Classical group

In mathematics, the classical groups are the matrix groups arising from finite-dimensional vector spaces and from nondegenerate bilinear, sesquilinear, quadratic, and Hermitian forms. In the traditional setting of Lie groups, this includes the real, complex, and quaternionic general linear, special linear, orthogonal, unitary, and symplectic groups, together with their indefinite analogues. In the language of linear algebraic groups, the connected classical groups are the connected reductive groups of Dynkin types A_n , B_n , C_n , and D_n , together with their forms over arbitrary fields. Over \mathbb R and \mathbb C this recovers the familiar classical Lie groups, while over finite fields one obtains the finite classical groups. The term goes back to Hermann Weyl's book The Classical Groups. Among the simple Lie groups, the classical groups are in contrast to the exceptional Lie groups, G 2 , F 4 , E 6 , E 7 , E 8 , which share their abstract properties, but not their familiarity. This article begins with the classical Lie groups over \mathbb R , \mathbb C , and \mathbb H , and later discusses the more general formulation over arbitrary fields.

In mathematics, the classical groups are the matrix groups arising from finite-dimensional vector spaces and from nondegenerate bilinear, sesquilinear, quadratic, and Hermitian forms. In the traditional setting of Lie groups, this includes the real, complex, and quaternionic general linear, special linear, orthogonal, unitary, and symplectic groups, together with their indefinite analogues. In the language of linear algebraic groups, the connected classical groups are the connected reductive groups of Dynkin types A_n , B_n , C_n , and D_n , together with their forms over arbitrary fields. Over \mathbb R and \mathbb C this recovers the familiar classical Lie groups, while over finite fields one obtains the finite classical groups. The term goes back to Hermann Weyl's book The Classical Groups. Among the simple Lie groups, the classical groups are in contrast to the exceptional Lie groups, G 2 , F 4 , E 6 , E 7 , E 8 , which share their abstract properties, but not their familiarity. This article begins with the classical Lie groups over \mathbb R , \mathbb C , and \mathbb H , and later discusses the more general formulation over arbitrary fields. For F=\mathbb R or \mathbb C , the special linear group is \mathrm{SL}(n,F)=\{g\in \mathrm{GL}_n(F)\mid \det g=1\}. Its Lie algebra is \mathfrak{sl}(n,F)=\{X\in M_n(F)\mid \operatorname{tr}(X)=0\}. Thus \mathfrak{sl}(n,\mathbb R) consists of all real traceless n\times n matrices, and \mathfrak{sl}(n,\mathbb C) consists of all complex traceless n\times n matrices. \operatorname{Aut}(\varphi)=\{g\in \operatorname{GL}(V)\mid \varphi(gv,gw)=\varphi(v,w)\text{ for all }v,w\in V\}. After a choice of basis, \varphi is represented by a Gram matrix \Phi , and \operatorname{Aut}(\varphi) becomes a matrix group defined by one of the equations g^{\mathrm T}\Phi g=\Phi,\qquad g^*\Phi g=\Phi, according to whether \varphi is bilinear or sesquilinear. The Lie algebra of \operatorname{Aut}(\varphi) is \mathfrak{aut}(\varphi)=\{X\in \operatorname{End}(V)\mid \varphi(Xv,w)+\varphi(v,Xw)=0\text{ for all }v,w\in V\}, or, in matrix form, X^{\mathrm T}\Phi+\Phi X=0,\qquad X^*\Phi+\Phi X=0. symmetric if \varphi(v,w)=\varphi(w,v) ; alternating (or skew-symmetric, when \operatorname{char}F\neq 2 ) if \varphi(v,v)=0 for all v , equivalently \varphi(v,w)=-\varphi(w,v) . A sesquilinear form h is: Hermitian if h(v,w)=\overline{h(w,v)} ; skew-Hermitian if h(v,w)=-\overline{h(w,v)} . Over \mathbb R , nondegenerate symmetric bilinear forms are classified by their signature (p,q) . Over \mathbb C , all nondegenerate symmetric bilinear forms of a given dimension are equivalent. Nondegenerate alternating forms exist only in even dimension, and over both \mathbb R and \mathbb C all such forms are equivalent. On a complex vector space, multiplying a skew-Hermitian form by i yields a Hermitian form, so the two cases lead to the same isometry groups up to a harmless change of convention. On a quaternionic vector space, by contrast, there are no nonzero bilinear forms, so only the Hermitian and skew-Hermitian cases occur. \varphi(x,y)=x^{\mathrm T}I_{p,q}y,\qquad I_{p,q}=\operatorname{diag}(I_p,-I_q), where p+q=n . Its automorphism group is the indefinite orthogonal group \mathrm O(p,q)=\{g\in \mathrm{GL}_n(\mathbb R)\mid g^{\mathrm T}I_{p,q}g=I_{p,q}\}. The subgroup of determinant 1 is the special orthogonal group \mathrm{SO}(p,q)=\mathrm O(p,q)\cap \mathrm{SL}_n(\mathbb R). When q=0 this is the compact orthogonal group \mathrm O(n) , with determinant- 1 subgroup \mathrm{SO}(n) . Over \mathbb C , every nondegenerate symmetric bilinear form is equivalent to the standard form \varphi(x,y)=x^{\mathrm T}y. Its automorphism group is the complex orthogonal group \mathrm O(n,\mathbb C)=\{g\in \mathrm{GL}_n(\mathbb C)\mid g^{\mathrm T}g=I_n\}, with determinant- 1 subgroup \mathrm{SO}(n,\mathbb C) . The corresponding Lie algebras are \mathfrak{o}(p,q)=\{X\in M_n(\mathbb R)\mid X^{\mathrm T}I_{p,q}+I_{p,q}X=0\}, \mathfrak{o}(n,\mathbb C)=\mathfrak{so}(n,\mathbb C)=\{X\in M_n(\mathbb C)\mid X^{\mathrm T}+X=0\}. If g= \begin{pmatrix} A&B\\ C&D \end{pmatrix} with block sizes p\times p , p\times q , q\times p , and q\times q , then the defining relation for \mathrm O(p,q) is equivalent to A^{\mathrm T}A-C^{\mathrm T}C=I_p,\qquad D^{\mathrm T}D-B^{\mathrm T}B=I_q,\qquad A^{\mathrm T}B=C^{\mathrm T}D. Writing X= \begin{pmatrix} P&Q\\ R&S \end{pmatrix}, one obtains the block form \mathfrak{o}(p,q)= \left\{ \begin{pmatrix} P&Q\\ Q^{\mathrm T}&S \end{pmatrix} \;|\; P^{\mathrm T}=-P,\ S^{\mathrm T}=-S \right\}. Thus \mathfrak{so}(n,\mathbb C) is the space of complex skew-symmetric n\times n matrices. \omega(x,y)=x^{\mathrm T}J_ny,\qquad J_n=\begin{pmatrix}0&I_n\\-I_n&0\end{pmatrix}. Its automorphism group is the symplectic group \mathrm{Sp}(2n,F)=\{g\in \mathrm{GL}_{2n}(F)\mid g^{\mathrm T}J_ng=J_n\},\qquad F=\mathbb R,\mathbb C. Many authors write \mathrm{Sp}(n,\mathbb R) and \mathrm{Sp}(n,\mathbb C) for these groups. Its Lie algebra is \mathfrak{sp}(2n,F)=\{X\in M_{2n}(F)\mid X^{\mathrm T}J_n+J_nX=0\}. Equivalently, every element has block form \begin{pmatrix}A&B\\C&-A^{\mathrm T}\end{pmatrix} with B and C symmetric. h(z,w)=z^*I_{p,q}w,\qquad I_{p,q}=\operatorname{diag}(I_p,-I_q), where p+q=n . Its