1.Alternative algebras. 2.Moufang identities. 3.Bruck-Kleinfeld map. 4.Artin’s The-orem. 5.Multiplications in alternative algebras. 6.Generating sets in alternative al-gebras. 7.Module finite alternative alal-gebras. 8.Simple alternative alal-gebras. 9.Jordan algebras. 10.Special Jordan algebras. 11.Identity. 12.Generating sets in Jordan al-gebras. 13.Module finite Jordan alal-gebras. 14.Lie alal-gebras. 15.Generating sets in Lie algebras. 16.Special Lie algebras. 17.Malcev algebras. 18.Simple Malcev algebras.
19.Exercises.
In this section we study classes of algebras defined by some identities replacing associativity. One of our purposes is to show how such identities can be used to get information about the multiplication algebra.
3.1 Alternative algebras. Def inition. An algebra A is called left alternative if a2b=a(ab),
right alternative if ab2 = (ab)b,
flexible if a(ba) = (ab)a for all a, b∈A,
alternative if it is left and right alternative (and flexible).
Using the associator defined in 1.11 we can express these identies by (a, a, b) = 0, (a, b, b) = 0, (a, b, a) = 0.
Hence alternative algebras are characterized by the fact that the associator yields an alternating map
(−,−,−) :A×A×A→A.
The identities above are of degree 2 ina orb. We can derive multilinear identities from these by linearization. In our case this can be achieved by replacing a bya+d in the first identity
(a+d)2b = (a+d)[(a+d)b].
Evaluating this equality and referring again to the first identity in the forma2b=a(ab) and d2b =d(db) we obtain the multilinear identity
(ad+da)b=a(db) +d(ab).
Puttingb =a we obtain a new relation for left alternative rings (ad+da)a=a(da) +da2.
Now applying the second identiy (right alternative) we have (ad)a=a(da) and thus:
Left and right alternative implies flexible.
In an alternative algebra we also have
[a(ab)]b−a[(ab)b] =a2b2−a2b2 = 0, which means (a, ab, b) = 0 for all a, b∈A.
Linearizing with respect to b, i.e., replacing b by b+c, we derive 0 = (a, ab, c) + (a, ac, b) = −(ab, a, c)−(a, b, ac)
=−[(ab)a]c+ (ab)(ac)−(ab)(ac) +a[b(ac)]
=a[b(ac)]−[(ab)a]c.
This identity is named afterRuth Moufangwho studied alternative fields in the context of geometric investigations (see [207]). By similar computations we obtain further identities of this kind:
3.2 Moufang identities.
In an alternative algebra we have the
left Moufang identity (aba)c=a[b(ac)], right Moufang identity c(aba) = [(ca)b]a, middle Moufang identity a(bc)a= (ab)(ca).
Using the associator we have for all a, b, c∈A:
(a, b, ca) = a(b, c, a), (ab, c, a) = (a, b, c)a,
(b, a2, c) = a(b, a, c) + (b, a, c)a.
It was observed in Bruck-Kleinfeld [92] that the following map is helpful for proving our next theorem:
3.3 Bruck-Kleinfeld map.
Let A be an alternative algebra. For the map
f :A4 →A, (a, b, c, d)7→(ab, c, d)−(b, c, d)a−b(a, c, d), we have (1) f(a, b, c, d) =−f(d, a, b, c) and
(2) f(a, b, c, d) = 0 in case any two arguments coincide.
Proof. It is easy to verify that for a, b, c, din any algebra A,
(ab, c, d)−(a, bc, d) + (a, b, cd) = a(b, c, d) + (a, b, c)d.
For alternative algebras this yields
(ab, c, d)−(bc, d, a) + (cd, a, b) = a(b, c, d) + (a, b, c)d.
Forming the map
F(a, b, c, d) := f(a, b, c, d)−f(b, c, d, a) +f(c, d, a, b)
= [a,(b, c, d)]−[b,(c, d, a)] + [c,(d, a, b)]−[d,(a, b, c)], we conclude from above
0 = F(a, b, c, d) +F(b, c, d, a) = f(a, b, c, d) +f(d, a, b, c), proving (1).
