L - Cimat

Mercedes Siles Molina 

Universidad de Málaga

Part of this course follows the notes on a course on Jordan algebras 

delivered by Prof. Cris=na Draper Fontanals. I’m grateful to her for 

sharing her notes.

Bibliography 

N. Jacobson, Structure and representa=ons of Jordan 

algebras, Colloquium Publica=ons (American Mathema=cal 

Society) Vol. 39, 1968. 

N. Jacobson, Excep=onal Lie algebras, Lecture Notes in 

pure and applied Mathema=cs, 1971. 

O. Loos, Jordan pairs, Lecture Notes in Math., vol. 460, 

Springer-‐Verlag, Berlin, New York, 1975. 

K. McCrimmon, A taste of Jordan algebras, Springer-‐ 

Verlag, New York, 2003. 

R. D. Schafer, An introduc=on to non-‐associa=ve algebras, 

Academic Press, 1966. 

K. A. Zhevlakov, A. M. Slinko, I.P. Shestakov, A. 

I. Shirshov, Rings that are nearly associa=ve, Academic 

Press, 1982. 

3

F-algebra: 

FIRST DEFINITIONS 

F field 

An F-vector space A 

Equipped with a bilinear mul=plica=on A×A→A 

(x,y) à xy 

AssociaCve F-algebra: 

(xy)z = x(yz) for all x,y, z in A 

WE WILL CONSIDER NON-‐ASSOCIATIVE ALGEBRAS 

4

Examples 

Associa=ve F-algebras: 

Non associa=ve F-algebras: 

5

Jordan algebra: 

An F-algebra (J, •) is a Jordan algebra if: 

x • y = y • x (Commuta=vity) 

(x 2 • y) • x = x 2 • (y • x) (Jordan iden=ty) 

Note: x 2 :=x • x 

This is the defini=on of a LINEAR Jordan algebra valid for rings of 

scalars containing ½. There is a more modern defini=on of quadra=c 

Jordan algebra over arbitrary rings of scalars. 

6

Examples: 

A associa=ve and commuta=ve F-‐algebra 

A alterna=ve commuta=ve algebra 

A associa=ve F-‐algebra 

A + = (A, •) 

a • b = 1/2(ab+ba) 

Exercise: Show that these are examples of Jordan algebras 

7

alterna=ve algebra 

8

Suppose A is an algebra 

The commutator is [x,y]:= xy - yx 

The associator is (x, y, z):= (xy)z - x(yz) 

With this nota=on 

A is commuta=ve if [x, y]:= 0 for all x, y in A 

A is associa=ve if (x, y, z):= 0 for all x, y, z in A 

A is alterna=ve if 

A is a Jordan algebra if 

• It is commuta=ve: [x, y]=0 

(x, x, y):= 0 

(x, y, y):= 0 for all x, y in A 

• It sa=sfies the Jordan iden=ty (x 2 , y, x) = 0 

9

Examples 

A = Mn (F), A x • y = 1/2 (xy+yx) 

+ 

H(A, t ) = {x in M n (F) | x = x t } 

Is not a subalgebra of M n (F) 

Is a subalgebra of A + Is Jordan 

Is not associa=ve 

10

Examples 

H(M n (C ) , *) = {x in M n (C ) | x = x * } 


Is a subalgebra of A + 

x • y = 1/2 (xy+yx) 


Is Jordan 

H(A, *) = {x in M n (C ) | x = x * } ≤ A + 

x • y = 1/2 (xy+yx) 

x* = x t 

__ 

11

Examples 

H(M n (C ) , *) = {x in M n (C ) | x = x * } 


Is a subalgebra of A + 

(A, *) 

associa=ve with 

an involu=on 

x • y = 1/2 (xy+yx) 

x • y = 1/2 (xy+yx) 


Is Jordan 

H(A, *) = {x in M n (C ) | x = x * } ≤ A + 

x* = x t 

__ 

12

ObservaCons: 

(A, *) 

J ≤ A + 

A = H(A, *) + SK(A, *) 

Is Jordan Is Lie 

Special Jordan algebra 

Exceptional Jordan algebra 

13

WHEN and WHY did the Jordan algebras appear? 

