Restrcted Le Algebras Jared Warner
1. Defntons and Examples Defnton 1.1. Let k be a feld of characterstc p. A restrcted Le algebra (g, ( ) [p] ) s a Le algebra g over k and a map ( ) [p] : g g called the p-operaton such that for all a k and for all x, y g we have: (ax) [p] = a p x [p] ad x [p] =ad p x and (x + y) [p] = x [p] + y [p] + expresson ad tx+y(x) =1 s (x, y) where s (x, y) s the coeffcent of t 1 n the Recall that for any x g ad x : g g s the lnear map defned by ad x (y) = [x, y] for all y g. Example 1.2. Let g be any abelan Le algebra, e, a Le algebra such that [x, y] = 0 for all x, y g. Notce that blnearty and alternatvty ([x, x] = 0 for all x g) mply that any one-dmensonal Le algebra s abelan. The p-operaton defned by x [p] = 0 for all x g gves g the structure of a restrcted Le algebra. Notce that the condton that g be abelan s necessary, for f we consder gl n, the Le algebra of n n matrces wth entres n k whose bracket s gven by commutaton, t can be shown that the second and thrd condtons lsted above are not satsfed by the p-operaton A [p] = 0 for all matrces A gl n. Example 1.3. Let A be any assocatve, untal algebra over k (from here on, unless otherwse specfed, by algebra we mean an assocatve, untal algebra over k). Defne a bracket and p-operaton on elements of A by x [p] := x p and [x, y] := xy yx for all x, y A. We check that such defntons gve A the structure of a restrcted Le algebra. Let x, y A and a k. (ax) p = a p x p = a p x [p], so the frst condton above holds. ( ) n ad x [p](y) =ad x p(y) = [x p, y] = x p y yx p. Usng the combnatoral dentty + ( ) n n ( ) n along wth nducton, one can show that ad n 1 x(y) = ( 1) x n yx. ( ) =0 p Snce s dvsble by p for all = 1, 2,...,, t follows that ad p x(y) = x p y yx p, and hence the second condton above holds. The thrd condton s dffcult to check n generalty. Usng the above formula, we ( ) p 1 have that ad tx+y(x) = ( 1) (tx + y) x(tx + y). I clam that the =0 coeffcent of t j 1 n ths expresson s j (all words of length p n x and y that have j x s). From the formula, t can be seen that the coeffcent of t j 1 s equal ( ) p 1 (all to ( 1) words of length p n x and y wth j x s, whose (p ) th =0 entry s x). Gven a word of length p n x and y wth j x s, we would lke to know how many tmes ths word s counted n the gven sum. Fx a word w of the desred 3
type and let J {0, 1, 2,..., } be the subset of j elements such that for all J, w has an x n the (p ) th poston. Then ths word s counted ( ) p 1 ( 1) ( ) J p 1 many tmes. Snce ( 1) 1modp for all {0, 1, 2,..., p 1} as can be checked wth the above combnatoral dentty, t follows that each word s counted j tmes, provng the clam. Now (x + y) [p] = (x + y) p = x p + y p + (words n x and y of length p) = x [p] + y [p] + =1 s (x, y), and the thrd condton holds as well. Denote by A L the restrcted Le algebra assocated to the algebra A. We use the same notaton for the Le algebra (not restrcted) assocated to A (wth bracket gven by the commutator as defned above). 2. Envelopng Algebras Here we desre to dscuss the restrcted envelopng algebra u(g) of a restrcted Le algebra g. We frst brefly revew the defnton of the unversal envelopng algebra. Defnton 2.1. Let k be a feld of any characterstc, and let g be a Le algebra over k. The unversal envelopng algebra of g s an algebra U together wth a map of Le algebras h : g U L satsfyng the followng unversal property: gven any algebra A and any map of Le algebras f : g A L, there exsts a unque map of algebras g : U A such that f = g h. Note that the unversal property descrbed mples that f a unversal envelopng algebra exsts for g, then t s unque up to unque somorphsm. That s, f U and U both satsfy the unversal property, then there s a unque algebra somorphsm U U. Ths can be seen as follows. Let h : g U L and h : g U L both satsfy the unversal property. Hence, there exst unque maps g : U U and g : U U satsfyng h = g h and h = g h. Hence h = g g h. Snce we also have h = d U h, by unqueness, t follows that g g = d U. The same argument apples to g g, so that g and g are algebra somorphsms. We now show that the unversal envelopng algebra always exsts for for any Le algebra. Theorem 2.2. Let g be a Le algebra. exsts. The unversal envelopng algebra of g always Proof. Let T (g) = n 0 T n (g) be the tensor algebra of g where T n (g) = g g... g n-tmes, and multplcaton s defned by concatenaton of smple tensors. Notce that ths multplcaton turns T (g) nto a graded algebra. Notce that g T (g) by takng x g to tself n T 1 (g) = g T (g). Next, let I be the two-sded deal generated by elements of the form x y y x [x, y] where x, y g, and let U = T (g)/i. Let h : g U L take x g to the coset x + I n U. Notce ths s a map of Le algebras precsely because x y y x = [x, y] n U. We show that the par (U, h) satsfy the unversal property descrbed above. Suppose A s an algebra, and f : g A L s a map of Le algebras. Defne a map of algebras g : U A by mappng x + I to f(x) and extendng lnearly and multplcatvely. Thus, 4