By definition, f(a, b, c, d) =−f(a, b, d, c) and f(a, b, c, c) = 0. This together with
(1) yields (2). 2
3.4 Artin’s Theorem.
An algebra A is alternative if and only if any subalgebra of A generated by two elements is associative.
Proof. (1) (compare [92]) One implication is clear. Now assume thatAis alternative.
ForD:={a, b} we define in obvious notation K := {k ∈A|(D, D, k) = 0},
M := {m∈K|(D, m, K) = 0 and mK ⊂K}, S:= {s∈M|(s, M, K) = 0}.
It suffices to show that S is an associative subalgebra andD⊂S.
Obviously, K, M and S are R-submodules of A and (D, D, K) = 0. SinceD con-tains only two distinct elements, we have (D, D, D) = 0 and by 3.3, f(D, K, D, D) = 0. This yields (D, D, DK) = 0 and so
DK ⊂K, D⊂M and D⊂S.
Since S⊂M ⊂K this means (S, S, S) = 0, i.e., S is associative.
It remains to show that SS ⊂ S. From (S, M, K) = 0 and S ⊂ M we derive (K, S, S) = 0 = (M K, S, S). By 3.3,
f(s, s0, m, k) =f(m, k, s, s0) for all s, s0 ∈S, m∈M, k∈K.
This yields (SS, M, K) = 0, in particular, (D, SS, K) = −(SS, D, K) = 0. We have SS⊂K and (SS)K =S(SK)⊂K. Hence we conclude SS ⊂M and SS⊂S. 2
A different proof, using an induction argument, is given in [244].
The following generalized form of Artin’s Theorem is proved in [92]:
Let A be an alternative algebra and a, b, c ∈ A with (a, b, c) = 0. Then the subal-gebra of A generated by {a, b, c} is associative.
We see from Artin’s theorem that, in particular, in an alternative algebra every subalgebra generated by one element is associative. Algebras with this property are called power-associative.
We are now going to investigate the effect of identities of an algebra on the mul-tiplication algebra. For this we consider left and right mulmul-tiplications in alternative algebras:
3.5 Multiplications in alternative algebras.
For any elements a, bin an alternative algebra A we have:
left alternative La2 =LaLa, right alternative Ra2 =RaRa,
flexible RaLa=LaRa, left Moufang identity Laba =LaLbLa, right Moufang identity Raba =RaRbRa, middle Moufang identities RaLaLb =LabRa,
RaLaRb =RbaLa.
Recalling that the associator is alternating we derive from this Lab =LaLb−[Lb, Ra], Rab =RbRa−[La, Rb], LaLb+LbLa =Lab+ba, RaRb+RbRa=Rab+ba.
These identities show, for example, that Rab and Lab belong to the subalgebra generated byLa,Lb, Ra and Rb and we conclude:
3.6 Generating sets in alternative algebras.
Let A be an alternative R-algebra.
(1) If A is generated by {aλ}Λ as an R-algebra, then M(A) is generated as an R-algebra by
{Laλ, Raλ, idA|λ ∈Λ}.
(2) If A is finitely generated as an R-algebra, then M(A) is also finitely generated as an R-algebra.
We have seen in 2.5 that for an associative module finite R-algebra the multipli-cation algebra is also module finite. This remains true for alternative algebras:
3.7 Module finite alternative algebras.
LetAbe an alternativeR-algebra which is finitely generated as anR-module. Then M(A) is also finitely generated as an R-module.
Proof. Assume a1, . . . , an to generate A as an R-module. We want to show that M(A) is generated as an R-module by monomials of the form
Lεa1 substituting according to 3.5 (with appropriate sk, tl ∈R):
(i)RaiLaj =LajRai +Laiaj−LaiLaj;
(ii) LaiLaj =−LajLai +Laiaj+ajai =−LajLai+PskLak if i > j;
(iii) Lak
i =Lkai =PtlLal (by Artin’s Theorem).
Similar operations can be applied to the Rai’s. 2
The structure theory of alternative algebras is closely related to associative alge-bras partly due to the following observation (by Kleinfeld [180], see [41, Chap. 7]):
3.8 Simple alternative algebras.
Let A be a simple alternative algebra which is not associative. Then the centre of A is a field and A is a Cayley-Dickson algebra (of dimension 8) over its centre.
