Lie algebra

In mathematics, a Lie algebra (pronounced /liː/ 'Lee') is a vector space g {displaystyle {mathfrak {g}}} together with a non-associative operation called the Lie bracket, an alternating bilinear map g × g → g ,   ( x , y ) ↦ [ x , y ] {displaystyle {mathfrak {g}} imes {mathfrak {g}} ightarrow {mathfrak {g}}, (x,y)mapsto } , satisfying the Jacobi identity. x × ( y × z )   =   ( x × y ) × z   +   y × ( x × z ) . {displaystyle x imes (y imes z) = (x imes y) imes z + y imes (x imes z).} g = { X = c ′ ( 0 ) ∈ M n ( C )   ∣    smooth  c : R → G ,   c ( 0 ) = I } . {displaystyle {mathfrak {g}}={X=c'(0)in M_{n}(mathbb {C} ) mid { ext{ smooth }}c:mathbb {R} o G, c(0)=I}.} In mathematics, a Lie algebra (pronounced /liː/ 'Lee') is a vector space g {displaystyle {mathfrak {g}}} together with a non-associative operation called the Lie bracket, an alternating bilinear map g × g → g ,   ( x , y ) ↦ [ x , y ] {displaystyle {mathfrak {g}} imes {mathfrak {g}} ightarrow {mathfrak {g}}, (x,y)mapsto } , satisfying the Jacobi identity. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space 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 theorem). This correspondence allows one to study the structure and classification 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, directions of symmetry. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics. An elementary example is the space of three dimensional vectors g = R 3 {displaystyle {mathfrak {g}}=mathbb {R} ^{3}} with the bracket operation defined by the cross product [ x , y ] = x × y . {displaystyle =x imes y.} This is skew-symmetric since x × y = − y × x {displaystyle x imes y=-y imes x} , and instead of associativity it satisfies the Jacobi identity: This is the Lie algebra of the Lie group of rotations of space, and each vector v ∈ R 3 {displaystyle vin mathbb {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 [ x , x ] = x × x = 0 {displaystyle =x imes x=0} . Lie algebras were introduced to study the concept of infinitesimal transformations by Marius Sophus Lie in the 1870s, and independently discovered by Wilhelm Killing in the 1880s. The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group is used. A Lie algebra is a vector space g {displaystyle ,{mathfrak {g}}} over some field F {displaystyle mathbb {F} } together with a binary operation [ ⋅ , ⋅ ] : g × g → g {displaystyle :{mathfrak {g}} imes {mathfrak {g}} o {mathfrak {g}}} called the Lie bracket satisfying the following axioms: Using bilinearity to expand the Lie bracket [ x + y , x + y ] {displaystyle } and using alternativity shows that [ x , y ] + [ y , x ] = 0   {displaystyle +=0 } for all elements x, y in g {displaystyle {mathfrak {g}}} , showing that bilinearity and alternativity together imply It is customary to denote a Lie algebra by a lower-case fraktur letter such as g , h , b , n {displaystyle {mathfrak {g,h,b,n}}} . If a Lie algebra is associated with a Lie group, then the algebra is denoted by the fraktur version of the group: for example the Lie algebra of SU(n) is s u ( n ) {displaystyle {mathfrak {su}}(n)} .