g h(x) = g(x+i) = f(x), and g s unque because any map of algebras U A s determned by where t sends a set of generators, n ths case, {x + I} x g=t 1 (g). The dscusson before the proof justfes the notaton U(g) for the unversal envelopng algebra of a Le algebra g. Let s calculate some unversal envelopng algebras for some specfc Le algebras. Example 2.3. Let g = g m := k L, e, g m s a one dmensonal k vector space wth bracket gven by the commutator n k. Snce g m s one dmensonal, t s necessarly abelan. Hence the unversal envelopng algebra s commutatve. We have mplctly defned two functors n the precedng pages. The functor U( ) : {Le algebras} {algebras} takes a Le algebra to ts unversal envelopng algebra. The functor ( ) L : {algebras} {Le algebras} takes an algebra to ts assocated Le algebra. Notce that the unversal property states that any f Hom Le alg (g, A L ) unquely defnes a g Hom alg (U(g), A). Conversely, any algebra map U(g) A unquely defnes a map of Le algebras g A L. Hence there s a bjecton between the Hom sets Hom Le alg (g, A L ) and Hom alg (U(g), A) for any Le algebra g and algebra A. Moreover, ths bjecton s natural wth respect to g and A, meanng the correspondence of Hom sets s consstent wth Le algebra maps h g and algebra maps B A. In other words, for all maps g h, B A, the followng dagram commutes: (1) Hom Le alg (g, A L ) Hom alg (U(g), A) Hom Le alg (h, B L ) Hom alg (U(h), B) Here the vertcal arrows are nduced by the maps g h, and B A. To see why ths s true, start n the bottom rght wth a map of algebras U(h) B. Movng left gves us a map of Le algebras h B L nduced by restrcton. Movng up then yelds a map of Le algebras g h B L A L. If we ntally move up, we have the map of algebras U(g) U(h) B A, and then movng left, we restrct to g U(g), yeldng g h B L A L. Functors that gve such a correspondence on Hom sets are called adjont functors, and are now defned n greater generalty. Defnton 2.4. Let C and D be two categores, and G : C D and F : D C be two functors. Suppose for all objects X C and Y D there s a natural bjecton Hom C (F (Y ), X) Hom D (Y, G(X)), e, a bjecton such that for all morphsms Y Y and X X the followng dagram commutes: Hom D (Y, G(X)) Hom C (F (Y ), X) (2) Hom D (Y, G(X ) Hom C (F (Y ), X ) Then F s called left adjont to G and G s sad to be rght adjont to F. 5
In our defnton of adjont functors, we recover the case just dscussed f C s the category of algebras, D s the category of Le algebras, F = U( ) and G = ( ) L. Havng suffcently revewed the unversal envelopng algebra of a Le algebra, we now defne the restrcted envelopng algebra of a restrcted Le algebra. To do so, we need the noton of a map of restrcted Le algebras. Defnton 2.5. Let g and h be restrcted Le algebras. A k-lnear map f : g h s a homomorphsm of restrcted Le algebras f: f([x, y] g ) = [f(x), f(y)] h and f(x [p]g ) = f(x) [p] h for all x, y g. Ths s the clear defnton of a homomorphsm n any category respectng the algebrac structure. We now defne the restrcted envelopng algebra analagously to how we defned the unversal envelopng algebra, through a unversal property. As above, we wll show f t exsts, the restrcted envelopng algebra s unque up to unque somorphsm, and we wll construct t as a quotent of the unversal envelopng algebra. Defnton 2.6. Let g be a restrcted Le algebra. The restrcted envelopng algebra of g s an algebra u together wth a map of restrcted Le algebras h : g u L satsfyng the followng unversal property: gven any algebra A and any map of restrcted Le algebras f : g A L, there exsts a unque map of algebras g : u A such that f = g h. Theorem 2.7. The restrcted envelopng algebra of a restrcted Le algebra g exsts, and s unque up to unque somorphsm. Proof. The proof of unqueness s no dfferent than that for