Top Commutative Algeb
Top Commutative Algeb MATH 681
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This 4 page Class Notes was uploaded by Henderson Lind II on Tuesday September 8, 2015. The Class Notes belongs to MATH 681 at University of Oregon taught by Jon Brundan in Fall. Since its upload, it has received 44 views. For similar materials see /class/187182/math-681-university-of-oregon in Mathematics (M) at University of Oregon.
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Date Created: 09/08/15
LIE ALGEBRAS 7 EXAMPLES SHEET 4 Notation Always g is a semisimple Lie algebra Fix a maximal toral subalgebra E of g with corresponding root system I Let A 041 al be a base for I IJr the corresponding positive roots W the Weyl group and e W a i1 be the sign representation of W relative to the simple re ections 31 31 Let w1 w1 denote the corresponding fundamental dominant weights so that w 04739 6 Let P and PJr denote the integral and dominant integral weights respectively 1 Let g 52C with standard basis efh Let 739 be the endomorphism of g de ned by 739 expad eexpad7fexpad 5 Verify explicitly that the auto morphism 739 acts on g as conjugation by the matrix 71 1 Deduce that W f f fed1 h 2 If V and V are nite dimensional g modules show that chV 63 V chV chV and chV X V chVchV 1 3 Show that p 5 Z 04 can also be written as wl 1 w a6 5 5A H 1757 Z aw17p aelt1gt weW 5 For the root system of type B2 order the base so 041 is the short root and 042 is the long root Compute the corresponding fundamental dominant weights wh Lug in terms of 041 and 042 then write 041 and 042 in terms of wl and Lug Do the same for A2 and G2 hint what are the Cartan matrices7 6 Apply Weyl s character formula to compute the dimension of all weight spaces of the module V2w1 1 mg where g B2 and LUth are as in question 5 Verify that the sum of the dimensions of all weight spaces equal the dimension as computed by Weyl s dimension formula 7 Let g 53C with fundamental dominant weights wl Lug Abbreviate Vm1w1 mgwg by Vm1m2 Use Weyl s dimension formula to show 4 For E P prove for yourself that ch M i 1 d1m Vm1 m2 m1 mg 8 With notation as in question 7 show that V1 1 V1 2 V2 3 V3 1 63 V0 4 69 V1 2 69 V1 2 69 V20 69 V01 9 Use Weyl s dimension formula to show that a faithful irreducible nite di mensional g module of smallest possible dimension has highest weight equal to w for some i Hence verify that the smallest dimension of a faithful irreducible Gg module is 7 10 Labelling Bg s Dynkin diagram the way we usually do explain without making any calculations why dim Lw1 5 and dim Lw2 4 Hint for the second one think about 02 Lie algebras Examples sheet 1 All Lie algebras and modules are nite dimensional over an arbitrary eld k unless otherwise stated De nition chasing questions 1 Let g be the real vector space R3 De ne z x y cross product of vectors and verify that this makes g into a Lie algebra Write down the multiplication table for g relative to the usual basis for R3 2 Letelt8 1 h 1 71 andf 1 8 beanordered basis for 52k Write down the multiplication table for 52k relative to this basis Hence compute the matrices of ad eadh and adf with respect to this basis 3 a Show that there is precisely one Lie algebra of dimension 1 up to isomorphism b Show that there are precisely two non isomorphic Lie algebras of dimen sion 2 one is abelian the other is not c Let g be the Lie algebra over k with basis zy z and relations z m y compare with question 1 If k C show that g is isomorphic to 52k but that this is false if k R So the classi cation of 3 dimensional Lie algebras depends on the ground eld 4 Prove that the centre of gk is the set of scalar matrices Prove that 5k has centre 0 unless chark divides n in which case the centre is again the set of scalar matrices 5 Show 52k is simple if chark 7 2 What happens if chark 2 Hints Work in the basis of question 2 By applying ade twice or adf twice show that if 0 7 as bf ch lies in an ideal I of52k then one of e f or h lies in I hence I 52k Derivations 6 Prove that the set of all inner derivations ad m z E g is an ideal of Der g 7 Verify that the commutator of two derivations of a k algebra is again a derivation Is the ordinary product always a derivation The PBW theorem 8 If g is a free Lie algebra on a set X7 show that Ug is isomorphic to TV7 where V is the vector space with X as basis 9 Describe the free Lie algebra on the set X 10 Let g be an arbitrary nite dimensional Lie algebra Use the PBW theorem to show that Ug has no zero divisors Soluble and nilpotent Lie algebras 11 Let unk and bnk be the set of all strictly upper triangular ie zeros on the diagonal and upper triangular ie anything on the diagonal n x n matrices over k respectively Show that these are Lie subalgebras of gnk 12 Let g unk as in question 11 Show that the lower central series of g is gg0gtg1gtg2gtgtg70 where g9 equals M E gnklMlj 0 for all 1g Lj n with j 7 s g i g Deduce that unk is nilpotent 13 Using question 127 show that bnk is soluble 14 Show 52k is nilpotent if ChaTk 2 15 Let g be nilpotent and E be a proper subalgebra of Q Show that ngE is strictly larger than E 16 Let k be a eld of characteristic p gt 0 Let 71 6 gpk be the following p x p matrices 0 1 0 0 0 0 0 1 0 0 m g g 7ydiag017p71 0 0 0 0 1 1 0 0 0 0 Show that 71 generate a 2 dimensional soluble subalgebra of gpk but that they have no common eigenvector Hence7 Lie s theorem is false in general in non zero characteristic The Killing form 17 Using question 27 compute the Killing form explicitly for 52C and hence verify directly that it is non degenerate on 52C 18 Let k have characteristic 3 Show 53k modulo its centre is semisimple but has degenerate Killing form 19 Let chark p 7 0 Prove that g is semisimple if its Killing form is non degenerate the converse fails by example 18 Jon Brundan7 26905