A class of continuous non-associative algebras arising from algebraic groups including E 8
aa r X i v : . [ m a t h . R A ] J un A CLASS OF CONTINUOUS NON-ASSOCIATIVE ALGEBRASARISING FROM ALGEBRAIC GROUPS INCLUDING E MAURICE CHAYET AND SKIP GARIBALDI
Abstract.
We give a construction that takes a simple linear algebraic group G over a field and produces a commutative, unital, and simple non-associativealgebra A over that field. Two attractions of this construction are that (1)when G has type E , the algebra A is obtained by adjoining a unit to the3875-dimensional representation and (2) it is effective, in that the productoperation on A can be implemented on a computer. A description of thealgebra in the E case has been requested for some time, and interest has beenincreased by the recent proof that E is the full automorphism group of thatalgebra. The algebras obtained by our construction have an unusual Peircespectrum. Contents
1. Introduction 12. Background material 33. The representation A ( g ) 54. The commutative algebra A ( g ) 65. A ( g ) as an algebra obtained by adding a unit 86. Associativity of the bilinear form τ A ( g ) as a representation of G τ is nondegenerate and A ( g ) is simple 139. The group scheme Aut( A ( g )) 1410. Construction A ( g ) in End( V ) 1411. Final remarks 17Appendix A. Adjoining a unit to a k -algebra 17References 211. Introduction
We present a construction that takes an absolutely simple linear algebraic group G over a field k and produces a commutative, unital non-associative algebra thatwe denote by A ( g ). As a vector space, A ( g ) is a subspace of the symmetric squareS g of the Lie algebra g of G . We give an explicit formula (4.1) for the producton A ( g ), which makes our construction effective in the sense that one can performcomputer calculations ( § A ( g ), which we show isassociative ( §
6) and nondegenerate ( §
8) and positive-definite in case k = R and G Mathematics Subject Classification.
Primary 17B25; Secondary 17D99, 20G41. is compact. We leverage this and the structure of A ( g ) as a representation of G toshow that it is a simple k -algebra (Cor. 8.4).This work may be viewed in the context of the general problem of describingexceptional groups as automorphism groups, which dates back to Killing’s 1889paper [Kil89]. As an example, the Lie group G can be viewed as the automorphismgroup of the octonions (E. Cartan [Car14]), the stabilizer of a cross product on R (F. Engel, [Eng00], [Her83]), or the symmetry group for a ball of radius 1 rolling ona fixed ball of radius 3 without slipping or twisting (E. Cartan, [BH14]). For E , itis known from [GG15] that it is the identity component of the stabilizer of an octicform on the Lie algebra e and that it is the automorphism group of the E -invariantalgebra on its 3875-dimensional irreducible representation. (See also [Gar16, §
3] or[GG15, §
16] for broader discussions of other realizations.) The latter description of E is known to be true even though this algebra is not well-understood; this papergives explicit and effective formulas for calculating in the algebra. We note herethat Aut( A ( e )) = E , see Prop. 9.1.The algebras A ( g ) constructed here are “non-generic” in the sense of [KT19],meaning that A ( g ) ⊗ k contains infinitely many idempotents, for k an algebraicclosure of k . Moreover, the Peirce spectrum of A ( g ) ⊗ k , i.e., the union of the setof eigenvalues for left multiplication by u as u varies over idempotents of A ( g ) ⊗ k ,is infinite, see Example 4.9. In case k = R and apart from types A and A , thiscollection of eigenvalues includes the unit interval, and consequently one might callthese algebras “continuous” as we have done in the title of the article. We remarkthat this kind of situation — where a popular property holds for generic casesbut fails for a structure naturally associated with a simple algebraic group G — isfamiliar from the study of homogeneous G -invariant polynomials. In that setting,a generic homogeneous polynomial is non-singular, yet G -invariant polynomials ofdegree ≥ n -by- n matrices.Ignoring some very small cases, the algebras A ( g ) are not power-associative. Thisis not a defect of our construction. We show that even if one alters the one choice wemade in the construction, the resulting algebra would still not be power-associative(Prop. 5.3(2)).In the penultimate section, §
10, we give an alternative realization of A ( g ) insideEnd( B ) where B is the natural module for G of type A , G , F , E , or E . Weuse this alternative realization to explicitly compute A ( sl ) (Example 10.8). Weconclude with an appendix ( § A) giving various results about adding a unit to anon-associative algebra that we refer to in the body of the paper.We work over a rather general field k and do not assume that G is split, althoughour results are new already in the case where k is the complex numbers C . Theadditional generality comes at hardly any cost due to the tools we use. Readerswho are not interested in the full generality are invited to assume throughout that k = C and identify the symbols H ( λ ) = V ( λ ) = L ( λ ).An unusual feature of our work is that the case where G is of type E is lesscomplicated than other G in several ways, at least when k = C . For E , onehas extra formulas to use, such as Okubo’s Identity Tr( π ( X ) ) = α π K ( X, X ) (Lemma 10.1, which holds for all G of exceptional type) and a similar identity forTr( π ( X ) ) (which holds for type E ). Another way that E is less complicated isthat the Molien series 1 + t + t + 3 t + 3 t + 10 t + 16 t + · · · for E acting onits 3875-dimensional representation V has coefficients no greater than the Molien CLASS OF CONTINUOUS NON-ASSOCIATIVE ALGEBRAS 3 series for the corresponding representations of other groups of type E , F , or G .Yet another way is that the second symmetric power S V is a sum of 6 irreducibleterms, which is minimal among the types E , F , and G .Our original approach to the material in this paper was to focus on the case of E and leverage these tools. In this way, we discovered the product formula on A ( g ),and only in hindsight did we see that it was a general construction that worked forall simple G . Due to this inverted approach, preparing this document took morethan three years. Just before we intended to release this work on the arxiv, thepaper [DV20] appeared, which studies algebras that are almost the same, albeitrestricted to the cases where the root system of G is simply laced and G is split andchar k = 0, see Remark 4.6 below. Both that article and this one view the algebrasas subspaces of S g and provide an associative symmetric bilinear form (we say A ( g ) is metrized, whereas they say Frobenius), but from there our approaches andresults diverge. 2. Background material
Let k be a field of characteristic different from 2 and that g is a Lie algebraover k whose Killing form, K , is nondegenerate. Then the k -algebra of lineartransformations of g , denoted End( g ), has a “transpose” operator ⊤ given by K ( T ( X ) , Y ) = K ( X, T ⊤ ( Y )) for T ∈ End( g ) and X, Y ∈ g . Identification of representations.
Another way to view the nondegeneracy of K is that it provides a g -equivariant isomorphism of representations(2.1) g ∼ −→ g ∗ via X K ( X, ) . This identification extends to an isomorphism of g -modules(2.2) g ⊗ g ∼ −→ g ⊗ g ∗ = End( g ) . As char k = 2, the natural surjection of g ⊗ g onto the 2nd symmetric power S g is split by the map(2.3) S g ֒ → g ⊗ g given by XY
12 ( X ⊗ Y + Y ⊗ X ) . Definition 2.4.
Define P : S g ֒ → End( g ) as the composition of (2.2) with (2.3).It is G -equivariant and its image is the space H ( g ) := { T ∈ End( g ) | T ⊤ = T } of symmetric operators. We have: P ( XY ) = 12 [ X ⊗ K ( Y, ) + Y ⊗ K ( X, )] for X, Y ∈ g . Example 2.5.
For { X i } a basis of g and { Y i } the dual basis with respect to K ,set e ⊗ := P X i ⊗ Y i ∈ g ⊗ g and e S := P X i Y i , the image of e ⊗ in S g . Neither e ⊗ nor e S depend on the choice of the X i ’s. Moreover, the identification (2.2) sends e ⊗ Id g , so P ( e S ) = Id g . Example 2.6.