the unversal envelopng algebra. To construct the restrcted envelopng algebra, let J U(g) be the two-sded deal generated by elements of the form x x... x x [p] + I, where the tensor occurs p tmes. Let u = U(g)/J. We can stll thnk of g as sttng nsde of u by sendng x g to the element (x + I) + J u. Notce that ths embeddng nduces a map of restrcted Le algebras h : g u L precsely because quotentng by J forces the p-operaton n u L to agree wth that of g nsde of u L. We show that the par (u, h) satsfy the unversal property descrbed above. Suppose A s an algebra, and f : g A L s a map of restrcted Le algebras. Defne a map of algebras g : u A by mappng (x + I) + J to f(x) and extendng lnearly and multplcatvely. Thus, g h(x) = g((x + I) + J) = f(x), and g s unque because any map of algebras u A s determned by where t sends a set of generators, n ths case, {(x + I) + J} x g u. Agan, by unqueness, we may use the notaton u(g) when speakng of the restrcted envelopng algebra assocated to g. 3. Representatons Before movng on to our desred dscusson of the connecton between restrcted Le algebras and heght 1 nfntesmal group schemes, t makes sense that we dscuss representatons of Le algebras and restrcted Le algebras, havng just spent a consderable amount of tme developng the concept of envelopng algebras. 6
Defnton 3.1. Let g be a Le algebra. A representaton of g s a map of Le algebras ρ : g gl(v ) where V s a k vector space. If g s a restrcted Le algebra, then a representaton of g s a map of restrcted Le algebras ρ : g gl(v ) where the p-operaton n gl(v ) s gven by p th -powers of matrces. We now show that a representaton of g s equvalent to a module over the approprate envelopng algebra. Workng wth modules s often easer than workng wth maps of Le algebras, so ths correspondence s qute useful. Theorem 3.2. A representaton of a (restrcted) Le algebra g defnes a (u(g)-) U(g)- module structure on V. Smlarly, a (u(g)-) U(g)-module V defnes a representaton of g. Proof. We start wth the case of a Le algebra g. Let ρ : g gl(v ) be a representaton. Defne a U(g)-module structure on V by x v := ρ(x)v for all x g U(g) and all v V, and extend multplcatvely and lnearly. To show ths scalar multplcaton s well-defned, we must show that x y y x and [x, y] n U(g) yeld the same acton on v for all x, y g. Ths follows from the fact that ρ s a map of Le algebras, and that the bracket n gl(v ) s gven by commutaton. Now suppose we are gven a U(g) module V. Defne a map ρ : g gl(v ) by ρ(x)v := x v. Lnearty follows from the module structure, and ρ preserves the bracket because x y y x = [x, y] n U(g). For the case of restrcted Le algebras, everythng from above holds, wth one extra step to deal wth the p-operaton. The detals are nearly dentcal to how we dealt wth the bracket. Example 3.3. Let g be a restrcted Le algebra. Consder the map: ρ : g gl(g) x ad x We check that ths s ndeed a map of restrcted Le algebras. Lnearty follows from that of the bracket. The Jacob dentty shows that the above map s a map of Le algebras (e, the bracket s preserved). Snce restrcted Le algebras satsfy ad x [p] =ad p x, the above map also preserves the p-operaton. Hence we have a representaton of g called the adjont representaton. Notce that f we forget the p-operaton, we stll have a representaton of a Le algebra. 4. Hopf Algebra Structure 5. Infntesmal Groups Schemes of Heght 1 In ths secton we would lke to establsh an equvalence of categores between heght 1 nfntesmal group schemes G and fnte-dmensonal restrcted Le algebras g. Let C be the category of heght 1 nfntesmal group schemes, and let D be the category of restrcted Le algebras. We begn by defnng a functor E : C D. Let G C. There are many ways to defne E, and all such are naturally somorphc. We take the vew of k-lnear ɛ-dervatons, e, E(G) := Der(k[G], k), where for a commutatve rng R and an R-module M, Der(R, M) s all k-lnear maps satsfyng f(rs) = rf(s) + sf(r) for all r, s R. Here we vew k as a k[g]-module va the augmentaton map ɛ. To gve Der(k[G], k) the structure of a restrcted Le algebra, frst notce that Der(k[G], k) s a k-sapce va pontwse addton and scalar multplcaton. Turn Der(k[G], k) nto a untal, assocatve k-algebra by defnng a multplcaton as follows: f g := (f g). It can be shown 7