It’s appearance was not to study matrices and symmetric 

matrices… 

14

1932 

15

Matrix Interpreta=on 

of Quantum Mechanics 

Physical observables represented 

Observable opera=ons 

λx à mul=plica=on by a real 

x+y à addi=on 

x n à powers of matrices 

Hermi=an matrices (x*= x) 

Matrix opera=ons 

λx à mul=plica=on by a complex 

x+y à addi=on 

xy à mul=plica=on of matrices 

x* à adjoint operator 

xy is observable only if x and y commute (are simultaneously observable) 

x* is “invisible” 

16

The Jordan program 

hkp://en.wikipedia.org/wiki/ 

File:Jordan,Pascual_1963_Kopenhagen.jpg 

Pascual Jordan 

(1902-‐1980) 

17

The Jordan program 

In 1932, Jordan proposed a program to discover a new 

algebraic senng for quantum mechanics 

Ø To study the intrinsic algebraic proper=es of hermi=an 

matrices, without reference to the underlying 

(unobservable) matrix algebra. 

Ø To capture the algebraic essence of the physical 

situa=on by formal algebraic proper=es that seemed 

essen=al and physically significant. 

Ø To consider abstract systems axioma=zated by these 

formal proper=es and see which other new (non-‐matrix) 

systems sa=sfied the axioms. 

18

The Jordan opera=ons 

Which should be the basic observable opera=ons? 

How to combine hermi=an matrices to obtain 

another hermi=an matrix? 

The most natural way is linearizing (forming 

polynomials): 

product by a real scalar + addi=on + powers 

x2 linearization 

½(xy+yx) = x • y 

The dot product appears 

19

Is the dot enough? 

Jordan thought that all the products could be expressed using • 

More… 

x 2 = ½ x • x 

x n 

That’s why we use 

½ in the formula 

xyz + zxy (Jordan triple product, obtained linearizing x3 ) 

x -1 

Not xy! (fortunately) 

There are opera=ons with hermi=an elements that cannot 

be built from the •, e.g. 

xyzt + tzyx 

Jordan didn’t realise and that was good luck! 

20

The Jordan axioms 

Which law would they obey? 

A property of the hermi=an matrices: 

power-‐associa=ve x n • x m = x n+m 

The Jordan iden=ty (x 2 • y)• x = x 2 • (y• x) 

Another basic property of the hermi=an matrices: 

posi=ve-‐definite formally real 

The -‐Jordan algebra is called formally real if 

x 1 2 +…+ xn 2 = 0 implies x1 = … = x n = 0 

In fact, there are some laws sa=sfied by • for hermi=an 

elements that cannot be obtained from the Jordan iden=ty. 

21


Which law would they obey? 


power-‐associa=ve x n • x m = x n+m 








22


The first iden=ty 

Which law would 

that 

they obey? 


power-‐associa=ve xn • xm = xn+m cannot be built from the 

Jordan product was 


founded in 1963. 



It was needed to wait for 



30 years 



23

There were three basic examples of linear Jordan algebras 

known by Jordan 

The FIRST example: 

Full algebras 

A à -associa=ve algebra, A + = (A, •) is a Jordan 

algebra 

x • y = 1/2 (xy+yx) 

It is not necessarily formally real, for instance, for 

A = M n (D), n > 1 

24



The SECOND example: 

HERMITIAN algebras 

(A, *) à - associa=ve algebra with involu=on, 

H(A, *) := {x in A | x* = x} 

is a Jordan algebra x • y = 1/2 (xy+yx) 

*: A→A, linear map 

(a*)* = a 

(ab)*=b*a* 

H(A, *) ≤ A+ 

25



The SECOND example: 

HERMITIAN algebras 

(A, *) à - associa=ve algebra with involu=on, 

H(A, *) := {x in A | x* = x} 

is a Jordan algebra 

A parCcular case: 

à (D,-) unital associa=ve algebra with involu=on a ā 

à A= M n (D) with * the conjugate traspose, that is, if x = (a ij ), 

then x*=(ā ji ). H(A,*) = H n (D, -) 

If D = R, C or H with their usual involu=ons then Hn (D, -‐) is 

formally real. 