The spaces End( g ) and H ( g ) are Jordan algebras under the Jordanproduct • defined by(2.7) T • U := 12 ( T U + U T ) for
T, U ∈ End( g ) . MAURICE CHAYET AND SKIP GARIBALDI
The same is true for P (S l ) for each subspace l of g . If K ( X, X ) = 0, then theelement P ( X ) /K ( X, X ) is an idempotent. Suppose that furthermore l has anorthonormal basis X , . . . , X r . Then for i = j , P ( X i ) • P ( X j ) = 0 and P ( X i ) • P ( X i X j ) = P ( X i X j ). In particular, P P ( X i ) is the identity element in P (S l ). Global hypotheses.
We now add hypotheses that will be assumed until the startof the appendix, section A. We will assume that g is the Lie algebra of an absolutelysimple linear algebraic group G over k . That is, G is a smooth affine group schemeof finite type over k , and G × k is simple, i.e., G × k is connected, semisimple (=has trivial radical), is = 1, and its associated root system is irreducible.We write h for the Coxeter number and h ∨ for the dual Coxeter number of(the root system of) G ; some examples are given in Table 1 below. It is true thatrank G < h ∨ ≤ h , and the root system of G is simply laced if and only if h ∨ = h .We additionally assume until the start of the appendix that char k is zero or atleast h + 2 . Consequently: The integers 2, rank G , h ∨ , h ∨ + 1 are not zero in k , sothe same is true for dim G = (rank G )( h + 1). The characteristic is “very good” for G , the Killing form K on g is nondegenerate, and g is a simple Lie algebra that isan irreducible representation of G . Moreover, the isomorphism class of g dependsonly on G up to isogeny (and not up to isomorphism). Representations.
Suppose G is split and put h for the Lie algebra of a splitmaximal torus T . For a dominant weight λ ∈ T ∗ , we write L ( λ ) for the irreduciblerepresentation with highest weight λ . The dimension and character of L ( λ ) maydepend on the characteristic of k and not just on root system data. However, thereare modules H ( λ ) and V ( λ ) with highest weight λ , which equal L ( λ ) when char k is zero or “big enough” (where what counts as big enough depends on G and λ ),and whose character is the same as the character of the irreducible representationover C with highest weight λ . See [Jan03] for background on these representations.We use the fact that these representations are defined over Z , see [Jan03, II.8.3] or[Ste16]. Casimir operator.
Recall that the Killing form induces a nondegenerate bilinearform on h ∗ as follows. For λ ∈ h ∗ , set H ′ λ ∈ h such that K ( H ′ λ , h ) = λ ( h ) for all h ∈ h . Then set h λ | µ i := K ( H ′ λ , H ′ µ ) for λ, µ ∈ h ∗ . Recall that h α | α i = 1 /h ∨ foreach long root α , and that h e α | e α + 2 δ i = 1 for e α the highest weight and δ half thesum of the positive roots. Lemma 2.8.
Suppose that the representation π : G → GL( V ) is equivalent to H ( λ ) or V ( λ ) over the algebraic closure of k . Then:(1) For { X i } a basis of g and { Y i } the dual basis with respect to K , P π ( X i ) π ( Y i ) = h λ | λ + 2 δ i Id V where δ is half the sum of the positive roots.(2) For all x, y ∈ g we have Tr( π ( x ) π ( y )) = h λ | λ + 2 δ i dim V dim G K ( x, y ) . In the statement, we have abused notation by writing π for the differential g → gl ( V ) of π . Sketch of proof.
In case k is algebraically closed of characteristic zero, this result isabout an irreducible representation and the claims are part of the usual theory of the CLASS OF CONTINUOUS NON-ASSOCIATIVE ALGEBRAS 5 quadratic Casimir operator P X i Y i ∈ U ( g ) as in, for example, [Bou05, § VIII.6.4,Cor.] or [Dyn57, Th. 2.5].In case char k = 0, it suffices to verify the claims over an extension field, forwhich we take the algebraic closure of k .Now suppose that char k is a prime p and G is split. Put R := Z ( p ) . There is asplit group G R and representation π R , both defined over R , whose base change to k is equivalent to G , π . As the claims amount to certain polynomials over R beingzero, and those polynomials are zero over the field of fractions Q of R , they are alsozero over the quotient field F p and therefore over k .Finally, if char k is prime, again it suffices to verify the claims over the algebraicclosure of k , where G is split. (cid:3) The representation A ( g )Define a map g × g → End( g ) via X ⊗ Y h ∨ (ad X )(ad Y ) + XK ( Y, ). It isbilinear, so provides a G -equivariant linear map g ⊗ g → End( g ). Composing thiswith (2.3), we find a G -equivariant linear map S : S ( g ) → End( g ) such that(3.1) S ( XY ) := h ∨ ad( X ) • ad( Y ) + P ( XY ) , where P is as in Definition 2.4 and • denotes the Jordan product (2.7).Since (ad X ) ⊤ = − ad X for all X ∈ g , we find that S ( XY ) belongs to H ( g ).Since S is linear in X and in Y and symmetric in the two terms, it extends linearlyto all of S ( g ). We set:(3.2) A ( g ) := im S ⊆ H ( g ) . Example 3.3.
For X ∈ g we haveTr( S ( X )) = h ∨ K ( X, X ) + Tr( XK ( X, )) = ( h ∨ + 1) K ( X, X ) . Linearizing this shows that Tr( S ( XY )) = ( h ∨ + 1) K ( X, Y ) for
X, Y ∈ g . Example 3.4.
For S ( e S ), we have P ( e S ) = Id g as in Example 2.5 and P (ad X i )(ad Y i ) =Id g as in Lemma 2.8(1). Therefore, S ( e S ) = ( h ∨ + 1) Id g . The split case.
Suppose that G is split, i.e., contains a split maximal torus T defined over k . (This is automatic if k is algebraically closed.) Fix a Chevalleybasis of g with respect to h := Lie( T ) in the sense of [Ste16], [SS70], or [DG11, § XX.2.11]. That is, for each root α , put H α = h α | α i H ′ α ∈ h for the elementcorresponding to the coroot α ∨ and pick a basis element X α for the α -weight spaceso that [ X α , X − α ] = H α . (This convention differs by a sign from the one used in[Bou05, § VIII.2.2].)
Lemma 3.5.
Maintain the notation of the preceding paragraph. Suppose that α and β are roots of G such that α + β is not a root.(1) If h α | β i = 0 , then S ( X α X β ) = 0 in A ( g ) .(2) Suppose h α | β i > . Then S ( X α X β ) = 0 in A ( g ) if and only if there aretwo root lengths and α and β are both short.Proof. Since X α , X β commute in g , so do ad X α , ad X β in End( g ). We note thatfor any root α , K ( X α , X − α ) = K ( H α , H α ) / νh ∨ by the formulas in [SS70,pp. E-14, E-15], where ν = 1 if α is long and is the square length ratio of a longroot to a short root if G has two root lengths and α is short. (In any case, ν is notzero in k .) MAURICE CHAYET AND SKIP GARIBALDI If h α | β i = 0, then [ X β H α ] = 0, so S ( X α X β ) X − α = 2 νh ∨ X β = 0, verifying (1).Now suppose h α | β i >
0. We have(3.6) S ( X α X β ) X − α = ( ν − α ∨ ( β )) h ∨ X β + 12 K ( X β , X − α ) X α . If β = α , then α ∨ ( β ) = 2 and (3.6) equals 2( ν − h ∨ X β , which is not zero ifand only if α is not long. If β = α and both roots are not long, then (3.6) equals h ∨ ( ν − X β = 0. This proves one direction of (2). For the remainder of the proofwe will suppose that α is long (and we may even assume it is the highest root e α );our goal is to show that S ( X α X β ) = 0. Note that if β = α then α ∨ ( β ) = 1 andso (3.6) is zero. Repeating the argument earlier in the paragraph, we find that S ( X α X β ) X − β = 0.Because S ( X α X β ) H = 0 for all H ∈ h , it remains to evaluate(3.7) S ( X α X β ) X − γ = h ∨ [ X β , [ X α , X − γ ]] for γ = α, β .Note that if ρ is any root orthogonal to α , then since at least one of α ± ρ is nota root, neither can be. If α − γ is not a root, then (3.7) is zero. Otherwise, α − γ is a root, whence γ is positive, so α + γ is not a root, and h α | γ i ≥
0. If h α | γ i = 0,then α − γ cannot be a root. We are reduced to assuming h α | γ i >
0. As γ = α wehave h α | β − γ i = 0, and again α + β − γ cannot be a root, verifying that (3.7) iszero. (cid:3) Corollary 3.8. e α is not a weight of A ( g ) .Proof. The 2 e α weight space in S g is spanned by X e α , yet S ( X e α ) = 0 by Lemma3.5(2). (cid:3) The commutative algebra A ( g )Recall the vector space A ( g ) defined in (3.2). Define, for A, B, C, D ∈ g : S ( AB ) ⋄ S ( CD ) = h ∨ S ( A, (ad C • ad D ) B ) + S ((ad C • ad D ) A, B ))+ h ∨ S ( C, (ad A • ad B ) D ) + S ((ad A • ad B ) C, D ))+ h ∨ S ([ A, C ] , [ B, D ]) + S ([ A, D ] , [ B, C ]))(4.1) + 14 ( K ( A, C ) S ( B, D ) + K ( A, D ) S ( B, C ))+ 14 ( K ( B, C ) S ( A, D ) + K ( B, D ) S ( A, C ))in A ( g ). Lemma 4.2.
