(S) is the largest subalgebra such that S is an ideal of \mathfrak_(S).


Examples

For \mathfrak(2) \subset \mathfrak(2), the commutator of two elements
\begin \left \begin a & b \\ c & d \end, \begin x & 0 \\ 0 & y \end \right&= \begin ax & by\\ cx & dy \\ \end - \begin ax & bx\\ cy & dy \\ \end \\ &= \begin 0 & b(y-x) \\ c(x-y) & 0 \end \end
shows \mathfrak(2) is a subalgebra, but not an ideal. In fact, every one-dimensional linear subspace of a Lie algebra has an induced abelian Lie algebra structure, which is generally not an ideal. For any simple Lie algebra, all abelian Lie algebras can never be ideals.


Direct sum and semidirect product

For two Lie algebras \mathfrak and \mathfrak, their
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
Lie algebra is the vector space \mathfrak\oplus\mathfrakconsisting of all pairs \mathfrak(x,x'), \,x\in\mathfrak, \ x'\in\mathfrak, with the operation : x,x'),(y,y')( ,y ',y', so that the copies of \mathfrak g, \mathfrak g' commute with each other: x,0), (0,x')= 0. Let \mathfrak be a Lie algebra and \mathfrak an ideal of \mathfrak. If the canonical map \mathfrak \to \mathfrak/\mathfrak splits (i.e., admits a section), then \mathfrak is said to be a
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product of groups, direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product ...
of \mathfrak and \mathfrak/\mathfrak, \mathfrak=\mathfrak/\mathfrak\ltimes\mathfrak. See also semidirect sum of Lie algebras.
Levi's theorem In Lie theory In mathematics, the researcher Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact (mathematics), contact of spheres that have come to be called Lie theory. For ...
says that a finite-dimensional Lie algebra is a semidirect product of its radical and the complementary subalgebra ( Levi subalgebra).


Derivations

A ''derivation'' on the Lie algebra \mathfrak (or on any
non-associative algebra A non-associative algebra (or distributive algebra) is an algebra over a field where the binary operation, binary multiplication operation is not assumed to be associative operation, associative. That is, an algebraic structure ''A'' is a non-ass ...
) is a
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

linear map
\delta\colon\mathfrak\rightarrow \mathfrak that obeys the Leibniz law, that is, :\delta ( ,y = delta(x),y+ , \delta(y)/math> for all x,y\in\mathfrak g. The ''inner derivation'' associated to any x\in\mathfrak g is the adjoint mapping \mathrm_x defined by \mathrm_x(y):= ,y/math>. (This is a derivation as a consequence of the Jacobi identity.) The outer derivations are derivations which do not come from the adjoint representation of the Lie algebra. If \mathfrak is semisimple, every derivation is inner. The derivations form a vector space \mathrm(\mathfrak g), which is a Lie subalgebra of \mathfrak(\mathfrak); the bracket is commutator. The inner derivations form a Lie subalgebra of \mathrm(\mathfrak g).


Examples

For example, given a Lie algebra ideal \mathfrak \subset \mathfrak the adjoint representation \mathfrak_\mathfrak of \mathfrak acts as outer derivations on \mathfrak since ,i\subset \mathfrak for any x \in \mathfrak and i \in \mathfrak. For the Lie algebra \mathfrak_n of upper triangular matrices in \mathfrak(n), it has an ideal \mathfrak_n of strictly upper triangular matrices (where the only non-zero elements are above the diagonal of the matrix). For instance, the commutator of elements in \mathfrak_3 and \mathfrak_3 gives
\begin \left \begin a & b & c \\ 0 & d & e \\ 0 & 0 & f \end, \begin 0 & x & y \\ 0 & 0 & z \\ 0 & 0 & 0 \end \right&= \begin 0 & ax & ay+bz \\ 0 & 0 & dz \\ 0 & 0 & 0 \end - \begin 0 & dx & ex+yf \\ 0 & 0 & fz \\ 0 & 0 & 0 \end \\ &= \begin 0 & (a-d)x & (a-f)y-ex+bz \\ 0 & 0 & (d-f)z \\ 0 & 0 & 0 \end \end
shows there exist outer derivations from \mathfrak_3 in \text(\mathfrak_3).