Interest in alternative rings arose first in axiomatic geometry. For an incidence plane in which Desargues’ theorem holds the coordinate ring is an associative division ring. A weaker form of Desargues’ theorem, the Satz vom vollst¨andigen Vierseit, is equivalent to the coordinate ring being an alternative division ring (see [206]).
The motivation for the investigation of the next type of algebras came from quan-tum mechanics. In 1932 the physicist Pascual Jordan draw attention to this class of algebras. Later on they turned out to be also useful in analysis. For example, in [177]
the reader may find an account of their role for the description of bounded symmetric domains.
3.9 Jordan algebras. Definition.
An algebraAover a ringRwith 2 invertible inRis called a(linear) Jordan algebra if it is commutative and, for alla, b,∈A,
a(a2b) =a2(ab) (Jordan identiy).
In a commutative algebra A the left multiplication algebra L(A) coincides with the multiplication algebra M(A) and the Jordan identity can also be written in the following forms
(a, b, a2) = 0, LaLa2 =La2La, [La, La2] = 0.
The condition 12 ∈ R is not necessary for the definition of a Jordan algebra.
However, especially for producing new identities by linearization it is quite often useful to divide by 2. Hence it makes sense to include the condition 12 ∈R already in the definition. Over rings with 2 not invertible it is preferable to consider quadratic Jordan algebras instead of linear Jordan algebras (e.g., Jacobson [20]). Important examples of Jordan algebras can be derived from associative algebras:
3.10 Special Jordan algebras.
Let (B,+,·) be an associative algebra over the ring R with 2 invertible. Then the new product for a, b∈B,
a×b = 1
2(a·b+b·a), turns (B,+,×) into a Jordan algebra.
Algebras isomorphic to a subalgebra of an algebra of type (B,+,×) are called spe-cial Jordan algebras.
Proof. Obviously, the new multiplication is commutative. Denoting by La the left multiplicationb 7→a×b, it is easy to check that
LaLa2(b) =La2La(b) for all a, b∈B.
2 Not every Jordan algebra is a special Jordan algebra.
Consider a Jordan algebra A. In the Jordan identity [La, La2] = 0 the element a occurs with degree 3. Let us try to gain linear identities from this.
Replacing a by a+rbwith r ∈R, b∈A, we obtain 0 = [La+rb, L(a+rb)2]
=r(2[La, Lab] + [Lb, La2]) +r2(2[Lb, Lab] + [La, Lb2]).
Puttingr = 1 andr = 12 and combining the resulting relations we derive [Lb, La2] + 2[La, Lab] = 0.
Again linearizing by replacinga by a+cwe obtain
[La, Lbc] + [Lb, Lac] + [Lc, Lab] = 0.
Applying this to an x∈A we have
a[(bc)x] +b[(ac)x] +c[(ab)x] = (bc)(ax) + (ac)(bx) + (ab)(cx).
Interpreting this as a transformation on a for fixed b, c, x ∈ A and renaming our elements we conclude
3.11 Identity. In any Jordan algebra we have the identity
L(ab)c=LabLc+LbcLa+LcaLb−LaLcLb−LbLcLa.
This shows that L(ab)c belongs to the subalgebra of M(A) generated by left mul-tiplications with one, or products of two, of the elements {a, b, c}. Hence, similar to the alternative case considered in 3.6, we have the following relationship between A and M(A):
3.12 Generating sets in Jordan algebras.
Let A be a Jordan algebra over the ring R in which 2 is invertible.
(1) If A is generated by {aλ}Λ as an R-algebra, then M(A) is generated as an R-algebra by
{Laλaµ, Laλ, idA|λ, µ∈Λ}.
(2) If A is finitely generated as R-algebra, then M(A) is also finitely generated as an R-algebra.
Proof. Every a ∈ A is a linear combination of finite product of aλ’s. By 3.11, La is a linear combination of products of Laλaµ’s and Laν’s and hence belongs to the
subalgebra generated by these elements. 2
3.13 Module finite Jordan algebras.
Let A be a Jordan algebra over the ring R with 12 ∈ R. If A is finitely generated as an R-module, then M(A) is also finitely generated as an R-module.