The formula (4.1) extends to a symmetric bilinear map ⋄ : A ( g ) × A ( g ) → A ( g ) .Proof. Since both sides of (4.1) are linear in each of A , B , C , D and symmetricunder swapping A , B and C , D , it remains only to check that ⋄ is well defined, i.e.,that the expression given for S ( AB ) ⋄ S ( v ) is zero for all v ∈ ker S . It is sufficientto check this over an algebraic closure of k , where we are reduced to the followingcomputation. CLASS OF CONTINUOUS NON-ASSOCIATIVE ALGEBRAS 7
Let
Y, X , . . . , X r ∈ g be such that S ( P X i ) = 0. The expression for S ( Y ) ⋄ P S ( X i ) is(4.3) h ∨ X S (((ad Y ) X i ) X i ) + h ∨ X S (((ad X i ) Y ) Y )+ h ∨ X S ([ Y, X i ][ Y, X i ]) + X K ( Y, X i ) S ( X i Y ) . As P S ( X i ) = 0, P P ( X i ) = − h ∨ P (ad X i ) , so the second and fourth terms in(4.3) cancel.Furthermore, as S is G -equivariant, we have(4.4) [ad Z, S ( AB )] = S ([ Z, A ] B ) + S ( A [ Z, B ]) for
A, B, Z ∈ g .Adding the first and third term in (4.3), dividing by h ∨ , and applying this identitygives h ad Y, X S ([ Y, X i ] X i ) i = 12 [ad Y, [ad Y, X S ( X i )]] = 0 . In summary, (4.3) is zero. Therefore, if we write a, a ′ ∈ A ( g ) as a = S ( w ) and a ′ = S ( w ′ ) for w, w ′ ∈ S g , the value of a ⋄ a ′ given by (4.1) does not depend onthe choice of w, w ′ . (cid:3) With Lemma 4.2 in hand, we view A ( g ) as a commutative k -algebra with theproduct ⋄ defined by (4.1). Lemma 4.5.
The identity transformation e of g is the multiplicative identity in A ( g ) , i.e., e ⋄ a = a for all a ∈ A ( g ) .Proof. First note that e is in A ( g ) by Example 3.4. We may enlarge our basefield and so assume that k is algebraically closed and in particular that g has anorthonormal basis { X i } . Combining (4.1) and (4.4), we obtain S ( X i ) ⋄ S ( Y ) = h ∨ Y, [ad Y, S ( X i )]] + h ∨ S ((ad X i ) Y, Y )+ K ( X i , Y ) S ( X i Y ) . If we sum both sides over i , we have ( h ∨ + 1) e ⋄ S ( Y ) on the left by Example2.5 and 0 + h ∨ S ( Y ) + S ( Y ) on the right. Consequently S ( Y ) ⋄ e = S ( Y ), asrequired. (cid:3) Remark . The paper [DV20] constructs an algebra A similar to A ( g ) that isalso a subspace of H ( g ), but with a different product, which we denote by ∗ forthe moment. It defines a ∗ a ′ := proj A ( a • a ′ ), which differs from our productdefined in (4.1). The analog of (4.1) for their multiplication ∗ has additional terms.This difference is minor. For the case of most interest, where G has type E , bothalgebras can be viewed as different ways of adding a unit to the irreducible 3875-dimensional representation, i.e., if our A ( g ) is written as U ( V, f ) in the notation ofsection A, then theirs is U ( V, cf ) for some invertible c = 1 in k . A Jordan subalgebra.
Suppose that l is an abelian subalgebra of g . (For exam-ple, one could take l = h .) Define a k -linear map(4.7) i : P (S l ) → A ( g ) via i ( P ( xy )) := S ( xy ) . Writing out (4.1), we find that i ( P ( xy ) • P ( zw )) = S ( xy ) ⋄ S ( zw ) , MAURICE CHAYET AND SKIP GARIBALDI i.e., i is an algebra homomorphism, and the image of P (S l ) is a Jordan subalgebraof A ( g ). (Note that the identity element of P (S l ) need not map to the identityelement of A ( g ), see the proof of Prop. 5.3.) Lemma 4.8. If l is an abelian subalgebra of g and the Killing form K restricts tobe nondegenerate on l , then the homomorphism (4.7) is injective. Note that when K | l is nondegenerate, the isomorphism g ⊗ g ∼ −→ g ⊗ g ∗ restrictsto an isomorphism ℓ ⊗ ℓ ∼ −→ ℓ ⊗ ℓ ∗ which identifies P (S l ) with the Jordan algebra H ( l ) of symmetric elements in End( l ). Proof.
The definition of S shows that i ( P (S l )), as a subspace of End( g ), acts on l via i ( P ( X ))( Y ) = P ( X )( Y ) for all X, Y ∈ l . The nondegeneracy of K thenidentifies i ( P (S l )) with the symmetric elements in End( l ). (cid:3) Example 4.9.
Suppose G is split and not of type A nor A . Fix a Chevalleybasis for G as in §
3. For H ∈ h such that K ( H, H ) is not zero, the element u H := i ( P ( H )) /K ( H, H ) is an idempotent in A ( g ). Therefore, if k is infinite,there are infinitely many idempotents in A ( g ).Now, there is a positive root γ that is orthogonal to the highest root e α . For theelement S ( X e α X γ ), which is nonzero by Lemma 3.5(1), we have u H ⋄ S ( X e α X γ ) = λ H S ( X e α X γ ) for λ H = h ∨ (( e α + γ )( H )) K ( H, H ) . The map H λ H is a rational function h k that is not constant and thereforeis dominant. In particular, the collection of eigenvalues of the maps x u ⋄ x as u varies over the idempotents of A ( g ) is not contained in { , , } , and therefore A ( g ) is not power-associative, cf. [Sch94, Ch. V].5. A ( g ) as an algebra obtained by adding a unit The usual trace form Tr : End( g ) → k is linear and G -invariant. We use it todefine a counit, in the sense of the appendix, as ε := G Tr so that ε ( e ) = 1,for e = Id g the identity element in A ( g ) (Lemma 4.5). Thus we obtain a bilinearform τ on A ( g ) via (A.5), τ ( a, a ′ ) := ε ( a ⋄ a ′ ). The form τ is evidently G -invariant(because Tr and ⋄ are), symmetric (because ⋄ is commutative), and bilinear. Example 5.1.