Split Lie algebra

Let ''V'' be a finite-dimensional vector space over a field ''F'', \mathfrak(V) the Lie algebra of linear transformations and \mathfrak \subseteq \mathfrak(V) a Lie subalgebra. Then \mathfrak is said to be split if the roots of the characteristic polynomials of all linear transformations in \mathfrak are in the base field ''F''. More generally, a finite-dimensional Lie algebra \mathfrak is said to be split if it has a Cartan subalgebra whose image under 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 map, linear transformations of the group's Lie algebra, considered as a vector space. For example, if ' ...
\operatorname: \mathfrak \to \mathfrak(\mathfrak g) is a split Lie algebra. A
split real form In mathematics, the notion of a real form relates objects defined over the Field (algebra), field of Real number, real and Complex number, complex numbers. A real Lie algebra ''g''0 is called a real form of a complex Lie algebra ''g'' if ''g'' is t ...
of a complex semisimple Lie algebra (cf. #Real form and complexification) is an example of a split real Lie algebra. See also
split Lie algebra In the mathematical field of Lie theory, a split Lie algebra is a pair (\mathfrak, \mathfrak) where \mathfrak is a Lie algebra and \mathfrak < \mathfrak is a splitting Cartan subalgebra, where "splitting" means that for all x \in \mat ...
for further information.


Vector space basis

For practical calculations, it is often convenient to choose an explicit
vector space basis In mathematics, a Set (mathematics), 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 referre ...
for the algebra. A common construction for this basis is sketched in the article
structure constant In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting prod ...
s.


Definition using category-theoretic notation

Although the definitions above are sufficient for a conventional understanding of Lie algebras, once this is understood, additional insight can be gained by using notation common to
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, that is, by defining a Lie algebra in terms of
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

linear map
s—that is,
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

morphism
s of the
category of vector spaces In Abstract algebra, algebra, given a Ring (mathematics), ring ''R'', the category of left modules over ''R'' is the Category (mathematics), category whose Object (category theory), objects are all left Module (mathematics), modules over ''R'' and w ...
—without considering individual elements. (In this section, the
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
over which the algebra is defined is supposed to be of
characteristic Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
different from two.) For the category-theoretic definition of Lie algebras, two braiding isomorphisms are needed. If is a vector space, the ''interchange isomorphism'' \tau: A\otimes A \to A\otimes A is defined by :\tau(x\otimes y)= y\otimes x. The ''cyclic-permutation braiding'' \sigma:A\otimes A\otimes A \to A\otimes A\otimes A is defined as :\sigma=(\mathrm\otimes \tau)\circ(\tau\otimes \mathrm), where \mathrm is the identity morphism. Equivalently, \sigma is defined by :\sigma(x\otimes y\otimes z)= y\otimes z\otimes x. With this notation, a Lie algebra can be defined as an
object Object may refer to: General meanings * Object (philosophy) An object is a philosophy, philosophical term often used in contrast to the term ''Subject (philosophy), subject''. A subject is an observer and an object is a thing observed. For mo ...
A in the category of vector spaces together with a
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

morphism
: cdot,\cdotA\otimes A\rightarrow A that satisfies the two morphism equalities : cdot,\cdotcirc(\mathrm+\tau)=0, and : cdot,\cdotcirc ( cdot,\cdototimes \mathrm) \circ (\mathrm +\sigma+\sigma^2)=0.


Examples


Vector spaces

Any vector space V endowed with the identically zero Lie bracket becomes a Lie algebra. Such Lie algebras are called abelian, cf. below. Any one-dimensional Lie algebra over a field is abelian, by the alternating property of the Lie bracket.


Associative algebra with commutator bracket

* On an
associative algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
A over a field F with multiplication (x, y) \mapsto xy, a Lie bracket may be defined by the
commutator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
,y= xy - yx. With this bracket, A is a Lie algebra. The associative algebra ''A'' is called an ''enveloping algebra'' of the Lie algebra (A, ,\cdot\, , \cdot \,. Every Lie algebra can be embedded into one that arises from an associative algebra in this fashion; see
universal enveloping algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. * The associative algebra of the endomorphisms of an ''F''-vector space V with the above Lie bracket is denoted \mathfrak(V). *For a finite dimensional vector space V = F^n, the previous example is exactly the Lie algebra of ''n'' × ''n'' matrices, denoted \mathfrak(n, F) or \mathfrak_n(F), and with bracket ,YXY-YX where adjacency indicates matrix multiplication. This is the Lie algebra of the
general linear group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, consisting of invertible matrices.


