Classical group

In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term 'classical group' was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups. φ ( A x , A y ) = φ ( x , y ) , ∀ x , y ∈ V . {displaystyle varphi (Ax,Ay)=varphi (x,y),quad forall x,yin V.}     (1) φ ( A x , y ) = φ ( x , A φ y ) , x , y ∈ V . {displaystyle varphi (Ax,y)=varphi (x,A^{varphi }y),qquad x,yin V.}     (2) Aut ⁡ ( φ ) = { A ∈ GL ⁡ ( V ) : A φ A = 1 } . {displaystyle operatorname {Aut} (varphi )={Ain operatorname {GL} (V):A^{varphi }A=1}.}     (3) A φ = Φ − 1 A T Φ {displaystyle A^{varphi }=Phi ^{-1}A^{mathrm {T} }Phi }     (4) a u t ( φ ) = { X ∈ M n ( V ) : Φ − 1 X T Φ = − X } {displaystyle {mathfrak {aut}}(varphi )={Xin M_{n}(V):Phi ^{-1}X^{mathrm {T} }Phi =-X}}     (5) a u t ( φ ) = { X ∈ M n ( V ) : Φ − 1 X ∗ Φ = − X } . {displaystyle {mathfrak {aut}}(varphi )={Xin M_{n}(V):Phi ^{-1}X^{*}Phi =-X}.}     (6) q = a 1 + b i + c j + d k = α + j β ↔ [ α − β ¯ β α ¯ ] = Q , q ∈ H , a , b , c , d ∈ R , α , β ∈ C . {displaystyle q=amathrm {1} +bmathrm {i} +cmathrm {j} +dmathrm {k} =alpha +jeta leftrightarrow {egin{bmatrix}alpha &-{overline {eta }}\eta &{overline {alpha }}end{bmatrix}}=Q,quad qin mathbb {H} ,quad a,b,c,din mathbb {R} ,quad alpha ,eta in mathbb {C} .}     (7) ( Q ) n × n = ( X ) n × n + j ( Y ) n × n ↔ ( X − Y ¯ Y X ¯ ) 2 n × 2 n . {displaystyle left(Q ight)_{n imes n}=left(X ight)_{n imes n}+mathrm {j} left(Y ight)_{n imes n}leftrightarrow left({egin{matrix}X&-{ar {Y}}\Y&{ar {X}}end{matrix}} ight)_{2n imes 2n}.}     (8) Q = ( X p × p Z p × q Z ∗ Y q × q ) , X ∗ = − X , Y ∗ = − Y {displaystyle {mathcal {Q}}=left({egin{matrix}{mathcal {X}}_{p imes p}&{mathcal {Z}}_{p imes q}\{mathcal {Z}}^{*}&{mathcal {Y}}_{q imes q}end{matrix}} ight),qquad {mathcal {X}}^{*}=-{mathcal {X}},{mathcal {Y}}^{*}=-{mathcal {Y}}}     (9) − Φ V ∗ Φ = − V ⇔ V ∗ = j n V j n . {displaystyle -Phi V^{*}Phi =-VLeftrightarrow V^{*}=mathrm {j} _{n}Vmathrm {j} _{n}.}     (9) In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term 'classical group' was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups. The classical groups form the deepest and most useful part of the subject of linear Lie groups. Most types of classical groups find application in classical and modern physics. A few examples are the following. The rotation group SO(3) is a symmetry of Euclidean space and all fundamental laws of physics, the Lorentz group O(3,1) is a symmetry group of spacetime of special relativity. The special unitary group SU(3) is the symmetry group of quantum chromodynamics and the symplectic group Sp(m) finds application in hamiltonian mechanics and quantum mechanical versions of it. The classical groups are exactly the general linear groups over R, C and H together with the automorphism groups of non-degenerate forms discussed below. These groups are usually additionally restricted to the subgroups whose elements have determinant 1, so that their centers are discrete. The classical groups, with the determinant 1 condition, are listed in the table below. In the sequel, the determinant 1 condition is not used consistently in the interest of greater generality. The complex classical groups are SL(n, C), SO(n, C) and Sp(m, C). A group is complex according to whether its Lie algebra is complex. The real classical groups refers to all of the classical groups since any Lie algebra is a real algebra. The compact classical groups are the compact real forms of the complex classical groups. These are, in turn, SU(n), SO(n) and Sp(m). One characterization of the compact real form is in terms of the Lie algebra g. If g = u + iu, the complexification of u, then if the connected group K generated by exp(X): X ∈ u is a compact, K is a compact real form. The classical groups can uniformly be characterized in a different way using real forms. The classical groups (here with the determinant 1 condition, but this is not necessary) are the following: For instance, SO∗(2n) is a real form of SO(2n, C), SU(p, q) is a real form of SL(n, C), and SL(n, H) is a real form of SL(2n, C). Without the determinant 1 condition, replace the special linear groups with the corresponding general linear groups in the characterization. The algebraic groups in question are Lie groups, but the 'algebraic' qualifier is needed to get the right notion of 'real form'. The classical groups are defined in terms of forms defined on Rn, Cn, and Hn, where R and C are the fields of the real and complex numbers. The quaternions, H, do not constitute a field because multiplication does not commute; they form a division ring or a skew field or non-commutative field. However, it is still possible to define matrix quaternionic groups. For this reason, a vector space V is allowed to be defined over R, C, as well as H below. In the case of H, V is a right vector space to make possible the representation of the group action as matrix multiplication from the left, just as for R and C. A form φ: V × V → F on some finite-dimensional right vector space over F = R, C, or H is bilinear if It is called sesquilinear if