For
X, Y ∈ g , Example 3.3 gives(5.2) τ ( e, S ( XY )) = h ∨ + 1dim G K ( X, Y ) for
X, Y ∈ g .We also note for future reference: τ ( S ( X ) , S ( Y )) = (cid:16) h ∨ +1dim G (cid:17) (cid:0) − h ∨ K ([ X, Y ] , [ X, Y ]) + K ( X, Y ) (cid:1) = (cid:16) h ∨ +1dim G (cid:17) K ( S ( X ) Y, Y )Using this counit, the algebra A ( g ) can be viewed as an algebra U ( V, f ) as inthe appendix, where V is the vector space ker ε endowed with the commutativeproduct · and f as defined in (A.4). With this notation, we prove: Proposition 5.3. If G is not of type A nor A , then:(1) The multiplication · on V is not zero.(2) Neither V nor U ( V, cf ) are power-associative for any c ∈ k . CLASS OF CONTINUOUS NON-ASSOCIATIVE ALGEBRAS 9
For the excluded cases of A and A , see Examples 7.1 and 10.8 respectively. Proof.
For each claim, we may enlarge k and so assume that the Lie algebra h of some maximal torus in G has an orthonormal basis X , . . . , X ℓ . We set B := i ( P (S h )).We begin with (1). By (5.2), for i = j , S ( X i X j ) is in V . On the other hand, if r ≥ S ( X X ) ⋄ S ( X X ) = i ( P ( X X ) • P ( X X )) = S ( X X ) = 0and we are done. If r = 2, then e ′ := S ( X + X ) is the identity element in B byExample 2.6, yet s := τ ( e, e ′ ) = 2 h ∨ + 1dim G = h ∨ + 1 h + 1is not 1 because G is not of type A . Then e ′ − se is in V and ( e ′ − se ) · S ( X X ) =(1 − s ) S ( X X ) = 0.We have already observed in Example 4.9 that U ( V, f ) is not power-associative,so we fix c = 1 and verify that U ( V, cf ) is not strictly power-associative. Aschar k = 2 , , U ( V, cf ) is commutative, it will follow that U ( V, cf ) is notpower-associative. Put r := ( h ∨ + 1) / (dim G ), a rational number whose denom-inator is not divisible by char k . Since h ∨ ≤ h , 0 < r ≤ /
2. Define a map S + : S g → V by S + ( p ) = S ( p ) − ε ( S ( p )) e . Applying Example 5.1, we find:(5.4) τ ( S + ( X ) , S + ( X )) = r (1 − r ) K ( X, X ) for X ∈ g .Therefore τ (equivalently, f ) is not zero on the irreducible representation V .Set b := i ( P ( X ) + tP ( X )) where t ∈ k is neither 0 nor 1, so (1 , b , b arelinearly independent (Lemma 4.8). The subalgebra B ′ of U ( V, f ) generated by B and (1 ,
0) is also power-associative and B ′ = k (1 , ⊕ ( B ′ ∩ V ).We argue that equation (A.12), a ( a ( aa )) − ( aa )( aa ) = 0, fails for some a ∈U ( V, cf ). Writing b as ( b , b ) with b ∈ k and b ∈ V , we have that b and b arelinearly independent, so a , a are linearly independent for generic a ∈ V . Fromthis it follows, as in the proof of Proposition A.13, that the polynomial function y in (A.14) is not identically zero on B ′ ∩ V . But x + y is zero, so x + cy is notidentically zero for c = 1, i.e., U ( V, cf ) is not power-associative. Moreover, x is notidentically zero, whence V is not power-associative. (cid:3) As opposed to defining the product on A ( g ) via (4.1), one could build A ( g ) “frombelow” by starting with a G -invariant commutative product · on a representation V and a G -invariant bilinear form f and setting A ( g ) to be U ( V, f ). In case G hastype E and V is the irreducible 3875-dimensional representation, both · and f areuniquely determined up to a factor in k × . But only the scalar factor on f matters(Remark A.3), and (2) says that the resulting algebra is not power-associative, nomatter what choice one makes for that parameter.Similarly, the conclusion of Lemma 6.1 below would be unchanged by multiplying f by a scalar factor, as is clear from the proof of Proposition A.7.6. Associativity of the bilinear form τ The following property of the symmetric bilinear form τ on A ( g ) is sometimesdescribed as saying that “ τ is associative”, especially in the context of Dieudonn´e’sLemma as in [Jac68, pp. 199, 239]. Lemma 6.1.
The bilinear form τ on A ( g ) satisfies (6.2) τ ( a ⋄ a ′ , a ′′ ) = τ ( a, a ′ ⋄ a ′′ ) for all a, a ′ , a ′′ ∈ A ( g ) .Proof. It suffices to verify this in the case a = S ( X ), a ′ = S ( Y ), and a ′′ = S ( Z )for X, Y, Z ∈ g . One can use the Jacobi identity to verify the following, where weabbreviate ψ for the alternating trilinear form ψ ( A, B, C ) = K ([ AB ] , C ) on g :(6.3) dim Gh ∨ + 1 τ ( a ⋄ a ′ , a ′′ ) = − h ∨ ) ψ ([ XZ ] , [ XY ] , [ Y Z ]) + h ∨ ψ ( X, Y, Z ) − h ∨ E + K ( X, Y ) K ( X, Z ) K ( Y, Z )with E = ψ ( X, Y, [ XZ ]) K ( Y, Z ) + ψ ( X, Z, [ Y Z ]) K ( X, Y ) + ψ ( Y, X, [ Y Z ]) K ( X, Z ) . Each of the four terms on the right side of (6.3) is unchanged when we swap X and Z , and therefore the claim is verified. (cid:3) Remark . Here is another argument to show associativity of τ that works when G has type E . In that case, A ( g ) = ke ⊕ V where V is an irreducible representationof G (Lemma 7.2), the restriction f of τ to V is nondegenerate, and the space( V ∗ ⊗ V ∗ ⊗ V ∗ ) G of G -invariant trilinear forms on V is 1-dimensional. It followsthen that the linear maps defined by sending v ⊗ v ′ ⊗ v ′′ ∈ ⊗ V to f ( v · v ′ , v ′′ ) and f ( v, v ′ · v ′′ ) agree up to a scalar factor, where f is the restriction of τ to V . Thetwo cubic forms are nonzero (Prop. 5.3(1)) and agree when v = v ′ = v ′′ is a genericelement of V , so the two forms agree in general, i.e., f is associative with respectto the product · , whence τ is associative with respect to the product ⋄ on A ( g ) byProp. A.7. 7. A ( g ) as a representation of G The counit ε gives a direct sum decomposition A ( g ) = ke ⊕ V as a representationof G . In this section, we describe V as a representation of G and show that itsdimension and character depend only on the root system of G and not on the field k nor even the characteristic of k . We use the notion of Weyl module recalled in § Example 7.1 ( A ( sl )) . Suppose G is split of type A , so g = sl . By hypothesis,char k is zero or at least 5, so the Weyl module V (4) of G with highest weight 4 isirreducible over k . It is a submodule of H ( g ) generated by P ( X e α ) and H ( g ) /k is V (4) by dimension count. As A ( g ) does not meet V (4) (Cor. 3.8), it follows that A ( sl ) = k as a vector space, spanned by Id g , i.e., A ( sl ) is identified with k as a k -algebra.The notion of Weyl module still makes sense when G is not assumed to be split.In that case, one still picks a maximal torus T defined over k and has a notion ofdominant weight λ ∈ T ∗ . In case λ is in the root lattice and is fixed by the actionof Aut( k/k ) on the dominant weights, there is a unique representation of G over k that becomes isomorphic to V ( λ ) (respectively, H ( λ ); resp. L ( λ )) over k [Tit71],and it makes sense to use the same notation to refer to that representation of G . Proposition 7.2.
As a representation of G , A ( g ) is a direct sum of pairwise non-isomorphic irreducible modules. The dimension of A ( g ) depends only on the typeof G (i.e., the root system) and not on the field k . Furthermore: CLASS OF CONTINUOUS NON-ASSOCIATIVE ALGEBRAS 11 (1) If G and λ are as in Table 1, then A ( g ) = k ⊕ L ( λ ) .(2) If char k = 0 or G is one of the groups listed in Table 1, then H ( g ) = A ( g ) ⊕ L (2 e α ) . type of G A G F E E E Dual Coxeter number h ∨ h λ ω + ω ω ω ω + ω ω ω Dim. of irred. rep. L ( λ ) 8 27 324 650 1539 3875 Table 1.