Exercise 

26



The THIRD example: 

SPIN factor 

< , > euclidean inner product on n 

JSpinn := 1 n 

is a formally real Jordan algebra: Jordan spin factor 

1 is the unit element 

v • w = 1 for all v, w in 

n 

27

P. Jordan, J. v. Neumann, E. Wigner, On an algebraic 

generaliza=on of the quantum mechanical formalism, Annals 

of Math. Vol. 35 (1), 29-‐64 (1934). 

Theorem 

CLASSIFICATION (I) 

hkp://www.jstor.org/stable/1968117 

Every finite-‐dimensional formally real Jordan algebra is a direct sum 

of a finite number of simple ideals. 

Every finite-‐dimensional simple formally real Jordan algebra is isomorphic to 

one of: 

H n (R, -), H n (C, -), H n (H, -), H 3 (O, -) and JSpin n 

28

Special and excep=onal Jordan algebras 

J à Jordan algebra 

J is special if it is isomorphic to a subalgebra of A + , 

for some associa=ve algebra A. 

Examples 

- 

- 

- 

J is excepConal if it is not special. 

H n (R, -‐), H n (C, -‐), H n (H, -‐) are special Jordan algebras. 

Jspin n is a special Jordan algebra. 

H 3 (O,-‐) seems to be exceptional, is it? 

Exercise: 

Show that Jspinn is a subalgebra of A + , for A the 

associative) Clifford algebra generated by elements 

v1 ,…,vn such that v 2 

i = 1, vivj + vjvi = 0). 

29

Albert’s Theorem 

The surprise 

The algebra H 3 (O, -‐) is an excep=onal Jordan algebra 

of dimension 27. 

Now it is called the Albert algebra (as well as some 

extensions). 

Exercise: 

Show it. 

30

Jordan’s idea 

The end 

(Physical end) 

Jordan wanted to find a formally real Jordan algebra of infinite 

dimension, not special (since for him the (associa=ve) product xy was 

not relevant). 

His idea when classifying the simple finite-‐dimensional ones was to find 

a family of excep=onal formally real Jordan algebras depending on a 

parameter, so that they could provide a clue to find one infinite-‐ 

dimensional formally real Jordan algebras (perhaps by taking some kind 

of limit with the parameter). 

But with only one example in the list, this list was not of help to find 

the excep=onal searched system. 

The reality 

(1979) Zelmanov proved that such an algebra does not exist. In fact, he 

proved that the examples of Jordan algebras in arbitrary dimension are 

essen=ally generaliza=ons of this one finite dimensional). 

31

Jordan’s idea 

The end 

(Physical end) 

With the only exception of the 

Jordan wanted to find a formally real Jordan algebra of infinite 

dimension, not special (since for him the (associa=ve) product xy was 

not relevant). 

Albert algebra, Jordan algebras 

His idea when classifying the simplefinite-‐dimensional ones was to find 

a family of excep=onal formally real Jordan algebras depending on a 

parameter, so that they could provide a clue to find one infinite-‐ 

dimensional formally real Jordan algebras (perhaps by taking some kind 

of limit with the parameter). 

But with only one example in the list, this list was not of help to find 

the excep=onal searched system. 

hide an associative structure: 

Its Jordan product comes from 

The reality 

an associative one!! 

(1979) Zelmanov proved that such an algebra does not exist. In fact, he 

proved that the examples of Jordan algebras in arbitrary dimension are 

essen=ally generaliza=ons of this one finite dimensional). 

