Help solve Lie algebra question ii. I have added the reference of 1.2 below it. Thanks There is a correspondence between matrices and bilinear forms given by B(v, w):v' Aw.(i) Find necessary and sufficient conditions on A so that the correspondingbilinear form B is symmetric i.e. B(v, w) = B(w, v).(ii) The Lie algebra B, is defined as the set of a € gl1(C) leaving the formf invariant:f(xv, w) + f(v, xw) = 0Vv, w e C2n+1.Explain why a has the form stated in (1.2).B,: Let dim V = 2l+1 be odd, and take f to be the nondegenerate sym-(1 0 0metric bilinear form on V whose matrix is s = 0 0 I,The orthogonalalgebra o(V), or o(26+1, F), consists of all endomorphisms of V satisfying(x(v), w) = -(v, x(w)) (the same requirement as for C,). If we partition x in/a b, baC, mn\C2 P qtranslates into the following set of conditions: a = 0, c, = -b, c2 =q = -m', n' = -n, p' -p. (As in the case of C,, this shows that Tr(x)= 0.) For a basis, take first the diagonal matrices eu-e+i,t+i (2 sisl+1). Add the 2l matrices involving only row one and column one:the same form as s, say x =then the condition sx = -x's- b,e1,+i+1-e+1,1 and e,1+1-e+i+1,1 (1 s is). Corresponding toq = -m', take (as for C) e+1J+1-++1,t+i+1 (I

There is a correspondence between matrices and bilinear forms given by B(v, w) = v' Aw. (i) Find necessary and sufficient conditions on A so that the corresponding bilinear form B is symmetric i.e. B(v, w) = B(w,v). (ii) The Lie algebra B is defined as the set of a E gln+1(C) leaving the form f invariant: f(xv, w) + f(v, xw) = 0 Vv, w e C2n+1.

There is a correspondence between matrices and bilinear forms given by B(v, w) = v' Aw. (i) Find necessary and sufficient conditions on A so that the corresponding bilinear form B is symmetric i.e. B(v, w) = B(w,v). (ii) The Lie algebra B is defined as the set of a E gln+1(C) leaving the form f invariant: f(xv, w) + f(v, xw) = 0 Vv, w e C2n+1.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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