CHAPTER 2 Hopf Algebras, Algebraic, Formal, and Quantum Groups 43 44 2. HOPF ALGEBRAS, ALGEBRAIC, FORMAL, AND QUANTUM GROUPS 3. Ane Algebraic Groups We apply the preceding considerations to the categories K - cAlg and K - cCoalg. Consider K - cAlg, the category of commutative K -a lgebras. Let A; B 2 K - cAlg. Then A B is again a commutative K -a lgebra with comp onentwise multiplication. In fact this holds also for non-commutative K -a lgebras A.5.3, but in K - cAlg we have Prop osition 2.3.1. The tensor product in K - cAlg is the categorical coproduct. Proof. Let f 2 K - cAlgA; Z ;g 2 K - cAlgB; Z . The map [f; g]: A B ! Z de ned by[f; g]a b:= f ag b is the unique algebra homomorphism such that [f; g]a 1 = f a and [f; g]1 b=g b or such that the diagram A B - A A B B [f;g] @ g f @ @ @R ? Z commutes, where a=a 1 and b=1 b are algebra homomorphisms. A B So the category K - cAlg has nite copro ducts and also an initial ob ject K . A more general prop erty of the tensor pro duct of arbitrary algebras was already considered in 1.2.13. Observe that the following diagram commutes 1 2 - A A A A r @ 1 1 A A @ @ ? @R A where r is the multiplication of the algebra and by the diagram the co diagonal of the copro duct. De nition 2.3.2. An ane algebraic group is a group in the category of com- mutative geometric spaces. By the dualitybetween the categories of commutative geometric spaces and com- mutative algebras, an ane algebraic group is represented by a cogroup in the cate- gory of K - cAlg of commutative algebras. For an arbitrary ane algebraic group H we get by Corollary 2.2.7 = 2 K - cAlgH; H H ; 1 2 1 " = e 2 K - cAlgH; K ; and S = id 2 K - cAlgH; H : These maps and Corollary 2.2.7 lead to Prop osition 2.3.3. Let H 2 K - cAlg. H is a representing object for a functor K - cAlg ! Gr if and only if H is a Hopf algebra. 3. AFFINE ALGEBRAIC GROUPS 45 Proof. Both statements are equivalent to the existence of morphisms in K - cAlg :H ! H H " : H ! K S : H ! H such that the following diagrams commute - H H H coasso ciativity 1 ? ? 1 - H H H H H - H H H P P P P counit P 1 " 1 P P P P ? ? Pq " 1 - H H K H H H K = = - - H H K 6 coinverse r ? S id - H H H H id S This Prop osition says two things. First of all each commutative Hopf algebra H de nes a functor K - cAlgH; - : K - cAlg ! Set that factors through the category of groups or simply a functor K - cAlgH; - : K - cAlg ! Gr. Secondly each repre- sentable functor K - cAlgH; - : K - cAlg ! Set that factors through the category of groups is represented by a commutative Hopf algebra. Corollary 2.3.4. An algebra H 2 K - cAlg represents an ane algebraic group if and only if H isacommutative Hopf algebra. The category of commutative Hopf algebras is dual to the category of ane alge- braic groups. In the following lemmas we consider functors represented by commutative alge- bras. They de ne functors on the category K - cAlg as well as more generally on K - Alg.We rst study the functors and the representing algebras. Then we use them to construct commutative Hopf algebras. + Lemma 2.3.5. The functor G : K - Alg ! Ab de nedbyG A:=A , the a a underlying additive group of the algebra A,isarepresentable functor representedby the algebra K [x] the polynomial ring in one variable x. Proof. G is an underlying functor that forgets the multiplicative structure of a the algebra and only preserves the additive group of the algebra. Wehave to determine + natural isomorphisms natural in A G A K - AlgK [x];A. Each element a 2 A = a is mapp ed to the homomorphism of algebras a : K [x] 3 px 7! pa 2 A. This is a homomorphism of algebras since a px+ q x = pa+ q a= a px + a q x 46 2. HOPF ALGEBRAS, ALGEBRAIC, FORMAL, AND QUANTUM GROUPS and a pxq x = paq a= a pxa q x. Another reason to see this is that K [x] is the free commutative K -algebra over fxg i.e. since