32

Only in the sense that he has not had capture all the proper=es of 

the hermi=an matrices.

IDENTITIES 

An idenCty in an algebra A is a polynomial f(x 1 , x 2 ,…, x n ) € F[x 1 , x 2 ,…, x n ] 

such that f(a 1 , a 2 ,…, a n ) = 0 for all a 1 , a 2 ,…, a n € A. 

Example: xy-yx and (x 2 y)x-x 2 (yx) are iden==es in any Jordan algebra (char 

F≠ 2). (Moreover, the variety of Jordan algebras is defined by such iden==es.) 

Luiz A. Peresi will speak about 

iden==es in free algebras.

IDENTITIES 





Shirshov and Macdonald’s Theorem 

If f(x 1 , x 2 , x 3 ) € F[x 1 , x 2 , x 3 ] is of degree 0 or 1 in x 3 , and is 

an iden=ty in every special Jordan algebra with 1, then f 

is an iden=ty in every Jordan algebra. 

Corollary 

Every Jordan algebra generated by two elements is special. 

35

IDENTITIES 





Shirshov and Macdonald’s Theorem 

If f(x1 , x2 , x3 ) € F[x1 , x2 , x3 ] is of degree 0 or 1 in x3 , and is 

anExercise: iden=ty in every special Jordan algebra with 1, then f 

2 

is an iden=ty Show in every it. (UJordan x =2Lx -‐ L algebra. x ) 2 

Immediate applicaCon: we may reduce to special Jordan algebras to prove 

iden==es such as Macdonald’s iden=ty: 

36

An s-‐iden=ty 

2{xzx}•{y{zy 2 z}x}-2{yzy}•{x{zx 2 z}y} = 

={x{z{x{yzy}y}z}x}-{y{z{y{xzx}x}z}y} 

This iden=ty is valid in any special Jordan algebra. 

4{xzx} • {y{zy 2 z}x} -‐ 4{yzy} • {x{zx 2 z}y} 

= xzx(yzy 2 zx + xzy 2 zy) + (yzy 2 zx + xzy 2 zy)xzx -‐ yzy(yzx 2 zy + yzx 2 zx) -‐ (xzx 2 zy + yzx 2 zx)yzy 

= xzxyzy 2 zx + xzy 2 zyxzx -‐ yzyxzx 2 zy -‐ yzx 2 zxyzy 

= xz(xyzy 2 + y 2 zyx)zx -‐ yz(yxzx 2 + x 2 zxy)zy 

= 2{x{z{x{yzy}y}z}x} -‐ 2{y{z{y{xzx}x}z}y} 

But not in any Jordan algebra! 

That kind of iden==es are called s-‐idenCCes. 

Concretely, that one is Glennie’s idenCty G 9 , discovered in 1963. 

37

Excep=onality of Albert’s Algebra 

It is enough to check that it does not sa=sfy: 

2{xzx}•{y{zy 2 z}x}-2{yzy}•{x{zx 2 z}y} = {x{z{x{yzy}y}z}x}-{y{z{y{xzx}x}z}y} 

Exercise: 

Take u,v,w € O such that (u,v,w) ≠ 0 and check that the 

following choice of elements in Albert’s algebra does not 

sa=sfy the iden=ty 

38

(From McCrimmon’s book) 

Jordan’s program 

Jordan thought that there were no s-‐iden==es 

Jordan’s goal was to capture the algebraic behavior of hermi=an operators in 

the Jordan axioms. But Albert’s algebra does not sa=sfies all the algebraic 

proper=es of the Copenhagen model. Jordan was wrong in thinking that his 

axioms had captured the hermi=an essence – he overlooked some algebraic 

proper=es of hermi=an matrices. 