automorphism group is the unitary group \mathrm U(p,q)=\{g\in \mathrm{GL}_n(\mathbb C)\mid g^*I_{p,q}g=I_{p,q}\}. The subgroup of determinant 1 is the special unitary group \mathrm{SU}(p,q)=\mathrm U(p,q)\cap \mathrm{SL}_n(\mathbb C). When q=0 this is the compact unitary group \mathrm U(n) , with determinant- 1 subgroup \mathrm{SU}(n) . Its Lie algebra is \mathfrak{u}(p,q)=\{X\in M_n(\mathbb C)\mid X^*I_{p,q}+I_{p,q}X=0\}. If g= \begin{pmatrix} A&B\\ C&D \end{pmatrix}, then the defining relation is equivalent to A^*A-C^*C=I_p,\qquad D^*D-B^*B=I_q,\qquad A^*B=C^*D. In block form, \mathfrak{u}(p,q)= \left\{ \begin{pmatrix} P&Q\\ Q^*&R \end{pmatrix} \;|\; P^*=-P,\ R^*=-R \right\}. The special unitary Lie algebra is the traceless subalgebra \mathfrak{su}(p,q)=\{X\in \mathfrak{u}(p,q)\mid \operatorname{tr}(X)=0\}. \mathbb H=\mathbb C\oplus j\mathbb C and to represent a quaternion \alpha+j\beta by the complex matrix \begin{pmatrix}\alpha&-\overline{\beta}\\[2pt]\beta&\overline{\alpha}\end{pmatrix}. This extends to an embedding M_n(\mathbb H)\hookrightarrow M_{2n}(\mathbb C),\qquad X+jY\mapsto \begin{pmatrix}X&-\overline{Y}\\Y&\overline{X}\end{pmatrix}. \mathfrak{gl}(n,\mathbb H)= \left\{ \begin{pmatrix} A&-\overline{B}\\ B&\overline{A} \end{pmatrix} \;|\; A,B\in M_n(\mathbb C) \right\}. The corresponding real form of \mathrm{SL}(2n,\mathbb C) is denoted \mathrm{SU}^*(2n) , and as a Lie group it is isomorphic to the group traditionally written \mathrm{SL}(n,\mathbb H) , and is the subgroup of \mathrm{GL}(n,\mathbb H) of elements of reduced norm 1. Its Lie algebra is \mathfrak{sl}(n,\mathbb H)\cong \mathfrak{su}^*(2n)= \left\{ \begin{pmatrix} A&-\overline{B}\\ B&\overline{A} \end{pmatrix} \;|\; \operatorname{Re}\operatorname{tr}(A)=0 \right\}. Let V=\mathbb H^{p+q} and let B(w,z)=w^*I_{p,q}z,\qquad I_{p,q}=\operatorname{diag}(I_p,-I_q), be a nondegenerate quaternionic Hermitian form. Its isometry group is the quaternionic unitary group \mathrm{Sp}(p,q)=\{g\in \mathrm{GL}(p+q,\mathbb H)\mid g^*I_{p,q}g=I_{p,q}\}. When q=0 this is the compact group usually written \mathrm{Sp}(n) . Viewed as a subgroup of \mathrm{GL}(2n,\mathbb C) , the group \mathrm{Sp}(p,q) preserves both a complex Hermitian form of signature (2p,2q) and a nondegenerate complex alternating form. Its Lie algebra is \mathfrak{sp}(p,q)=\{X\in M_{p+q}(\mathbb H)\mid X^*I_{p,q}+I_{p,q}X=0\}. If g= \begin{pmatrix} A&B\\ C&D \end{pmatrix} with quaternionic blocks, then the defining relation is equivalent to A^*A-C^*C=I_p,\qquad D^*D-B^*B=I_q,\qquad A^*B=C^*D. In block form, \mathfrak{sp}(p,q)= \left\{ \begin{pmatrix} P&Q\\ Q^*&R \end{pmatrix} \;|\; P^*=-P,\ R^*=-R \right\}. Let V=\mathbb H^n and consider the quaternionic skew-Hermitian form C(x,y)=x^*jy. Its isometry group is the real Lie group \mathrm{SO}^*(2n), which is a real form of \mathrm{SO}(2n,\mathbb C) . Equivalently, if J_n=\begin{pmatrix}0&I_n\\-I_n&0\end{pmatrix}, then \mathrm{SO}^*(2n) may be realized as the subgroup \mathrm{SO}^*(2n)=\{g\in \mathrm{SO}(2n,\mathbb C)\mid \theta(g)=g\}, where \theta(g)=-J_n\overline{g}J_n is the involution defining this real form. Its Lie algebra is denoted \mathfrak{so}^*(2n) . A standard complex realization of \mathrm{SO}^*(2n) is