Data for some exceptional groups G . The fundamentaldominant weights in the formula for λ are numbered as in [Bou02]. Proof.
We first address the case where k is algebraically closed of characteristiczero. Then H ( g ) ∼ = k ⊕ J ⊕ L (2 e α ) where k is the span of e and L (2 e α ) is the G -submodule generated by P ( X e α ), which does not belong to A ( g ) by Corollary3.8. Writing J as a sum of irreducible representations ⊕ i L ( λ i ), the values of λ i are known. If G is from Table 1, then J = L ( λ ) is described in [Cd96], where it isdenoted by Y ∗ . Otherwise, J is a sum of three irreducible components for type D or two for the other types, see [Vog99] and [LM06] for more on this decompositionand related subjects. In all cases, the λ i are distinct and not zero.To complete the proof for this k , we must verify that J ⊆ A ( g ). The bulk of the λ i ’s are of the form e α + β for a root β obtained by the following procedure. Take theDynkin diagram for G , delete all simple roots that are not orthogonal to the highestroot e α , and select one of the connect components that remains. It corresponds toa subsystem of the root system of G and is the subsystem for a regular subalgebra g ′ of g ; put β for the highest root of g ′ (in the ordering induced from the chosenordering on the weights of G ). The element S ( X e α X β ) is not zero by Lemma 3.5(1)and e α + β is a maximal weight of J , so we conclude that S ( X e α X β ) is a highestweight vector and L ( e α + β ) ⊆ A ( g ).For types A n and C n with n ≥
2, one component of J is of the form consideredin the previous paragraph (and so we have shown that it belongs to A ( g )) and theother is λ for λ the highest short root. For type A n , we set β j := α + α + · · · + α j for 1 ≤ j < n , γ j := α j + α j +1 + · · · + α n for 1 < j ≤ n , and p := 2 h ∨ X α H ′ ω − ω n − n − X j =1 [ X e α , X − β j ] X β j + n X j =2 [ X e α , X − γ j ] X γ j ∈ S g . For type C n , λ = e α − β for β the simple root not orthogonal to e α , and we set p := 2 X e α X − β − X µ ∈ Φ S [ X − µ , X e α ][ X µ , X − β ] ∈ S g for Φ S the set of short roots. In either case, p has weight λ and a lengthy verificationshows that S ( p ) is not zero and is fixed by the unipotent subgroup, verifying that L ( λ ) is a summand of A ( g ) and completing the proof for this k . Next suppose that k is algebraically closed of characteristic p = 0. We transferthe results proved over C to k via R := Z ( p ) . We use subscripts C , k , R to denotecorresponding objects over these three rings. For example, let G R denote the uniquesplit reductive group scheme over R with the same root datum as G , so G R × k ∼ = G ,and put g R := Lie( G R ). Writing A ( g R × C ) = A ( g C ) as k ⊕ ( ⊕ i L C ( λ i )), each L C ( λ i ) is obtained by base change from a Weyl module V R ( λ i ) of G R defined over R . We claim that the Weyl module V k ( λ i ) of G , i.e., the base change V R ( λ i ) × k ,is irreducible, due to our global hypothesis that p ≥ h + 2.When G has type A ℓ and λ i is the highest short root, then V k ( λ i ) is the adjointrepresentation sl ℓ +1 , which is irreducible. When G has type C ℓ and λ i is the highestshort root, then p > ℓ , so V k ( λ i ) is irreducible, see [PS83, Th. 2(iv)] or [L¨ub01, Table2]. The remaining λ i are of the form e α + β . For G of type A ℓ with ℓ ≥ B ℓ with ℓ ≥ C ℓ with ℓ ≥ D , D ℓ with ℓ ≥ E , E , E , F , or G , g ′ has type A ℓ − ; A or B ℓ − ; C ℓ − ; A ; A or D ℓ − ; A ; D ; E ; C ; or A respectively.The representation V k ( λ i ) restricts to the representation sl ⊗ g ′ of the subalgebra sl × g ′ of g [Jan03, I.3.8], which is irreducible because in each case the Coxeternumber of g ′ is less than the Coxeter number of g . In summary, the Weyl module V k ( λ i ), whose dimension and character depend only on λ i and the type of G , isirreducible over k , i.e., it is L k ( λ i ).The map S : S g → H ( g ) is defined over R and the size of its image over C is atleast as large as its image over k by upper semicontinuity. As the arguments aboveshow that the irreducible representation L k ( λ i ) belongs to A ( g ) over k for all i andthere are no nontrivial extensions among the L k ( λ i ) [Jan03, II.4.13], we concludethat A ( g ) ∼ = k ⊕ ( ⊕ i L k ( λ i )) as a representation of G .For G from Table 1, the representation V k (2 e α ) is irreducible, equivalently, theirreducible representation L k (2 e α ) has the same dimension as L C (2 e α ) [L¨ub01]. Itfollows by dimension count that H ( g ) = A ( g ) ⊕ L k (2 e α ), i.e., (2).Finally, allow k to be arbitrary (apart from our global hypothesis on the char-acteristic from § G need not be split. We may view G and therepresentation A ( g ) as being obtained from a representation A ( g ) of the uniquesplit form G of G over k by twisting by a 1-cocycle in Galois cohomology as in[Ser02]. Each λ i is fixed by the automorphisms of the Dynkin diagram of G exceptwhen G has type D , in which case the three λ i ’s are permuted by that group. Inparticular, regardless of whether G is split, (2) holds. (cid:3) The hypothesis in Prop. 7.2(2) is there so that we may ignore whether the Weylmodule V (2 e α ) is irreducible in the excluded cases. One easy way around thisquestion would be to assume, for G not in Table 1, that char k is zero or greaterthan (cid:0) dim G +12 (cid:1) / (rank G ), in which case [McN98, Cor. 1.1.1] would say that H ( g )is a semisimple representation of G . (Note that this lower bound on char k growslike (rank G ) , so it is somewhat more restrictive than our global hypothesis thatchar k = 0 or at least h + 2, because h + 2 grows like rank G .)The proposition and its proof immediately provide the following easy formulafor the dimension of A ( g ): CLASS OF CONTINUOUS NON-ASSOCIATIVE ALGEBRAS 13
Corollary 7.3.
Let g C be a split Lie algebra over C with the same root system as G , and put X for the irreducible representation of g C with highest weight e α . Then dim k A ( g ) = (cid:18) dim G + 12 (cid:19) − dim X. Recall that dim X is given by the Weyl dimension formula.8. τ is nondegenerate and A ( g ) is simple Proposition 8.1. τ is nondegenerate on A ( g ) . Some authors would summarize Lemma 6.1 and Proposition 8.1, which say that A ( g ) has an associative and nondegenerate symmetric bilinear form, by saying“ A ( g ) is metrized”.The proof leverages the following. Example 8.2.
Suppose we are in the situation of Lemma 3.5(1), i.e., G is splitand α , β are orthogonal roots and α + β is not a root. Recall from the proof ofLemma 3.5 that K ( X γ , X − γ ) is not zero in k for every root γ (and is positive when k ⊆ R ) and that S ( X α X β ) X − α = ( K ( X α , X − α ) / X β . Bilinearizing Example 5.1,we have τ ( S ( X α X β ) , S ( X − α X − β ) = (cid:16) h ∨ +1dim G (cid:17) K ( S ( X α X β ) X − α , X − β ) = 0 ∈ k. Moreover, in case k ⊆ R , the left side is positive. Proof of Proposition 8.1.