First, he missed some algebraic opera=ons which could not be built from the 

bullet: the symmetric n-‐tad products {x 1 ,…,x n } := x 1 … x n + x n … x 1 cannot be 

expressed in terms of the bullet for n ≥ 4. (Their inclusion in the axioms would 

have excluded both Albert’s algebras and spin factors, landing back in 

Copenhagen with nothing but hermi=an algebras.) 

Secondly, Jordan missed some laws for the bullet which cannot be derived 

from the Jordan iden=ty, the s-‐iden==es, which are in fact just the algebraic 

iden==es sa=sfied by all hermi=an matrices which are not consequences of 

the Jordan axioms. 

39

Algebraists become interested... 

A big amount of unexpected connec=ons to other branches of 

mathema=cs appeared! 

Jordan -‐ von Neumann – Wigner’s Theorem was 

extended to finite-‐dimensional Jordan algebras over an 

arbitrary algebraically closed field of characterisCc ≠ 2: 

none new algebra really appeared.

Let A be an F-‐algebra 

Some defini=ons 

A is unital if there is a unit element 1€A such that 1x = x1 = x for all x. 

Any algebra A can be embedded in a unital algebra F1 A=:Â 

A subalgebra B of A is an F-vector subspace such that BB B. 

An ideal I A is an F-vector subspace such that IA+AI I. 

If I A, we may consider the quotient algebra =A/I: all cosets [x]=x+I 

with operations α[x]:=[αx], [x]+[y]:=[x+y] and [x][y]:=[xy]. 

A is simple if it has not proper ideals and AA ≠ 0. 

A is semisimple if it is a finite direct sum of simple ideals. 

The center Cent(A) = {c € A| c x = x c, c(xy)=(cx)y=x(cy), x, y € A} 

It is a ring of scalars, that is, associative and commutative. 

If A is unital, A can be considered as an algebra over its center 

A is simple-central if it is simple, unital and Cent(A)=F1. 

IMPORTANT BUILDING BLOCKS 

FOR THE STRUCTURE THEORY 

41

Let F be an algebraically closed field, char F ≠ 2. 

Let J be a finite-‐dimensional Jordan algebra. 

Then: 

• There is an unique maximal nilpotent ideal, Rad J. The algebra J is 

semisimple iff Rad J=0. The quo=ent J/Rad(J) is semisimple. 

• If J is semisimple, it has 1, and its decomposi=on as a direct sum of 

simple ideals is unique: the simple summands are precisely the minimal 

ideals. 

• If J is simple, it is simple-‐central over F. 

• J is simple if and only if it is one of this list: 

- 

- 

- 

CLASSIFICATION (II) 

F, the ground field 

JSpin n (F), the spin factor 

H n (C), hermitian matrices (=H(C,*), for C a (split) 

composition algebra (there are exactly 4) (n=3 if dim C=8)) 

42

Some details about the list 

The spin factor (direct generalization): if is a nondegenerate 

symmetric bilinear form in F n , then we define JSpin n (F)=F1 F n as the 

Jordan algebra given by: 

◊ 1 is the unit element 

◊◊ If v,w € F n , v • w:= 1 

Exercise: Check that the spin factor is special. 

Show that it is simple if n≠ 1. 

Concerning the hermitian matrices, which is the coordinate 

algebra, the composition algebra? Which are they? (The 4 

ones jointly their involutions…) 

More about composi=on algebras 

43

The n-‐Squares Problem 

Everything started by a number Theory problem… 

It was known from centuries that the sums of two squares can be composed: 

For us, it is enough to think that the module of the product of two complex 

numbers is the product of the modules, and apply to z 1 =x 0 +ix 1 and z 2 =y 0 +iy 1 ). 

Euler found (in 1748) a formula for the sum of four squares. 

And Cayley (in 1845) found another one for sums of eight squares, but there is no 

enough place here! (neither for copying nor for remembering, except if we look 

for some more conceptual approach to understand these formulae!) 