each map fxg! A can b e uniquely extended to a homomorphism of algebras K [x] ! A. The map A 3 a 7! a 2 K - AlgK [x];A is bijective with the inverse map K - AlgK [x];A 3 f 7! f x 2 A. Finally this map is natural in A since - - A K - AlgK [x];A g K-AlgK [x];g ? ? - B K - AlgK [x];B - commutes for all g 2 K - AlgA; B . + Remark 2.3.6. Since A has the structure of an additive group the sets of ho- momorphisms of algebras K - AlgK [x];A are also additive groups. Lemma 2.3.7. The functor G = U : K - Alg ! Gr de nedbyG A:=U A, m m the underlying multiplicative group of units of the algebra A,isarepresentable functor 1 represented by the algebra K [x; x ]= K [x; y ]=xy 1 the ring of Laurent polynomials in one variable x. Proof. Wehave to determine natural isomorphisms natural in A G A = m 1 K - AlgK [x; x ];A. Each element a 2 G A is mapp ed to the homomorphism of m 1 algebras a := K [x; x ] 3 x 7! a 2 A. This de nes a unique homomorphism of 1 algebras since each homomorphism of algebras f from K [x; x ]=K [x; y ]=xy 1 to A is completely determined by the images of x and of y but in addition the images have to satisfy f xf y = 1, i.e. f xmust b e invertible and f y must b e the inverse to f x. This mapping is bijective. Furthermore it is natural in A since - 1 - A K - AlgK [x; x ];A 1 g K -AlgK [x;x ];g ? ? 1 - B K - AlgK [x; x ];B - for all g 2 K - AlgA; B commute. Remark 2.3.8. Since U A has the structure of a multiplicative group the sets 1 K - AlgK [x; x ];A are also groups. Lemma 2.3.9. The functor M : K - Alg ! K - Alg with M A the algebraof n n n n-matrices with entries in A is representable by the algebra K h x ;x ;::: ;x i, 11 12 nn the non commutative polynomialring in the variables x . ij Proof. The p olynomial ring K h x i is free over the set fx g in the category of ij ij non commutative algebras, i.e. for each algebra and for each map f : fx g! A ij 3. AFFINE ALGEBRAIC GROUPS 47 there exists a unique homomorphism of algebras g : K hx ;x ;::: ;x i! A such 11 12 nn that the diagram - fx g K hx i ij ij @ @ g f @ @ ? @R A commutes. So each matrix in M A de nes a unique a homomorphism of algebras n K hx ;x ;::: ;x i! A and conversely. 11 12 nn Example 2.3.10. 1. The ane algebraic group called additive group G : K - cAlg ! Ab a + with G A:= A from Lemma 2.3.5 is represented by the Hopf algebra K [x]. We a determine comultiplication, counit, and antip o de. By Corollary 2.2.7 the comultiplication is = 2 K - cAlgK [x]; K [x] 1 2 G K [x] K [x]. Hence K [x] = a x= x+ x=x 1+1 x: 1 2 G K hence The counit is " = e =02 K - cAlgK [x]; K = a K "x= 0: 1 G K [x] hence The antipode is S =id 2 K - cAlgK [x]; K [x] = a K[x] S x= x: 2. The ane algebraic group called multiplicative group G : K - cAlg ! Ab m with G A:=A = U A from Lemma 2.3.7 is represented by the Hopf algebra m 1 K [x; x ]= K [x; y ]=xy 1. We determine comultiplication, counit, and antip o de. By Corollary 2.2.7 the comultiplication is 1 1 1 1 1 = 2 K - cAlgK [x; x ]; K [x; x ] K [x; x ] G K [x; x ] K [x; x ]: = 1 2 m Hence x= x x=x x: 1 2 1 The counit is " = e =12 K - cAlgK [x; x ]; K G K hence = K m "x= 1: 1 1 1 1 The antipode is S =id 2 K - cAlgK [x; x ]; K [x; x ] G K [x; x ] = 1 a K[x;x ] hence 1 S x= x : 3. The ane algebraic group called additive matrix group + M : K - cAlg ! Ab ; n 48 2. HOPF ALGEBRAS, ALGEBRAIC, FORMAL, AND QUANTUM GROUPS + with M A the additive group of n n-matrices with co ecients in A is represented n + by the commutative algebra M = K [x j1 i; j n] Lemma 2.3.9. This algebra ij n must b e a Hopf algebra. + + + + + + The comultiplication is = 2 K - cAlgM ;M M M M M . = 1 2 n n n n n n Hence x = x + x =x 1+1 x : ij 1 ij 2 ij ij ij + + The counit is " = e = 0 2 K - cAlgM ; K M K hence = K n n "x = 0: ij 1 + + + + 2 K - cAlgM ;M M M hence The antipode is S =id = + n n n n M n S x =x : ij ij 4.