We may enlarge k and so assume that G is split. Recallfrom § A ( g ) = ke ⊕ ( ⊕ i L ( λ i )) for a set of dominant weights { λ i } . This sumis an orthogonal sum with respect to τ , and therefore it suffices to verify the claimsfor the restriction of τ to each L ( λ i ).It suffices to verify that τ is nondegenerate in case k is algebraically closed, inwhich case G is split. Pick p ∈ S g such that S ( p ) is a highest weight vectorin L ( λ i ). In case λ i = e α + β for some positive root β orthogonal to e α , we take p := S ( X e α X β ). Otherwise, λ is the highest short root and G has type A n for n ≥ C n for n ≥
2; in that case we take p to be as in the proof of Proposition 7.2.Define θ to be the automorphism of g such that θ | h = − θ ( X γ ) = X − γ for each root γ . Then S ( θp ) is a lowest weight vector in L ( λ i ). We verify that τ ( S ( p ) , S ( θp )) is not zero; in the first case this is Example 8.2, and in the secondcase a calculation is required. Therefore, the restriction of τ to L ( λ i ) is not zero,so it is nondegenerate, verifying the claim. (cid:3) Corollary 8.3. If k = R and G is compact, then τ is positive-definite on A ( g ) .Proof. We continue the notation of the proof of Proposition 8.1. We view g asthe subalgebra of the split complex Lie algebra consisting of elements fixed bythe Cartan involution obtained by composing θ with complex conjugation as in[Bou05, § IX.3.2]. Then v := p + θp is in S g , S ( v ) is in L ( λ i ), and τ ( S ( v ) , S ( v )) =2 τ ( S ( p ) , S ( θp )) >
0. As G is compact, every nonzero G -invariant bilinear form on L ( λ i ) is definite, so τ is positive definite on L ( λ i ). (cid:3) In contrast to the result of the corollary, when G is split (whether or not k ⊆ R ), τ is isotropic. Indeed, for every nonzero weight ω of A ( g ), vectors of weight ± ω span a hyperbolic space. Alternative proofs for exceptional groups.
Here are very short proofs of Proposition8.1 and Corollary 8.3 in case G belongs to Table 1. By (5.4), τ is not zero on theirreducible representation V , so it is nondegenerate on V , hence on all of A ( g ).Suppose G is a compact real form, so every nonzero G -invariant bilinear form on theirreducible representation V is definite, as can be seen by averaging. In particular τ is definite on V , so positive definite on V by (5.4). Corollary 8.3 follows. (cid:3) Corollary 8.4. A ( g ) is a simple k -algebra.Proof. The nondegeneracy of τ and Proposition 7.2 verify the hypotheses of Propo-sition A.10. (cid:3) The group scheme
Aut( A ( g ))There is a natural homomorphism G → Aut( A ( g )). It has a finite kernel thecenter of G , and it is injective if and only if G is adjoint. The point of the followingresult is that in some cases this homomorphism is an isomoprhism. Proposition 9.1. If G has type F or E , then Aut( A ( g )) = G . It follows trivially that for G , G ′ of type F or E , we have: G ∼ = G ′ if and onlyif A ( g ) ∼ = A ( g ′ ). Proof of Proposition 9.1.
The number dim A ( g ) is not zero in k , so as in Exam-ple A.6 Aut( A ( g )) is the sub-group-scheme of GL( V ) preserving the commutativeproduct · on V (nonzero by Prop. 5.3(1)) as well as the G -invariant bilinear form.In case G has type F or E , it is known that G is the automorphism group of thisproduct by [GG15, Lemma 5.1, Remark 5.5, and § (cid:3) Here is what happens when the argument in the preceding proof is applied to G of the other types in Table 1: For G adjoint of type E , the argument showsthat G is the identity component of Aut( A ( g )). For G of type G or E , there is acopy of SO or Sp /µ in GL( V ) containing G and preserving a nontrivial linearmap V ⊗ V → V ; as G preserves a two-dimensional space of such products, theargument provided here is inconclusive in these cases.For type A , Aut( A ( g )) is the orthogonal group O ( g ), whose identity componenthas type D , see Example 10.8.In all of these cases except for A , when G is a split real form, one can argueon the level of Lie algebras that for every derivation D ∈ Lie(Aut( A ( g ))) and forgeneric X ∈ g , there exists a T ∈ End( g ) with T ⊤ = − T such that D ( S ( X )) =2 S ( XT ( X )). From this it can be shown that Lie(Aut( A ( g ))) = g , i.e., G is theidentity component of Aut( A ( g )).10. Construction A ( g ) in End( V )In this section, we leverage a common property of exceptional groups G observedby Okubo to describe A ( g ) inside of End( B ) for certain small B .Let π : G → GL( B ) be a representation. Recall that, for G simple, the ring k [ g ] G of G -invariant polynomial functions on g is a polynomial ring with homogeneousgenerators, where the generators are defined over a localization of Z mapping to k , hence have the same degree over k and C [Dem73, p. 297]. The generator ofsmallest degree is X K ( X, X ) of degree 2, and therefore an identity of the formTr( π ( X ) ) = c π K ( X, X ) for all X ∈ g , where c π depends on π , as in Lemma 2.8(2) CLASS OF CONTINUOUS NON-ASSOCIATIVE ALGEBRAS 15 is inevitable. Similarly, for G as in Table 1, the homogeneous generators of k [ g ] G are X K ( X, X ) of degree 2, for type A one of degree 3, and no generators ofdegree 4, and therefore there is an identity of the form Tr( π ( X ) ) = α π K ( X, X ) for X ∈ g , where α π depends only on π .In case k = C , Okubo calculated the value of α π in [Oku79], see also [Mey83],and we note that the same result holds over our more general k . Define R to bethe subring of Q obtained by adjoining to Z the inverses of all primes p such that2 ≤ p < h + 2. Lemma 10.1.
Suppose G is one of the types listed in Table 1 and that the repre-sentation π : G → GL( V ) is equivalent to H ( λ ) or V ( λ ) over the algebraic closureof k . Put µ π := h λ | λ + 2 δ i . If the rational number α π := (6 µ π − µ π dim B G )(dim G ) belongs to R , then Tr( π ( X ) ) = α π K ( X, X ) for all X ∈ g .Sketch of proof. Similar to the proof of Lemma 2.8. The key points are that themap X Tr( π ( X ) ) − α π K ( X, X ) is a polynomial function on a split form of g defined over R , and that it is the zero polynomial because that is true over C byOkubo. (cid:3) Example 10.2.
For the adjoint representation, we have α Ad = 52(2 + dim G ) , which belongs to R for G as in Table 1. (In case G has type A , α Ad = 1 /
4. Forthe other types, dim G + 2 is of the form 2 x y z for some x , y , z .) Massaging aformula for dim G in terms of h ∨ from [Cd96, p. 9] or examining the polynomial in[Oku79, 3.17] produces the remarkable formula:(10.3) 4 α Ad ( h ∨ ) = h ∨ + 6 . (This is just one example from many families of formulas, compare for example[Del96], [DG02], [LM02], and [LM06].)Here is the promised embedding. Proposition 10.4. If G has type A , G , F , E or E and π : G → GL( B ) isthe natural representation of dimension , , , , or respectively, then the G -equivariant homomorphism σ : A ( g ) ֒ → End( B ) defined via (10.5) σ ( S ( XY )) = 6 h ∨ π ( X ) • π ( Y ) − K ( X, Y ) Id B for X ∈ g satisfies (10.6) proj π ( g ) ( σ ( S ( X )) • π ( Y )) = π (cid:0) S ( X ) Y (cid:1) for Y ∈ g . Proof.