44

Forms adminng composi=on 

A composition algebra is a unital F-algebra A (for F an arbitrary field) with a 

nondegenerate quadratic form n: A→F admitting composition, that is, 

n(xy) = n(x)n(y) 

Nondegenerate means that n(x,A)=0 iff x=0, where n is here the polar form. 

The existence of an 

m-‐squares formula 

m=2 

m=4 

m=8 

The existence of an m-‐dimensional 

composi=on R-‐algebra 

C (complex numbers) 

H (quaternions) 

O (octonions) 

These three algebras are DIVISION algebras, which for 

composi=on algebras is equivalent to: 

** There are no (≠ 0) isotropic elements (n(x)=0) 

and to 

** There are no zero-‐divisors (x, y≠ 0 such that xy=0) 

45

Recalling H and O 

46

Recalling H and O 

47

C, H and O are real composi=on algebras, but not all of them. 

Anyway, we need composi=on algebras over algebraically closed fields. 

Thus we try to build composiCon algebras. 

The Cayley-‐Dickson process 

Assume that A is an associa=ve composi=on algebra: it is unital and n: A→F is 

a nondegenerate norm admitting composition. The standard conjugation given 

by ā = n(a,1)1- a verifies 

Take 0≠λ, and define in CD(A,λ) = A Au (two formal copies of A) with 

the following product and norm: 

The obtained algebra is AGAIN a composi=on algebra. 

The process is called the Cayley-‐Dickson process. 

48

The composi=on algebras 

Generalized Hurwitz Theorem 

Every composi=on algebra over a field F is isomorphic to one of the 

following types: 

q F (if char F≠2). It is a field. 

q K(µ) := F1+Fv with v 2 =v+µ and 4µ+1≠0. It is commuta=ve and 

associa=ve. 

q Q(µ, β):= CD(K(µ), β). These are called quaternion algebras. They 

are associa=ve but not commuta=ve. 

q C(µ, β, γ) := CD(Q(µ , β), γ). These are called Cayley algebras or 

octonion algebras. They are not associa=ve (but alterna=ve). 

49

The split composi=on algebras 

A composi=on algebra is split if it has zero divisors, or equivalently, isotropic 

elements. Over algebraically closed fields, every composiCon algebra is split. And 

it is not difficult to check that (over any F) there is only one split composi=on 

algebra of each dimension 1,2,4,8: 

♣ F 

♣ K(0). Isomorphic to F⊕F with the exchange involu=on and n(a):=a 2 

♣ Q(0,1). Isomorphic to M 2 (F) with involu=on the conjugate transpose 

and n(a)=det(a). 

♣ C(0,1,1). It is isomorphic to the Zorn matrices 

50

CLASSIFICATION (II REVISTED) 

The simple finite-‐dimensional Jordan algebras over an algebraically 

closed field F (char F ≠ 2) are: 

F 

JSpin n (F), n ≥2 

H n (F), n ≥3 

H n (F⊕F) n ≥3 

H n (M 2 (F)), n ≥3 

H 3 (C(0,1,1)) 

51

What happens if the characteris=c of the field is 2? 

Or simply, if we consider a ring of scalars instead of a field, like Z?

Some important Operators 

If x€J, the left and right mutiplication operators: 

L x : J → J , L x (y) = xy 

R x : J → J , R x (y) = yx 

In these terms, J is a Jordan algebra if and only if 

L x = R x 

[R x2 , R x ]=0 

Now take the quadratic operator 

U x : J → J , U x (y) = 2x • (x • y) - x 2 • y 

That is, U x = 2 L x 2 – Lx2 

Note: if J is special (there is a subjacent product xy), then U x (y) = xyx 

53

About these and other “natural” operators 

The U-operator satisfies the so called Fundamental Formula: 

We define the Jordan triple product: 

{x, y, z} = 2 (x • (y • z) + (x • y) • z - (x • z) • y) 

Note: if J is special (there is a subjacent product xy), then {x, y, z} = xyz + zyx 

We have a related operator: 

V x,y : J → J , V x,y (z) = {x,y,z} 

Which, at the same =me, can be wriken in terms of the U-‐operator: 

If U x,y := U x+y – U x – U y , then {x,y,z} = U x,z (y) 

54

The quadra=c program 

ObjecCve: give a defini=on of Jordan algebra valid also for fields of 

charateris=c 2, and for arbitrary rings of scalars, as the integers. 