as the subgroup of \mathrm{GL}(2n,\mathbb C) preserving both the symmetric bilinear form with Gram matrix S= \begin{pmatrix} 0&I_n\\ I_n&0 \end{pmatrix} and the Hermitian form with Gram matrix H= \begin{pmatrix} I_n&0\\ 0&-I_n \end{pmatrix}. Equivalently, \mathrm{SO}^*(2n)= \{g\in \mathrm{GL}(2n,\mathbb C)\mid g^{\mathrm T}Sg=S,\ g^*Hg=H\}. Its Lie algebra is \mathfrak{so}^*(2n)= \left\{ \begin{pmatrix} A&B\\ -\overline{B}&\overline{A} \end{pmatrix} \;|\; A^*=-A,\ B^{\mathrm T}=-B \right\}. \mathrm{GL}(V)=\operatorname{Aut}_k(V), and the special linear group is \mathrm{SL}(V)=\ker(\det\colon \mathrm{GL}(V)\to k^\times). After a choice of basis these become the matrix groups \mathrm{GL}_n(k) and \mathrm{SL}_n(k) . Their projective quotients are the projective general linear group \mathrm{PGL}(V) and the projective special linear group \mathrm{PSL}(V) . \mathrm{Sp}(V,\omega)=\{g\in \mathrm{GL}(V)\mid \omega(gv,gw)=\omega(v,w)\ \forall v,w\in V\}. For \dim V=2n , this is written \mathrm{Sp}_{2n}(k) after a choice of basis. If q is a nondegenerate quadratic form on V , its isometry group is the orthogonal group \mathrm{O}(V,q)=\{g\in \mathrm{GL}(V)\mid q(gv)=q(v)\ \forall v\in V\}. When \operatorname{char}(k)\neq 2 , this is equivalently the group preserving the associated symmetric bilinear form. In characteristic 2, orthogonal groups are still defined from quadratic forms, but the relation with the associated bilinear form is subtler. For orthogonal groups over general fields, one often also considers the subgroup \Omega(V,q) . In the isotropic case and in characteristic not 2, it may be described as the kernel of the spinor norm, a homomorphism from \mathrm{SO}(V,q) (or more generally from the appropriate index-2 subgroup of \mathrm{O}(V,q) ) to k^\times/(k^\times)^2 . In the theory of finite classical groups, the simple group is often \mathrm{P}\Omega(V,q) rather than \mathrm{PSO}(V,q) . If K/k is a quadratic field extension, or more generally if K is equipped with an involution \sigma , and h is a nondegenerate \sigma -hermitian form on a finite-dimensional K -vector space V , its isometry group is a unitary group \mathrm{U}(V,h)=\{g\in \mathrm{GL}_K(V)\mid h(gv,gw)=h(v,w)\ \forall v,w\in V\}. Its derived subgroup is the special unitary group \mathrm{SU}(V,h) . One also has the corresponding similitude groups \mathrm{GSp} , \mathrm{GO} , and \mathrm{GU} , whose elements preserve the relevant form up to multiplication by a scalar. Projective versions are obtained by quotienting by the center. When k=\mathbb F_q is a finite field, the groups of k -rational points of these algebraic groups yield the finite groups of Lie type. The classical families include groups such as \mathrm{PSL}_n(q) , \mathrm{PSU}_n(q) , \mathrm{PSp}_{2n}(q) , and the finite orthogonal groups. \operatorname{GL}_1(A)=A^\times for the group of invertible elements, and \operatorname{SL}_1(A)=\ker(\operatorname{Nrd}\colon A^\times\to k^\times) for the kernel of the reduced norm. These give the inner forms of type A . If (A,\tau) is of the first kind, then \operatorname{Iso}(A,\tau)=\{a\in A^\times\mid \tau(a)a=1\} is the group of isometries, and \operatorname{Sim}(A,\tau)=\{a\in A^\times\mid \tau(a)a\in