Special matrices

Two important subalgebras of \mathfrak_n(F) are: * The matrices of
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * The Trace (album), ''The Trace'' (album) Other ...
zero form the
special linear Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
\mathfrak_n(F), the Lie algebra of the
special linear group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
\mathrm_n(F). *The skew-hermitian matrices form the unitary Lie algebra \mathfrak u(n), the Lie algebra of the
unitary group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
 ''U''(''n'').


Matrix Lie algebras

A complex
matrix groupIn mathematics, a matrix group is a group (mathematics), group ''G'' consisting of invertible matrix, invertible matrix (mathematics), matrices over a specified field (mathematics), field ''K'', with the operation of matrix multiplication. A linear g ...
is a Lie group consisting of matrices, G\subset M_n(\mathbb), where the multiplication of ''G'' is matrix multiplication. The corresponding Lie algebra \mathfrak g is the space of matrices which are tangent vectors to ''G'' inside the linear space M_n(\mathbb): this consists of derivatives of smooth curves in ''G'' at the identity:
\mathfrak = \.
The Lie bracket of \mathfrak is given by the commutator of matrices, ,YXY-YX. Given the Lie algebra, one can recover the Lie group as the image of the
matrix exponential In mathematics, the matrix exponential is a matrix function on square matrix, square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exp ...
mapping \exp: M_n(\mathbb)\to M_n(\mathbb) defined by \exp(X) = I + X + \tfracX^2+\cdots, which converges for every matrix X: that is, G=\exp(\mathfrak g). The following are examples of Lie algebras of matrix Lie groups: * The
special linear group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
_n(\mathbb), consisting of all matrices with determinant 1. Its Lie algebra \mathfrak_n(\mathbb)consists of all matrices with complex entries and trace 0. Similarly, one can define the corresponding real Lie group _n(\mathbb) and its Lie algebra \mathfrak_n(\mathbb). * The
unitary group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
U(n) consists of ''n'' × ''n'' unitary matrices (satisfying U^*=U^). Its Lie algebra \mathfrak(n) consists of skew-self-adjoint matrices (X^*=-X). * The special
orthogonal group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
\mathrm(n), consisting of real determinant-one orthogonal matrices (A^=A^). Its Lie algebra \mathfrak(n) consists of real skew-symmetric matrices (X^=-X). The full orthogonal group \mathrm(n), without the determinant-one condition, consists of \mathrm(n) and a separate connected component, so it has the ''same'' Lie algebra as \mathrm(n). Similarly, one can define a complex version of this group and algebra, simply by allowing complex matrix entries.


Two dimensions

* On any field F there is, up to isomorphism, a single two-dimensional nonabelian Lie algebra. With generators ''x, y,'' its bracket is defined as \left , y\right = y. It generates the Affine group#Matrix representation, affine group in one dimension. :This can be realized by the matrices: :: x= \left( \begin 1 & 0\\ 0 & 0 \end\right), \qquad y= \left( \begin 0 & 1\\ 0 & 0 \end\right). Since : \left( \begin 1 & c\\ 0 & 0 \end\right)^ = \left( \begin 1 & c\\ 0 & 0 \end\right) for any natural number n and any c, one sees that the resulting Lie group elements are upper triangular 2×2 matrices with unit lower diagonal: :: \exp(a\cdotx+b\cdoty)= \left( \begin e^a & \tfrac(e^a-1)\\ 0 & 1 \end\right) = 1 + \tfrac\left(a\cdotx+b\cdoty\right).