Proof. Let theR-module Abe generated by a1, . . . , an. Every element in M(A) is a linear combination of products of Lai’s. We show that the finitely many products of the form
Lεa11Lεa22· · ·LεannLδa1
σ(1)Lδa2
σ(2)· · ·Lδan
σ(n), εi ∈ {0,2}, δi ∈ {0,1}, σ ∈ Sn,
where Sn denotes the group of permutations of n elements, generate the R-module M(A).
We see from 3.11 that a product of the form LaiLajLak can be replaced by LakLajLai plus expressions of lower degree in the Lai’s. Hence, if Lai occurs several times in a product, we finally arrive at partial products of the form L2ai,LaiLajLai or L3ai.
Again referring to 3.11, the last two expressions can be replaced by formulas of lower degree and the L2a
i can be collected at the left side with increasing indices. 2
Remark. The linear factors could be arranged such that the next but one factors have a higher index. In fact, by careful analysis one obtains that M(A) can be generated as anR-module by≤2n+1n elements (compare Theorem 13 in Chapter II of [19]).
The interest in the next class of algebras - named after the mathematicianSophus Lie - stems from their relationship to topological groups (Lie groups). For a detailled study of this interplay see, for example, [34].
3.14 Lie algebras. Definiton.
An algebra A over a ring R is called a Lie algebra if for all a, b, c∈A, a2 = 0 and a(bc) +b(ca) +c(ab) = 0 (Jacobi identity).
These properties imply in particular
0 = (a+b)2 =ab+ba, i.e., ab=−ba(anti-commutativity).
Hence the multiplication algebra M(A) coincides with the left multiplication algebra L(A) and the Jacobi identity can be written as
Lab =LaLb−LbLa, or [La, Lb] =Lab. From this the following is obvious:
3.15 Generating sets in Lie algebras.
Let A be a Lie algebra which is generated by {aλ}Λ as an R-algebra. Then M(A) is generated as an R-algebra by
{Laλ, idA|λ∈Λ}.
By our next observations Lie algebras are intimately related to associative algebras.
3.16 Special Lie algebras.
Let (B,+,·) be an associative algebra over the ring R. Then the new product for a, b∈B,
[a, b] =a·b−b·a, turns (B,+,[, ]) into a Lie algebra.
Algebras isomorphic to a subalgebra of an algebra of type (B,+,[ , ]) are called special Lie algebras.
Proof. It is straightforward to verify the identities required. 2
It follows from the Poincare-Birkhoff-Witt Theoremthat every Lie algebra A over R which is free as an R-module is in fact a special Lie algebra (e.g., [4]).
Let M be any finitely generated free R-module, i.e., M ' Rn, for n ∈ IN. Then the Lie algebra (EndR(M),+,[ , ]) is denoted by gl(M, R) or gl(n, R) (the general linear group). Subalgebras of gl(n, R) are called linear Lie algebras. The matrices with trace zero form such a subalgebra (denoted by sl(n, R)). Other subalgebras of gl(n, R) are (upper) triangular matrices (with trace zero, or diagonal zero) and skew symmetric matrices.
For any non-associative R-algebras (A,+,·), the commutator defines a new R-algebra (A,+,[ , ]) which is usually denoted by A(−). Obviously this is always an anti-commutative algebra but other identities depend on properties of (A,+,·).
In particular, the algebra A is calledLie admissible if A(−) is a Lie algebra.
As noticed in 3.16, any associative algebra is Lie admissible. An algebraAis called left symmetric ([155]) if
(a, b, c) = (b, a, c) for all a, b, c∈A.
Left symmetric algebras are Lie admissible (see Exercise (7)). See [61] for a connection between left symmetric products and flat structures on a Lie algebra.
Notice that alternative algebras need not be Lie admissible. For an alternative algebra A, the algebra A(−) satisfies identities which define a new class of algebras containing all Lie algebras:
3.17 Malcev algebras. Definiton.
An R-algebra A is called a Malcev algebraif for all a, b, c, d∈A, a2 = 0 and (ab)(ac) = ((ab)c)a+ ((bc)a)a+ ((ca)a)b.
