Recent content by arz2000

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    Graduate What Determines the Isomorphism of G/X in Group Theory?

    Thanks for your help.
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    Graduate Prove Perfect Lie Algebra of R^3 Euclidean Motions Isn't Semisimple

    Ok! many thanks
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    Graduate Prove Perfect Lie Algebra of R^3 Euclidean Motions Isn't Semisimple

    you're right. So what is the Lie algebra of transformation preserving distance? Since I'm learning Lie theory by myself, I have lots of questions even in definitions!
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    Graduate What Determines the Isomorphism of G/X in Group Theory?

    In fact I don't know what should I do?
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    Graduate What Determines the Isomorphism of G/X in Group Theory?

    Hi, Consider the simply connected group G of all 3 by 3 matrices [1 a b 0 1 c 0 0 1 ] where a,b,c are in C. The center of G is the subgroup Z(G)={ [1 0 b 0 1 0 0 0 1] ; b is in C} So Z(G) is isomorphic to C and therefore the discrete subgroups of Z(G) are just lattices X of rank 1...
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    Graduate Prove Perfect Lie Algebra of R^3 Euclidean Motions Isn't Semisimple

    does the lie group of Euclidean motions of R^3 means O(3) i.e. rotations?
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    Graduate Prove Perfect Lie Algebra of R^3 Euclidean Motions Isn't Semisimple

    Does anybody know the answer of the following problem? Show that the Lie group of Euclidean motions of R^3 has a Lie algebra g which is perfect i.e., Dg=g but g is not semisimple. By Dg I mean the commutator [g,g] and a semisimple lie algebra is one has no nonzero solvable ideals. Regards
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    Graduate Finding the Real Lie Algebra of SL(n,H) in GL(n,H)

    Thanks, how about sl(n,H) ?
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    Graduate Finding the Real Lie Algebra of SL(n,H) in GL(n,H)

    Many thanks, But how did you get the lie algebra is the set of matrices D satisfing Id+eD is in GL(n,H) mod e^2 by differentiation?
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    Graduate Group Theory: Proving Non-Isomorphism of G_n and G_m

    Hi all, Consider the groups G_n= {(a,b) in (C*) * C ; (a,b).(a',b')=( aa', b+(a^n)b' )} where n is in N. Show that if n and m are distinct, then the two groups G_n and G_m are not isomorphic. Ps. In fact G_n = G/ nZ , where G is the group of pairs (t,s) in C * C with group law (t,s)...
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    Graduate Group of 3x3 Matrices w/o Center - Complex & Real Numbers

    I mean all 3 by 3 matrices with the following rows (e^t, 0, u) (0, e^(tx), v) (0, 0, 1).
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    Graduate Group of 3x3 Matrices w/o Center - Complex & Real Numbers

    Hi all, can you show that the group of all 3 by 3 matrices [e^t 0 u 0 e^xt v 0 0 1] where t, u, v are in C (complex numbers) and x is in R (real number) has no center? Regards
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    Graduate Finding the Real Lie Algebra of SL(n,H) in GL(n,H)

    Actually, GL(n,H) is the set of all matrices A in GL(2n,C) such that AJ=Ja where a is complex conjugate of A and J is 2n by 2n matrice with the rows (0 -I) and (I 0), I is the n by n identity matrix. Thanks
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    Graduate Some questions on representation theory

    Hello all, Can anybody help me solve the following exercises from the book: Representation Theory, A first course by William Fulton and Joe Harris,1991 page 138, exercise 10.3 page 140, exercise 10.7 page 141, exercise 10.9 Thanks in Advance,