Three dimensions

* The Heisenberg algebra _3(\mathbb) is a three-dimensional Lie algebra generated by elements , , and with Lie brackets :: ,y= z,\quad [x,z] = 0, \quad [y,z] = 0. :It is usually realized as the space of 3×3 strictly upper-triangular matrices, with the commutator Lie bracket and the basis :: x = \left( \begin 0&1&0\\ 0&0&0\\ 0&0&0 \end\right),\quad y = \left( \begin 0&0&0\\ 0&0&1\\ 0&0&0 \end\right),\quad z = \left( \begin 0&0&1\\ 0&0&0\\ 0&0&0 \end\right)~.\quad :Any element of the Heisenberg group has a representation as a product of group generators, i.e.,
matrix exponential In mathematics, the matrix exponential is a matrix function on square matrix, square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exp ...
s of these Lie algebra generators, ::\left( \begin 1&a&c\\ 0&1&b\\ 0&0&1 \end\right)= e^ e^ e^~. * The Lie algebra \mathfrak(3) of the group SO(3) is spanned by the three matrices :: F_1 = \left( \begin 0&0&0\\ 0&0&-1\\ 0&1&0 \end\right),\quad F_2 = \left( \begin 0&0&1\\ 0&0&0\\ -1&0&0 \end\right),\quad F_3 = \left( \begin 0&-1&0\\ 1&0&0\\ 0&0&0 \end\right)~.\quad :The commutation relations among these generators are ::[F_1, F_2] = F_3, :: [F_2, F_3] = F_1, :: [F_3, F_1] = F_2. :The three-dimensional Euclidean space \mathbb^3 with the Lie bracket given by the
cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

cross product
of Vector (geometric), vectors has the same commutation relations as above: thus, it is isomorphic to \mathfrak(3). This Lie algebra is unitarily equivalent to the usual Spin (physics) angular-momentum component operators for spin-1 particles in
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
.


Infinite dimensions

* An important class of infinite-dimensional real Lie algebras arises in differential topology. The space of smooth vector fields on a differentiable manifold ''M'' forms a Lie algebra, where the Lie bracket is defined to be the Lie bracket of vector fields, commutator of vector fields. One way of expressing the Lie bracket is through the formalism of Lie derivatives, which identifies a vector field ''X'' with a first order partial differential operator ''L''''X'' acting on smooth functions by letting ''L''''X''(''f'') be the directional derivative of the function ''f'' in the direction of ''X''. The Lie bracket [''X'',''Y''] of two vector fields is the vector field defined through its action on functions by the formula: :: L_f=L_X(L_Y f)-L_Y(L_X f).\, *Kac–Moody algebra, Kac–Moody algebras are a large class of infinite-dimensional Lie algebras whose structure is very similar to the finite-dimensional cases above. * The Moyal bracket, Moyal algebra is an infinite-dimensional Lie algebra that contains all Classical Lie groups#Relationship with bilinear forms, classical Lie algebras as subalgebras. * The Virasoro algebra is of paramount importance in string theory.


Representations


Definitions

Given a vector space ''V'', let \mathfrak(V) denote the Lie algebra consisting of all linear endomorphisms of ''V'', with bracket given by ,YXY-YX. A ''representation'' of a Lie algebra \mathfrak on ''V'' is a Lie algebra homomorphism :\pi: \mathfrak g \to \mathfrak(V). A representation is said to be ''faithful'' if its kernel is zero. Ado's theorem states that every finite-dimensional Lie algebra has a faithful representation on a finite-dimensional vector space.


Adjoint representation

For any Lie algebra \mathfrak, we can define a representation :\operatorname\colon\mathfrak \to \mathfrak(\mathfrak) given by \operatorname(x)(y) =
, y The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
/math>; it is a representation on the vector space \mathfrak called the adjoint representation of a Lie algebra, adjoint representation.


Goals of representation theory

One important aspect of the study of Lie algebras (especially semisimple Lie algebras) is the study of their representations. (Indeed, most of the books listed in the references section devote a substantial fraction of their pages to representation theory.) Although Ado's theorem is an important result, the primary goal of representation theory is not to find a faithful representation of a given Lie algebra \mathfrak. Indeed, in the semisimple case, the adjoint representation is already faithful. Rather the goal is to understand ''all'' possible representation of \mathfrak, up to the natural notion of equivalence. In the semisimple case over a field of characteristic zero, Weyl's theorem on complete reducibility, Weyl's theorem says that every finite-dimensional representation is a direct sum of irreducible representations (those with no nontrivial invariant subspaces). The irreducible representations, in turn, are classified by a Lie algebra representation#Classifying finite-dimensional representations of Lie algebras, theorem of the highest weight.