The representation π is irreducible. Moreover, These choices of π have incommon that µ π = h ∨ + 1 h ∨ + 6 , which by (10.3) is the same as(10.7) h ∨ = 2 + d µ π −
1) = µ π d π α π d , where we have abbreviated d π := dim B and d := dim G .Recall that S g = k ⊕ L ( λ ) ⊕ L (2 e α ) as a representation of G whereas, at leastin case k = C , End( B ) and A ( g ) contain k and L ( λ ) with multiplicity 1 and donot contain L (2 e α ). It follows that any G -equivariant linear map S g → End( B )factors through S : S g → A ( g ). In particular, the map XY h ∨ π ( X ) • π ( Y ) − K ( X, Y ) Id B does so, whence the map σ from (10.5) is well defined. This σ isdefined over R , and so it is also well defined for k .We now verify (10.6). Linearizing Okubo’s identity givesTr( π ( X ) π ( Y ) ) = − µ π d π d K ([ X, Y ] , [ X, Y ])+ 2 α π K ( X, Y ) + α π K ( X, X ) K ( Y, Y ) , and linearizing this in Y givesTr(( π ( X ) • π ( Y )) π ( Z )) = − µ π d π d K ([ X, Y ] , [ X, Z ])+ 2 α π K ( X, Y ) K ( X, Z ) + α π K ( X, X ) K ( Y, Z ) , As K ( Y, Z ) = dµ π d π Tr( π ( Y ) π ( Z )) (Lemma 2.8), we have α π K ( X, X ) K ( Y, Z ) = Tr (cid:18)(cid:18) dα π µ π d π K ( X, X ) Id B • π ( Y ) (cid:19) π ( Z ) (cid:19) . We obtainTr (cid:18)(cid:18)(cid:18) π ( X ) − dα π µ π d π K ( X, X ) Id B (cid:19) • π ( Y ) (cid:19) π ( Z ) (cid:19) =2 α π K (cid:18)(cid:18) µ π d π α π d (ad X ) + P ( X ) (cid:19) Y, Z (cid:19) . Multiplying both sides by 6 h ∨ and applying (10.7) gives (10.6). (cid:3) Example 10.8 ( A ( sl )) . The case g = sl , which was included in Table 1 butexcluded from §
5, so we now use the preceding construction to describe A ( sl ). For X, Y ∈ sl , Tr( XY ) = K ( X, Y ) by Lemma 2.8, so the embeddings σ : A ( sl ) → M ( k ) is via σ ( S ( X )) = 18 X − X ) I and it is an isomorphism by dimension count. We define a product ∗ on M ( k ) via P ∗ Q := σ − ( P ) ⋄ σ − ( Q ). Putting ε := Tr for the counit and chasing throughthe formulas, we find:(10.9) P ∗ Q = (cid:2) ε ( P • Q ) − ε ( P ) ε ( Q ) (cid:3) I + ε ( Q ) P + ε ( P ) Q. That is, M ( k ) with the multiplication ∗ is of the form U ( sl , f ) with notation asin the appendix, where the multiplication on sl is taken to be identically zero and f ( P, Q ) = ε ( P • Q ). This is the Jordan algebra constructed from the bilinear form f as in [Jac68, pp. 13, 14], cf. Remark A.11. CLASS OF CONTINUOUS NON-ASSOCIATIVE ALGEBRAS 17
Final remarks
We have defined here a construction that takes a simple algebraic group G (equiv-alently, a simple Lie algebra g ) over a field k , with mild hypotheses on the field k , and gives an explicit formula (4.1) for the multiplication on a unital k -algebra A ( g ) on which G acts by automorphisms. We used the description of A ( g ) as arepresentation of G to show that it is a simple algebra, that the bilinear form onit is nondegenerate, and that for G of type F or E the automorphism group isexactly G . Computation.
One can construct A ( g ) in a computer in a way amenable tocomputations as follows. First, construct G or g together with its adjoint rep-resentation or, in the cases where Proposition 10.4 applies, its natural represen-tation. Pick a basis { X i } of g , and compute S ( X i X j ) ∈ End( g ) in the firstcase or σ ( S ( X i X j )) ∈ End( B ) in the second, for i ≤ j . Among these elements,select a maximal linearly independent subset; it is a basis for A ( g ). For eachpair of basis elements, one may calculate the product ⋄ using (4.1), and expressthe result in terms of the chosen basis. This gives the “structure constants” forthe algebra. Magma [BCP90] code implementing this recipe can be found at github.com/skipgaribaldi/chayet-garibaldi . Polynomial identities.
Among the algebras A ( g ) for G in Deligne’s exceptionalseries, the cases A ( sl ) and A ( sl ) are unusual for being Jordan algebras and inparticular power-associative, whereas A ( g ) is not power-associative for other choicesof g (Prop. 5.3(2)). It is natural, then, to ask what identities A ( g ) does satisfy inthose cases. It does not satisfy any polynomial identity of degree ≤ G = G , we verifiedusing a computer that A ( g ) and also U ( V, cf ) for every c = 1 do not satisfy anydegree 5 identity not implied by commutativity, leveraging the classification of suchidentities from [Osb65, Th. 5].In case G = G or E , the G -module S V has only 6 summands, which suggeststhe existence of an identity of degree ≤ G , experiments show that for random a ∈ A ( g ), the 26 nonassociative andcommutative monomials of degree ≤ a are linearly dependent. (Note that thisapproach aims to detect not just polynomial identities in the traditional sense, butalso those where the coefficients of the monomials may be functions of the variables,such as is the case for a generic minimum polynomial as in [Jac68] or the weightedpolynomial identities considered in [Tka18].) It would be interesting to have a cleardescription of the identities of degree ≤ A ( g ) and to verify whetherthe same identities hold for A ( e ). Appendix A. Adjoining a unit to a k -algebra We carefully record in this appendix some details concerning adjoining a multi-plicative identity to a k -algebra, because we do not know a sufficient reference forthis material. Suppose we are given a k -algebra V that may not contain a mul-tiplicative identity. That is, V is a vector space over k together with a k -bilinearmap · : V × V → V , which we call the multiplication on V . Given a bilinear form f on V , we define a unital k -algebra U ( V, f ) that has underlying vector space k ⊕ V and multiplication(A.1) ( x , x )( y , y ) = ( x y + f ( x , y ) , x y + y x + x · y )for x , y ∈ k and x , y ∈ V . Then (1 ,
0) is the multiplicative identity in U ( V, f )and V is a subalgebra. Remark
A.2 . The construction U ( V, f ) is discussed from a different point of view inFox’s paper [Fox20, § V , whose automorphism groupis the Monster. Fox points out (Example 5.7) that various choices of f are used inthe literature when authors add a unit to V .In the literature, one commonly finds the more restrictive recipe U ( V,
0) foradjoining a unit to V (i.e., where f is identically zero), see for example [Sch94,Ch. II]. This has the advantage of not introducing the parameter f , however ithas the disadvantage of always producing a non-simple algebra — V is an ideal in U ( V,
0) — and therefore it does not produce popular examples of simple algebraslike the n -by- n matrices over a field, the octonions, or Albert algebras. For moreon this, see Proposition A.10 below. Remark
A.3 . One could imagine generalizing the construction to add a furtherparameter µ ∈ k and defining U ( V, f, µ ) to have the same underlying vector spaceas U ( V, f ) but with multiplication rule( x , x )( y , y ) = ( x y + f ( x , y ) , x y + y x + µx y ) . It is easily seen, however, that U ( V, f, µ ) is isomorphic to U ( V, µ − f ), so no gener-ality would be gained.Throughout the remainder of this section, we assume that all algebras consideredare finite-dimensional . Counit.
For a k -algebra A with multiplicative identity e , we call a k -linear map ε : A → k such that ε ( e ) = 1 a counit . Such a map gives a direct sum decomposition A = ke ⊕ V as vector spaces where V := ker ε and furthermore expresses A as analgebra U ( V, f ) by setting(A.4) f ( v, v ′ ) := ε ( vv ′ ) and v · v ′ := vv ′ − f ( v, v ′ ) for v, v ′ ∈ V .Conversely, every algebra U ( V, f ) has a natural counit, namely the projection of k ⊕ V on its first factor. In this way, we may identify the notions of unital k -algebraswith a counit on the one hand and algebras of the form U ( V, f ) (with specified V and f ) on the other.Additionally, a counit defines a bilinear form τ on A by setting(A.5) τ ( a, a ′ ) := ε ( aa ′ ) for all a, a ′ ∈ A .Evidently, the direct sum decomposition A = k ⊕ V is an orthogonal sum withrespect to τ , i.e., τ ( e, v ) = 0 for all v ∈ V , and the restriction of τ to V is f .From this it follows that τ is symmetric (resp. nondegenerate) if and only if f issymmetric (resp. nondegenerate). Example A.6.
