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Appendix A Linear Algebraic Groups A.1 Linear Algebraic Groups Let us fix an algebraically closed field K. We work in the category of affine varieties over K. The purpose of this section is to give a brief exposition on the basic facts of algebraic groups. Definition A.1.1 A linear algebraic group is an affine algebraic variety G together with a unit element e 2 G and morphisms m W G G ! G and i W G ! G satisfying the group axioms (a) m.; e/ D m.e;/D for all 2 G; (b) m.; i.// D m.i./; / D e for all 2 G; (c) m.; m.; // D m.m.; /; / (associativity) for all ; ; 2 G. Often we will just write and 1 instead of m.; / and i./, respectively. Example A.1.2 The group GLn, the set of invertible nn matrices over K, is a linear algebraic group. Any Zariski closed subgroup G GLn is a linear algebraic group. In fact, every linear algebraic group is isomorphic to a Zariski closed subgroup of GLn (see Borel [1, Proposition I.1.10]). This justifies calling it a linear algebraic group. G The axioms for a linear algebraic group can also be stated in terms of the coordinate ring. The maps m W G G ! G and i W G ! G correspond to ring homomorphisms m W KŒG ! KŒG ˝ KŒG and i W KŒG ! KŒG. The unit element e 2 G corresponds to a ring homomorphism W KŒG ! K defined by f 7! f .e/. The axioms for a linear algebraic group translate to (a) .id ˝/ ı m D . ˝ id/ ı m D id; (b) .i ˝ id/ ı m D .id ˝i/ ı m D (using the inclusion K KŒG we can view W KŒG ! K as a map KŒG ! KŒG); (c) .id ˝m/ ı m D .m ˝ id/ ı m. © Springer-Verlag Berlin Heidelberg 2015 297 H. Derksen, G. Kemper, Computational Invariant Theory, Encyclopaedia of Mathematical Sciences 130, DOI 10.1007/978-3-662-48422-7 298 A Linear Algebraic Groups Example A.1.3 The additive group Ga is defined as the additive group of K.We define m.x; y/ D x C y, i.x/ Dx for all x; y 2 K and e D 0 2 K. The coordinate ring KŒGa is equal to the polynomial ring KŒt. The homomorphism m W KŒt ! KŒt ˝ KŒt is defined by m.t/ D t ˝ 1 C 1 ˝ t, i W KŒt ! KŒt is defined by t 7!t and W KŒt ! K is defined by f 7! f .0/. Example A.1.4 The group GLn is an algebraic group. The coordinate ring KŒGLn is isomorphic to KŒfzi;j j 1 Ä i; j Ä ng; T=. f /; . / 1 . /n Œ Œ where f D T det Z with ZPthe matrix zi;j i;jD1.Nowm W K GLn ! K GLn ˝ Œ n ; . // K GLn is defined by zi;j 7! kD1 zi;k ˝ zk;j for all i j. Notice that m det Z D det.Z/ ˝ det.Z/. Example A.1.5 Let T D .K/r. The coordinate ring KŒT is isomorphic to Œ ;:::; ; 1;:::; 1: K z1 zr z1 zr The ring homomorphisms m W KŒT ! KŒT ˝ KŒT, i W KŒT ! KŒT and Œ . / . / 1 . / 1 W K T ! K are given by m zi D zi ˝ zi, i zi D zi and zi D for all i. G Definition A.1.6 Suppose that X is an affine variety. A regular action of G on X is a morphism W G X ! X satisfying the axioms for an action (a) .e; x/ D x for all x 2 X; (b) .; .; x// D .m.; /; x/ for all ; 2 G and all x 2 X. We will often just write x instead of .; x/. Instead of saying that G acts regularly on X, we often just say that X is a G-variety. Definition A.1.7 Suppose that V is a (possibly infinite dimensional) vector space over K. A linear action W G V ! V of G on a vector spaceP is called rational ? Œ .; v/ l v ./ if there isP a map W V ! V ˝ K G satisfying D iD1 i fi whenever ?.v/ l v D iD1 i ˝ fi Lemma A.1.8 Suppose that a linear action of a linear algebraic group G on a vector space V is rational. For every v 2 V there exists a finite dimensional G-stable subspace W  V containing v. The restriction of the G-action to W is also rational. P ?.v/ l v ;:::; Proof Write D iD1 i ˝ fi with f1 fl linearly independent over K.Let W D K.G v/, the linear span of the orbit G v. It is not hard to see that W is a subspace of the space spanned by v1;:::;vl. This shows that dim W < 1. We can restrict ? to W to obtain a map W ! W ˝KŒG. Therefore the G-action on W is rational. ut A.2 The Lie Algebra of a Linear Algebraic Group 299 The content of Lemma A.1.8 can be expressed by saying that a rational action is “locally finite”. If G acts regularly on an affine variety X,thenG also acts on the coordinate ring KŒX.For 2 G and f 2 KŒX we define f 2 KŒX by . f /.x/ D f . 