Representation theory in physics

The representation theory of Lie algebras plays an important role in various parts of theoretical physics. There, one considers operators on the space of states that satisfy certain natural commutation relations. These commutation relations typically come from a symmetry of the problem—specifically, they are the relations of the Lie algebra of the relevant symmetry group. An example would be the angular momentum operators, whose commutation relations are those of the Lie algebra \mathfrak(3) of the rotation group SO(3). Typically, the space of states is very far from being irreducible under the pertinent operators, but one can attempt to decompose it into irreducible pieces. In doing so, one needs to know the irreducible representations of the given Lie algebra. In the study of the quantum Hydrogen-like atom, hydrogen atom, for example, quantum mechanics textbooks give (without calling it that) a classification of the irreducible representations of the Lie algebra \mathfrak(3).


Structure theory and classification

Lie algebras can be classified to some extent. In particular, this has an application to the classification of Lie groups.


Abelian, nilpotent, and solvable

Analogously to abelian, nilpotent, and solvable groups, defined in terms of the derived subgroups, one can define abelian, nilpotent, and solvable Lie algebras. A Lie algebra \mathfrak is ''abelian'' if the Lie bracket vanishes, i.e. [''x'',''y''] = 0, for all ''x'' and ''y'' in \mathfrak. Abelian Lie algebras correspond to commutative (or abelian group, abelian) connected Lie groups such as vector spaces \mathbb^n or torus, tori \mathbb^n, and are all of the form \mathfrak^n, meaning an ''n''-dimensional vector space with the trivial Lie bracket. A more general class of Lie algebras is defined by the vanishing of all commutators of given length. A Lie algebra \mathfrak is ''nilpotent Lie algebra, nilpotent'' if the lower central series : \mathfrak > [\mathfrak,\mathfrak] > \mathfrak,\mathfrak],\mathfrak] > [\mathfrak,\mathfrak],\mathfrak],\mathfrak] > \cdots becomes zero eventually. By Engel's theorem, a Lie algebra is nilpotent if and only if for every ''u'' in \mathfrak the adjoint endomorphism :\operatorname(u):\mathfrak \to \mathfrak, \quad \operatorname(u)v=[u,v] is nilpotent. More generally still, a Lie algebra \mathfrak is said to be ''solvable Lie algebra, solvable'' if the derived series: : \mathfrak > [\mathfrak,\mathfrak] > \mathfrak,\mathfrak],[\mathfrak,\mathfrak > [\mathfrak,\mathfrak],[\mathfrak,\mathfrak,\mathfrak,\mathfrak],[\mathfrak,\mathfrak] > \cdots becomes zero eventually. Every finite-dimensional Lie algebra has a unique maximal solvable ideal, called its radical of a Lie algebra, radical. Under the Lie correspondence, nilpotent (respectively, solvable) connected Lie groups correspond to nilpotent (respectively, solvable) Lie algebras.


Simple and semisimple

A Lie algebra is "Simple Lie algebra, simple" if it has no non-trivial ideals and is not abelian. (This implies that a one-dimensional—necessarily abelian—Lie algebra is by definition not simple, even though it has no nontrivial ideals.) A Lie algebra \mathfrak is called '' semisimple'' if it is isomorphic to a direct sum of simple algebras. There are several equivalent characterizations of semisimple algebras, such as having no nonzero solvable ideals. The concept of semisimplicity for Lie algebras is closely related with the complete reducibility (semisimplicity) of their representations. When the ground field ''F'' has characteristic (field), characteristic zero, any finite-dimensional representation of a semisimple Lie algebra is semisimple representation, semisimple (i.e., direct sum of irreducible representations.) In general, a Lie algebra is called reductive Lie algebra, reductive if the adjoint representation is semisimple. Thus, a semisimple Lie algebra is reductive.


