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Since B is nondegenerate, so(p, q) is a Lie subalgebra of sl(n, R). To obtain a basis-free definition of this family of Lie algebras, let B be a nondegenerate symmetric bilinear form on an n-dimensional vector space V over F. Let {v1 , . . 2). Let µ(T ) be the matrix of T ∈ End(V ) relative to this basis . When F = C, then µ defines a Lie algebra isomorphism of so(V, B) onto so(n, C). When F = R and B has signature (p, q), then µ defines a Lie algebra isomorphism of so(V, B) onto so(p, q). 2. We define sp(n, F) = {X ∈ M2n (F) : X t J = −JX} .

1. Let {v1 , . . , vn } and {w1 , . . , wn } be bases for an F vector space V . Suppose a / V has matrices A and B, respectively, relative to these linear map T : V bases. Show that det A = det B. 2. Determine the signature of the form B(x, y) = ∑ni=1 xi yn+1−i on Rn . 3. Let V be a vector space over F and let B be a skew-symmetric or symmetric nondegenerate bilinear form on V . Assume that W is a subspace of V on which B restricts to a nondegenerate form. Prove that the restriction of B to the subspace W ⊥ = {v ∈ V : B(v, w) = 0 for all w ∈ W } is nondegenerate.

Then F is a subalgebra of O[G], and we have just verified that the matrix entry functions and det−1 are in F. Since these functions generate O[G] as an algebra, it follows that F = O[G]. 31) as µ ∗ ( f ) = ∑ fi ⊗ fi . i This shows that µ is a regular map. Furthermore, Lg∗ ( f ) = ∑i fi (g) fi and R∗g ( f ) = ∑i fi (g) fi , which proves that Lg and Rg are regular maps. Examples 1. Let Dn be the subgroup of diagonal matrices in GL(n, C). The map (x1 , . . , xn ) → diag[x1 , . . , xn ] from (C× )n to Dn is obviously an isomorphism of algebraic groups.