Notice that the characterizing identity is quadratic in a. If 2 is invertible in R, Malcec algebras can be characterized by a multilinear identity (in 4 variables, see Exercise (6)).
Every Lie algebra is a Malcev algebra. The structure of Malcev algebras is closely related to Lie algebras based on the following fact (from [133]):
3.18 Simple Malcev algebras.
Let Abe a central simple Malcev R-algebra which is not a Lie algebra, and assume R is a field of characteristic 6= 2,3. Then A is of the form D(−)/R, where D is a Cayley-Dickson algebra over R.
An R-algebra Ais called Malcev admissibleif the attached algebra A(−) is a Mal-cev algebra. As mentioned above, every alternative algebra is MalMal-cev admissible (see [29, Proposition 1.4]). For a detailled study of these algebras we refer to [29].
3.19 Exercises.
(1) Let A be an alternative algebra. Prove that the product of any two ideals is again an ideal inA.
(2) For an alternative algebraA, consider the map (see 3.3)
f :A4 →A, (a, b, c, d)7→(ab, c, d)−(b, c, d)a−b(a, c, d).
Prove:
(i) 3f(a, b, c, d) = [a,(b, c, d)]−[b,(c, d, a)] + [c,(d, a, b)]−[d,(a, b, c)];
(ii) f(a, b, c, d) = ([a, b], c, d) + ([c, d], a, b).
(3) Let A be an R-module over a ring R, with 2 invertible in R.
Assume β : A ×A → R to be a symmetric bilinear form and e ∈ A satisfying β(e, e) = 1. Prove that the product of a, b∈A,
a·b :=β(e, a)b+β(e, b)a−β(a, b)e, turns A into a Jordan algebra with unite.
(4) Consider an eight dimensional algebraAwith basis1, a1, a2, . . . , a7, over a field K with charK 6= 2, given by the multiplication table ([41])
a1 a2 a3 a4 a5 a6 a7 a1 α a3 αa2 a5 αa4 −a7 −αa6 a2 −a3 β −βa1 a6 a7 βa4 βa5 a3 −αa2 βa1 −αβ a7 αa6 −βa5 −αβa4
a4 −a5 −a6 −a7 γ −γa1 −γa2 −γa3 a5 −αa4 −a7 −αa6 γa1 −αγ γa3 αγa2 a6 a7 −βa4 βa5 γa2 −γa3 −βγ −βγa1 a7 αa6 −βa5 αβa4 γa3 −αγa2 βγa1 αβγ
with non-zeroα, β, γ ∈K. Show thatAis an alternative central simple algebra which is not associative (Cayley-Dickson algebra).
(5) Let A be any R-algebra. An R-linear map D:A →A is called a derivationif D(ab) =D(a)b+aD(b), for any a, b∈A.
Show that the set of derivations ofA is a subalgebra of the Lie algebraEndR(A)(−).
(6) Let A be an anti-commutative R-algebra and assume 2 to be invertible in R.
Put
J(a, b, c) := (ab)c+ (bc)a+ (ca)b (Jacobian).
Prove that the following are equivalent ([238]):
(a) A is a Malcev algebra;
(b) J(a, b, ac) = J(a, b, c)a, for all a, b, c∈A;
(c) J(a, ab, c) = J(a, b, c)a, for all a, b, c∈A;
(d) (ab)(cd) =a((db)c) +d((bc)a) +b((ca)d) +c((ad)b), for all a, b, c, d∈A.
(7) Show that an algebra A is Lie admissible if and only if for all a, b, c∈A, (a, b, c) + (b, c, a) + (c, a, b)−(a, c, b)−(c, b, a)−(b, a, c) = 0.
References: Bakhturin [3], Baues [61], Bruck-Kleinfeld [92], Filippov [133, 134], Grabmeier-Wisbauer [147], Helmstetter [155], Jacobson [19, 20], Kaup [177], Klein-feld [180], Moufang [207], Myung [29], Schafer [37], Smiley [244], Wisbauer [267], Zhevlakov-Slinko-Shestakov-Shirshov [41].