k^\times\} is the group of similitudes. The scalar \mu(a)=\tau(a)a is called the multiplier of the similitude. (More generally, one can first define these by their associated group schemes. ) According to the type of \tau , we write: \mathrm{O}(A,\tau)=\operatorname{Iso}(A,\tau) , \mathrm{GO}(A,\tau)=\operatorname{Sim}(A,\tau) , and \mathrm{PGO}(A,\tau)=\operatorname{Aut}(A,\tau) in the orthogonal case; \mathrm{Sp}(A,\tau)=\operatorname{Iso}(A,\tau) , \mathrm{GSp}(A,\tau)=\operatorname{Sim}(A,\tau) , and \mathrm{PGSp}(A,\tau)=\operatorname{Aut}(A,\tau) in the symplectic case. If (B,\tau) is unitary, with center a quadratic étale algebra K/k , then \mathrm{U}(B,\tau)=\{b\in B^\times\mid \tau(b)b=1\} , \mathrm{GU}(B,\tau)=\{b\in B^\times\mid \tau(b)b\in k^\times\} , and \mathrm{PGU}(B,\tau)=\operatorname{Aut}_K(B,\tau). The kernel of the reduced norm on \mathrm{U}(B,\tau) is denoted \mathrm{SU}(B,\tau)=\ker(\operatorname{Nrd}\colon \mathrm{U}(B,\tau)\to K^\times), and gives the semisimple simply connected group of unitary type. In the symplectic case, \mathrm{Sp}(A,\tau) is the simply connected group and \mathrm{PGSp}(A,\tau) its adjoint quotient; in the unitary case, \mathrm{SU}(B,\tau) is the simply connected form and \mathrm{PGU}(B,\tau) the corresponding adjoint form. If A=\operatorname{End}_k(V) and \tau is the adjoint involution of a nondegenerate alternating bilinear form h on V , then \mathrm{Sp}(A,\tau)=\mathrm{Sp}(V,h) , and one recovers the ordinary symplectic group. If A=\operatorname{End}_k(V) and \tau is the adjoint involution of a nondegenerate symmetric bilinear form, equivalently of a nondegenerate quadratic form q , then \mathrm{O}(A,\tau)=\mathrm{O}(V,q) , and one recovers the ordinary orthogonal group. If K/k is a quadratic field extension, B=\operatorname{End}_K(V) , and \tau is adjoint to a nondegenerate Hermitian form on the K -space V , then \mathrm{U}(B,\tau)=\mathrm{U}(V,h) and \mathrm{SU}(B,\tau)=\mathrm{SU}(V,h) , so one recovers the usual unitary and special unitary groups. There is also a split description of the inner forms of type A . If the quadratic étale algebra is split, K\cong k\times k, then B is isomorphic to A\times A^{\mathrm{op}} with the exchange involution. In that case \mathrm{U}(B,\tau)\cong \operatorname{GL}_1(A),\qquad \mathrm{SU}(B,\tau)\cong \operatorname{SL}_1(A),\qquad \mathrm{PGU}(B,\tau)\cong \mathrm{PGL}_1(A). Thus the same formalism includes both the inner and outer forms of type A . if h is a nondegenerate Hermitian form on a right D -vector space V , and \tau_h is the adjoint involution on \operatorname{End}_D(V) , then \mathrm{Sp}(\operatorname{End}_D(V),\tau_h) is a classical group of type C_n ; over \mathbb R with D=\mathbb H this construction gives the groups \mathrm{Sp}(p,q) ; if s is a nondegenerate skew-Hermitian form over D , the adjoint involution on \operatorname{End}_D(V) is of orthogonal type, and the associated orthogonal and spin groups are nonsplit forms of types B_n or D_n ; over \mathbb R with D=\mathbb H , the even-dimensional case yields \mathrm{SO}^*(2n) .

Notes

References

E. Artin (1957) Geometric Algebra, chapters III, IV, & V via Internet Archive

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