1 x/ for all x 2 X. AmapX ! Y between two sets on which G acts is called G-equivariant if '. x/ D '.x/ for all x 2 X and all 2 G. Lemma A.1.9 Suppose that X is a G-variety. Then there exists a finite dimensional rational representation V and a G-equivariant closed embedding X ,! V. Proof Choose generators f1;:::;fr of KŒX and let W be a finite dimensional G- stable subspace of KŒX containing f1;:::;fr. The inclusion W ! KŒX extends to a surjective G-equivariant ring homomorphism S.W/ KŒX,whereS.W/ is the symmetric algebra. This ring homomorphism corresponds to a G-equivariant closed embedding X ,! W where W is the dual space of W. ut The following two propositions can be found in Borel [1, Proposition I.1.2]. Proposition A.1.10 A linear algebraic group G is smooth. Proposition A.1.11 Let Gı be the connected component of e 2 G. Then Gı is a normal subgroup of G of finite index. A.2 The Lie Algebra of a Linear Algebraic Group We will first study the dual vector space KŒG of KŒG. As we will see, KŒG contains the Lie algebra g as a finite dimensional subspace. For every 2 G we can define W KŒG ! K by f 7! f ./. In this way we can view G as a subset of KŒG. By Definition A.2.1 below, the group structure of G equips KŒG with an associative algebra structure with identity element D e. This even allows us to see KŒG as an enveloping algebra of g (containing the universal enveloping algebra of g). Our approach is similar to Borel [1,§I.3]. Definition A.2.1 Suppose that ı; 2 KŒG. Then we define the convolution ı 2 KŒG as the composition of ˝ ı W KŒG ˝ KŒG ! K ˝ K D KPand m W Œ Œ Œ Œ . / K G ! K G ˝ K G . In other words, suppose f 2 K G and m f D i gi ˝ hi, then X . ı/.f / D .gi/ı.hi/: i 300 A Linear Algebraic Groups Proposition A.2.2 ThespaceKŒG is an associative algebra with the multiplica- tion and unit element D e. Proof From the axiom m.m.; /; / D m.; m.; //; it follows that .m ˝ id/ ı m D .id ˝m/ ı m: This multiplication is associative, because .ı / ' D ...ı ˝ /ı m/ ˝ '/ ı m D .ı ˝ ˝ '/ ı .m ˝ id/ ı m D D .ı ˝ ˝ '/ ı .id ˝m/ ı m D .ı ˝ .. ˝ '/ ı m// ı m D ı . '/: (A.2.1) Notice that from m.e;/D it follows that . ˝ id/ ı m D id. We get ı D .ı ˝ / ı m D ı ı .id ˝/ ı m D ı ı id D ı; and similarly ı D ı. ut Example A.2.3 If f 2 KŒG and 2 G,wedefineL . f / and R . f / by L . f /./ D f ./ and R . f /./ D f ./. Notice that .id ˝ / ı m D R and . ˝id/ ı m D L . From this it follows that ı D ı ı .id ˝ / ı m D ı ı R and ı D ı ı . ˝ id/ ı m D ı ı L : G Definition A.2.4 We define the Lie algebra g of G as the set of all ı 2 KŒG satisfying ı. fg/ D ı. f /g.e/ C f .e/ı.g/: (A.2.2) for all f ; g 2 KŒG. An element ı 2 KŒG satisfying (A.2.2) is called a point derivation of KŒG at e.Letme be the maximal ideal of KŒG vanishing at e 2 G. The Zariski tangent . / . = 2/ = 2 space Te G of G at e is defined as me me which is the dual space of me me.If A.2 The Lie Algebra of a Linear Algebraic Group 301 ı . = 2/ .ı/ Œ 2 me me , we can define 2 K G by .ı/ ı. / 2/: f D f f e C me It is not hard to show that .ı/ is a point derivation at e and that ı 7! .ı/ is an isomorphism between Te.G/ and g.
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