In the special case where the integer dim A is not zero in k , thereis a natural counit ε : a A Tr( M a ), where we have written M a ∈ End( A )for the linear transformation b ab . Therefore there is a natural way of writing A as U ( V, f ) for V and f as in (A.4). Moreover, every algebra automorphism of CLASS OF CONTINUOUS NON-ASSOCIATIVE ALGEBRAS 19 A preserves ε , whence the group scheme Aut( A ) is identified with the sub-group-scheme of GL( V ) of transformations that preserve both the multiplication · andthe bilinear form f .Recall that a symmetric bilinear form on a k -algebra is called associative if itsatisfies (6.2). Proposition A.7.
In the notation of the preceding four paragraphs, suppose that τ is symmetric (e.g., this occurs if A is commutative). Then τ is associative (withrespect to the algebra A ) if and only if f is associative (with respect to the algebra V ).Proof. Write elements a, a ′ , a ′′ ∈ A as a = ( a , a ), etc. Then τ ( aa ′ , a ′′ ) − τ ( a, a ′ a ′′ ) = f ( a · a ′ , a ′′ ) − f ( a , a ′ · a ′′ ). (cid:3) The property of being metrized, i.e., of having a nondegenerate and associativebilinear form, has the following interesting consequence.
Proposition A.8.
Let A be a commutative k -algebra where char k = 2 , , andsuppose that A is metrized. If A satisfies an identity of degree ≤ not implied bycommutativity, then A satisfies the Jordan identity x ( x y ) = x ( xy ) and is power-associative.Proof. Writing ⊤ for the involution on End( A ) corresponding to the nondegenerateassociative bilinear form on A , we have M ⊤ a = M a and ( M a M b ) ⊤ = M b M a forall a, b ∈ A . Note that the Jordan identity is equivalent to the assertion that[ M a , M a ] = 0 for all a ∈ A .According to [Osb65, Th. 4], A satisfies (A.12) or(7) 2(( yx ) x ) x + yx = 3( yx ) x or(8) 2( y x ) x − yx ) y ) x − yx ) x ) y + 2( x y ) y − y x + ( yx ) = 0.Identity (7) is equivalent to the statement 2 M x + M x = 3 M x M x . Applying ⊤ tothis identity, subtracting it, and dividing by 3, we obtain [ M x , M x ] = 0.For (A.12), replacing a with x + y , expanding, and taking the terms of degree1 in y , we find M x + M x M x + 2 M x = 4 M x M x . Applying ⊤ to this identity,subtracting it, and dividing by 5 gives [ M x , M x ] = 0.Finally, if (8) holds, then replacing y with y + z and taking the terms of degree1 in y and z , replacing y with x , and applying the same procedure as in previouscases again gives [ M x , M x ] = 0. (cid:3) For comparison, the situation when A is not assumed to be metrized is morecomplicated, see [Osb68] and [CHP88].The following example provides a positive statement. Example A.9.
Let A be a commutative k -algebra that is metrized, and supposethat the Aut( A )-module S A has a composition series of length d . Define P e :S e A → End( A ) via P ( a a · · · a e ) := X permutations σ M a σ (1) M a σ (2) · · · M a σ ( e ) This is Aut( A )-equivariant and its image H e is contained in the space of symmetricoperators on A with respect to τ , which we identify with S A . Setting H := k Id A and I e := H + H + · · · + H e , we obtain an increasing chain of submodules 0 = I ( I ⊆ · · · so that I e = I e +1 for some e < d . That is, a symmetric expression X σ a σ (1) ( a σ (2) ( a σ (3) · · · ( a σ ( e +1) b )) · · · ) ∈ A, where each summand is a product of at most d + 1 terms, can be expressed in termsof symmetric expressions in the a ’s involving products of fewer terms. Simplicity. A k -algebra A is simple if the only two-sided ideals in A are 0 and A itself. We prove the following criterion for simplicity. Proposition A.10.
Let A be a unital k -algebra with counit ε . If(1) there is a connected group scheme G ⊆ Aut( A ) that stabilizes ε ;(2) k is not a composition factor of ker ε as a G -module;(3) τ as defined in (A.5) is nondegenerate,then A is simple.Remark A.11 . In the case where the multiplication on V := ker ε is identically zero,the algebra A is of the kind studied in [Jen60]. Proof of Proposition A.10.
Put V := ker ε . We first claim that every G -invariantsubspace I of A is a direct sum I = ( ke ∩ I ) ⊕ ( V ∩ I ). If the restriction of theprojection 1 − ε : A → V to I has a kernel, then ker(1 − ε ) = ke is containedin I and the claim is clear. Otherwise, 1 − ε is injective and I = { ( π ( w ) , w ) } for w ∈ W := (1 − ε )( I ) and some G -equivariant linear map π : W → ke . By (2),however, π must be zero, and the claim followsWe next verify that every nonzero and G -invariant ideal I of A is equal to A .By the preceding paragraph, we may suppose that there is a nonzero v ∈ V ∩ I .Since τ is nondegenerate, there is an a ∈ A so that 0 = τ ( v, a ) = ε ( va ). That is, va is a nonzero element of ke ∩ I , whence I = A .Now let I be a nonzero ideal in A . The sum of G -conjugates of I , P g gI is anonzero and G -invariant ideal, so it equals A . We conclude that I itself equals A by arguing as in the proof of [Pop95, Th. 5], which concerns the analogous case ofa non-unital algebra that is an irreducible representation of a connected group. (cid:3) Power-associativity. A k -algebra A is power-associative if the subalgebra gener-ated by any element a ∈ A is associative. It is strictly power-associative if A ⊗ k F ispower-associative for every field F containing k . We now focus on the case where A is commutative, as is the algebra A ( g ) elsewhere in this paper and as is the algebra U ( V, f ) when V is commutative and f is symmetric.If A is power-associative, then in particular(A.12) a ( a ( aa )) − ( aa )( aa ) = 0 for all a ∈ A .When char k = 2 , ,
5, (A.12) is equivalent to A being strictly power-associative[Alb48, Th. 1], cf. [Kok54, p. 364].The property of whether U ( V, f ) is strictly power-associative is rather con-strained.
Proposition A.13.
Suppose f is not alternating, V is commutative, and the multi-plication on V is not zero. If there is a group G ⊆ Aut( V ) so that Hom G ( V, k ) = 0 ,then there is at most one c ∈ k so that U ( V, cf ) is strictly power-associative. CLASS OF CONTINUOUS NON-ASSOCIATIVE ALGEBRAS 21
Proof.
We remark that the hypotheses imply that dim V ≥
2. For, if dim V = 1,then G acts by scalars, and the claim that Hom G ( V, k ) = 0 implies that G actsnontrivially.In this proof we write v to denote the square v · v of an element of V withrespect to the product · on V . The collection of v ∈ V such that v and v arelinearly dependent is the vanishing set of v v ∧ v ∈ ∧ V , so it is Zariski-closed.For sake of contradiction, suppose it is all of V . Fix a basis x , . . . , x n of V ∗ .The i -th coordinate x i | v of v is f i ( v ) for some homogeneous degree 2 polynomial f i ∈ k [ x , . . . , x n ]. There is a G -invariant function s : V \ { } → k defined implicitlyby s ( v ) v = v for nonzero v ∈ V .On the open set U i where x i does not vanish, s = f i /x i . For i = j , f i /x i and f j /x j agree on U i ∩ U j , so x i f j = x j f i in the polynomial ring. As x i does not divide x j ,it must divide f i . Setting ¯ f i := f i /x i , the polynomial function v v − ¯ f i ( v ) v iszero on U i , so it is zero on V , i.e., s : V → k is a G -invariant linear function, acontradiction. This proves that v and v are linearly independent for generic v .We now focus on (A.12) for a ∈ U ( V, cf ). Writing out a = ( a , a ) and expanding a ( a ( aa )) − ( aa )( aa ), we find (0 , x + cy ) for(A.14) x = a ( a a ) − a a and y = a f ( a , a ) − f ( a , a ) a . By the previous paragraph, a , a are linearly independent for generic a ∈ V .And f ( a , a ) is also nonzero for generic a ∈ V because f is not alternating, sowe conclude that y is not the zero polynomial on V . It follows that the polynomialfunction x + cy on V is identically zero for at most one value of c ∈ k . (cid:3) References [Alb48] A.A. Albert,
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