that Der(k[G], k) s closed under ths multplcaton. Havng defned an algebra structure on Der(k[G], k), we can now make t a restrcted Le algebra as n 1 by defnng the bracket to be the commutator and the p-operaton to be p th -powers. Now we construct a functor F : D C. Let g D. Then as shown prevously, u(g) s a local, commutatve Hopf algebra, satsfyng x p = 0 for all x m where m s the unque maxmal deal n u(g). Hence, g defnes a heght one nfntesmal group scheme G g := Hom k alg (u(g), ). Let φ : g h be a map of restrcted Le algebras. We make explct a number of computatons to further clarfy the above equvalence. Example 5.1. Let G = G a(1), the frst Frobenus kernel of G a. Then k[g = G a(1) = k[t ]/T p, where T s a prmtve element n the Hopf algebra structure. Usng the propertes of a k-lnear ɛ dervaton, tt can be shown that n ths case, f f Der(k[G], k), then f(a 0 + a 1 T +... + a T ) = a 1 f(t ), and s thus completely determned by where t maps T. Thus, Der(k[G], k) s a one-dmensonal k-algebra. Now suppose f(t ) = a and g(t ) = b for f, g Der(k[G], k). Then (f g)(t ) = a + b so that both the bracket and p-operaton are trval. Hence we see that E(G) = g a. In the other drecton, let s consder the trval restrcted Le algebra, g a, a one-dmensonal (necessarly abelan) Le algebra wth trval p- operaton, e, x [p] = 0 for all x g a. I clam that the heght one group scheme correspondng to g a s G a,(1), the frst Frobenus kernel of G a. We have that k[g a,(1) ] = k[t ]/(T p ) = kg m,(1), e, k[t ]/(T p ) s self-dual. The clam s a result of the computaton of the restrcted envelopng algebra of g a done above, whch s k[t ]/(T p ). Example 5.2. Generalzng the prevous example, let us consder g r a, the r-dmensonal trval restrcted Le-algebra. u(g r a ) = k[x 1,..., x r ]/(x p 1,..., x p r) as was shown above, and snce ths Hopf algebra s also self dual, we are lookng for a heght one group scheme whose coordnate algebra s k[x 1,..., x r ]/(x p 1,..., x p r). The desred group scheme s G r a,(1). Example 5.3. Let g = g m := k L, e, g m s a one dmensonal k vector space wth bracket and p-operaton gven by commutator and p th powers respectvely. The calculaton of u(g m ) was done above, yeldng the result u(g m ) = k[x]/(x p x). I clam that the heght one group scheme correspondng to g m s G m,(1), the frst Frobenus kernel of G m. We have that k[g m,(1) ] = k[t ]/(T p 1) and kg m,(1) = k Z/pZ so to prove the clam, t suffces to exhbt a Hopf algebra somorphsm f : k[x]/(x p x) k Z/pZ. The approprate map s gven by x e. Snce algebra maps preserve multplcatve denttes, we must have 1 e. =1 Extendng ths map multplcatvely, we see that x n n e because the e are mutually orthogonal dempotents. That x p x maps to zero s a result of Fermat s lttle theorem. It can also be shown that f preserves the count, comultplcaton, and the antpode. If we let {1, x, x 2,..., x } and {e 0, e 1, e 2,..., e } be ordered bases for k[x]/(x p x) and k Z/pZ respectvely, then the matrx of f, vewed as a lnear transformaton s gven by: 8 =1 =0
1 0 0 0... 0 1 1 1 1... 1 1 2 4 8... 2 (3) f = 1 3 9 27... 3........ 1 p 1 (p 1) 2 (p 1) 3... (p 1) where the entres are to be reduced mod p. Ths matrx has a non-zero determnant because t has the same determnant as ts p 1 p 1 Vandermonde sub-matrx excludng the frst row and column. Ths sub-matrx has non-zero determnant because each row s generated by powers of the dstnct numbers 1, 2, 3,..., p 1. Thus, the matrx s nvertble and f s an somorphsm. The nverse s gven by the followng matrx: (4) f 1 = 1 0 0 0... 0 0 1 2 p 2 3 p 2... (p 1) p 2 0 1 2 p 3 3 p 3... (p 1) p 3........ 0 1 4 9... (p 1) 2 0 1 2 3... (p 1) 1 1 1 1... 1 I ve wrtten the matrces wth entres that suggest the pattern nvolved, neglectng to reduce mod p. Here I wrte out the matrces for the case p = 5, reducng the entres approprately: (5) f = 1 0 0 0 0 1 1 1 1 1 1 2 4 3 1 1 3 4 2 1 1 4 1 4 1 f 1 = 1 0 0 0 0 0 4 2 3 1 0 4 1 1 4 0 4 3 2 1 4 4 4 4 4 Notce that f 1 gves a system of p mutually orthogonal dempotents n the k[x]/(x p x) as the mages of the e, a nontrval result. To be explct, we have: (6) e 0 1 x e n n 1 x p n = 1, 2,..., p 1 =1 Example 5.4. Combnng the prevous examples, we see that f g = g r m, then G g = G r m,(1). Example 5.5. In ths example we would lke to make explct the result suggested by notaton, that s, that the restrcted Le algebras gl n and sl n correspond to the heght one groups schemes GL n,(1) and SL n,(1) respectvely. Here k[gl n,(1) ] = k[x j ]/(x p j δ j) and k[sl n,(1) ] = 9