Cartan's criterion

Cartan's criterion gives conditions for a Lie algebra to be nilpotent, solvable, or semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on \mathfrak defined by the formula : K(u,v)=\operatorname(\operatorname(u)\operatorname(v)), where tr denotes the Trace (linear algebra), trace of a linear operator. A Lie algebra \mathfrak is semisimple if and only if the Killing form is nondegenerate form, nondegenerate. A Lie algebra \mathfrak is solvable if and only if K(\mathfrak,[\mathfrak,\mathfrak])=0.


Classification

The Levi decomposition expresses an arbitrary Lie algebra as a semidirect sum of its solvable radical and a semisimple Lie algebra, almost in a canonical way. (Such a decomposition exists for a finite-dimensional Lie algebra over a field of characteristic zero.) Furthermore, semisimple Lie algebras over an algebraically closed field have been completely classified through their root systems.


Relation to Lie groups

Although Lie algebras are often studied in their own right, historically they arose as a means to study
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s. We now briefly outline the relationship between Lie groups and Lie algebras. Any Lie group gives rise to a canonically determined Lie algebra (concretely, ''the tangent space at the identity''). Conversely, for any finite-dimensional Lie algebra \mathfrak g, there exists a corresponding connected Lie group G with Lie algebra \mathfrak g. This is
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; see the Baker–Campbell–Hausdorff formula. This Lie group is not determined uniquely; however, any two Lie groups with the same Lie algebra are ''locally isomorphic'', and in particular, have the same universal cover. For instance, the special orthogonal group SO(3) and the special unitary group SU(2) give rise to the same Lie algebra, which is isomorphic to \mathbb^3 with the cross-product, but SU(2) is a simply-connected twofold cover of SO(3). If we consider ''simply connected'' Lie groups, however, we have a one-to-one correspondence: For each (finite-dimensional real) Lie algebra \mathfrak g, there is a unique simply connected Lie group G with Lie algebra \mathfrak g. The correspondence between Lie algebras and Lie groups is used in several ways, including in the list of simple Lie groups, classification of Lie groups and the related matter of the representation theory of Lie groups. Every representation of a Lie algebra lifts uniquely to a representation of the corresponding connected, simply connected Lie group, and conversely every representation of any Lie group induces a representation of the group's Lie algebra; the representations are in one-to-one correspondence. Therefore, knowing the representations of a Lie algebra settles the question of representations of the group. As for classification, it can be shown that any connected Lie group with a given Lie algebra is isomorphic to the universal cover mod a discrete central subgroup. So classifying Lie groups becomes simply a matter of counting the discrete subgroups of the Center (group theory), center, once the classification of Lie algebras is known (solved by Élie Cartan, Cartan et al. in the Semisimple Lie algebra, semisimple case). If the Lie algebra is infinite-dimensional, the issue is more subtle. In many instances, the exponential map is not even locally a homeomorphism (for example, in Diff(S1), one may find diffeomorphisms arbitrarily close to the identity that are not in the image of exp). Furthermore, some infinite-dimensional Lie algebras are not the Lie algebra of any group.


Real form and complexification

Given a complex Lie algebra \mathfrak g, a real Lie algebra \mathfrak_0 is said to be a ''real form'' of \mathfrak g if the complexification \mathfrak_0 \otimes_ \mathbb \simeq \mathfrak is isomorphic to \mathfrak. A real form need not be unique; for example, \mathfrak_2 \mathbb has two real forms \mathfrak_2 \mathbb and \mathfrak_2. Given a semisimple finite-dimensional complex Lie algebra \mathfrak g, a ''split form'' of it is a real form that splits; i.e., it has a Cartan subalgebra which acts via an adjoint representation with real eigenvalues. A split form exists and is unique (up to isomorphisms). A ''compact form'' is a real form that is the Lie algebra of a compact Lie group. A compact form exists and is also unique.


Lie algebra with additional structures

A Lie algebra can be equipped with some additional structures that are assumed to be compatible with the bracket. For example, a graded Lie algebra is a Lie algebra with a graded vector space structure. If it also comes with differential (so that the underlying graded vector space is a chain complex), then it is called a differential graded Lie algebra. A simplicial Lie algebra is a simplicial object in the category of Lie algebras; in other words, it is obtained by replacing the underlying set with a simplicial set (so it might be better thought of as a family of Lie algebras).


Lie ring

A ''Lie ring'' arises as a generalisation of Lie algebras, or through the study of the lower central series of Group (mathematics), groups. A Lie ring is defined as a nonassociative ring with multiplication that is anticommutative and satisfies the
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. More specifically we can define a Lie ring L to be an abelian group with an operation [\cdot,\cdot] that has the following properties: * Bilinearity: :: [x + y, z] =
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+
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\quad [z, x + y] =
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+
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:for all ''x'', ''y'', ''z'' ∈ ''L''. * The ''Jacobi identity'': :: ,[y,z_+_[y,[z,x.html"_;"title=",z.html"_;"title=",[y,z">,[y,z_+_[y,[z,x">,z.html"_;"title=",[y,z">,[y,z_+_[y,[z,x_+_[z,[x,y.html" ;"title=",z">,[y,z_+_[y,[z,x.html" ;"title=",z.html" ;"title=",[y,z">,[y,z + [y,[z,x">,z.html" ;"title=",[y,z">,[y,z + [y,[z,x + [z,[x,y">,z">,[y,z_+_[y,[z,x.html" ;"title=",z.html" ;"title=",[y,z">,[y,z + [y,[z,x">,z.html" ;"title=",[y,z">,[y,z + [y,[z,x + [z,[x,y = 0 \quad :for all ''x'', ''y'', ''z'' in ''L''. * For all ''x'' in ''L'': :: ,x0 \quad Lie rings need not be
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s under addition. Any Lie algebra is an example of a Lie ring. Any associative ring can be made into a Lie ring by defining a bracket operator ,y= xy - yx. Conversely to any Lie algebra there is a corresponding ring, called the
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. Lie rings are used in the study of finite p-groups through the ''Lazard correspondence''. The lower central factors of a ''p''-group are finite abelian ''p''-groups, so modules over Z/''p''Z. The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be the
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of two coset representatives. The Lie ring structure is enriched with another module homomorphism, the ''p''th power map, making the associated Lie ring a so-called restricted Lie ring. Lie rings are also useful in the definition of a p-adic analytic groups and their endomorphisms by studying Lie algebras over rings of integers such as the p-adic integers. The definition of finite groups of Lie type due to Chevalley involves restricting from a Lie algebra over the complex numbers to a Lie algebra over the integers, and then reducing modulo ''p'' to get a Lie algebra over a finite field.


Examples

* Any Lie algebra over a general Ring (mathematics), ring instead of a Field (mathematics), field is an example of a Lie ring. Lie rings are ''not''
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s under addition, despite the name. * Any associative ring can be made into a Lie ring by defining a bracket operator :: ,y= xy - yx. * For an example of a Lie ring arising from the study of Group (mathematics), groups, let G be a group with ,y x^y^xy the commutator operation, and let G = G_0 \supseteq G_1 \supseteq G_2 \supseteq \cdots \supseteq G_n \supseteq \cdots be a central series in G — that is the commutator subgroup [G_i,G_j] is contained in G_ for any i,j. Then :: L = \bigoplus G_i/G_ :is a Lie ring with addition supplied by the group operation (which is abelian in each homogeneous part), and the bracket operation given by :: [xG_i, yG_j] = [x,y]G_\ :extended linearly. The centrality of the series ensures that the commutator ,y/math> gives the bracket operation the appropriate Lie theoretic properties.


See also

* Adjoint representation of a Lie algebra * Anyonic Lie algebra * Chiral Lie algebra * Free Lie algebra * Index of a Lie algebra * Lie algebra cohomology * Lie algebra extension * Lie algebra representation * Lie bialgebra * Lie coalgebra * Lie operad * Particle physics and representation theory * Lie superalgebra * Poisson algebra * Pre-Lie algebra * Quantum groups * Moyal bracket, Moyal algebra * Quasi-Frobenius Lie algebra * Quasi-Lie algebra * Restricted Lie algebra * Serre relations * Symmetric Lie algebra * Gelfand–Fuks cohomology


Remarks


References


Sources

* * * * Karin Erdmann, Erdmann, Karin & Wildon, Mark. ''Introduction to Lie Algebras'', 1st edition, Springer, 2006. * * * * * * * * * * * * *


External links

* * {{DEFAULTSORT:Lie Algebra Lie groups Lie algebras,