Idea: switch from the Jordan product x• y = ½ (xy+yx) 

to the product U x (y) = xyx 

Search 

A quadratic axiomatization such that: 

• Agrees with the usual one over scalars containing ½ 

• Admittes all basic examples in characteristic 2 

• Admittes essentially nothing new in simple algebras 

55

(Unital) quadra=c Jordan algebras 

McCrimmon, 1967 

A unital quadratic Jordan algebra J (over a ring of scalars Φ) is a 

Φ-module with 

A quadratic map U: J → End Φ (J) 

x U x 

and a unit element 1 such that the following operator identities hold in all 

scalar extensions 

This is equivalent to the fact that the linearizations are valid in J. 

56

About this connec=on 

Kantor : “There are no Jordan algebras, there are only Lie algebras.” 

58

About this connec=on 


K. McCrimmon: “Of course, this can be turned around: nine Cmes 

out of ten, when you open up a Lie algebra you find a Jordan 

algebra inside which makes it thick.” 

59

The deriva=on algebra 

A derivaCon of a Jordan algebra J is an endomorphism D of J such that 

D(xy) = D(x)y+xD(y). The deriva=ons form a Lie algebra, der(J). 

Example: If J=H(A,*) for A associa=ve with involu=on, each 

x € A such that x*=-‐x provides the deriva=on ad x= L x -‐ R x € 

der(J). 

60






der(J). 

In fact, Skew(A,*) ≈ ad{x € A| x*=-‐x} = der (J) 

Example: If J=Albert algebra, der(J)=F 4 ! now we 

check it 

61






der(J). 

In fact, Skew(A,*) ≈ ad{x € A| x*=-‐x} = der (J) 

Example: If J=Albert algebra, der(J)=F 4 ! 

The exceptional Lie algebras are related to 

the exceptional Jordan algebras 

now we 

check it 

62

Some deriva=ons 

In (x,y,x 2 )=0, replace x by x+λz: 

Now replace x by x+λw: 

Rewrite: 

Interchange x and y and subtract: 

Note that D € End(J) is a deriva=on iff [R x ,D]= R D(x) 

63




Rewrite: 


Note that D € End(J) is a deriva=on iff [R x ,D]= R D(x) , hence 

[R x ,R y ] € Der(J) 

64




Rewrite: 


Note that D € End(J) is a deriva=on iff [D,R x ]= R D(x) , hence 

[R x ,R y ] € Der(J), moreover, 

{Σ[R xi ,R yi ] |x i ,y i € J}=[R J ,R J ] is a Lie subalgebra of Der (J). 

65

Albert algebra 

An arbitrary element in the Albert algebra, say J, is, for α 1 , α 2 , α 3 € F 

and a,b,c € C: 

With the mul=plica=on table summarized as follows: 

66

for 

More proper=es of the Albert algebra 

Every element x € J as above verifies a cubic equa=on 

x 3 -‐ T(x) x 2 + Q(x) x-‐ N(x)1 = 0 

T(x) = α 1 + α 2 + α 3 

Q(x) = α 1 α 2 + α 2 α 3 + α 3 α 1 -‐ n(a) -‐ n(b) -‐ n(c) 

N(x) = α 1 α 2 α 3 -‐ α 1 n(α) -‐ α 2 n(b) -‐ α 3 n(c) + t(abc) 

As 1=e 1 +e 2 +e 3 is a decomposi=on into orthogonal idempotents, we have the 

generalized Peirce decomposi=on: 

J = J 11 + J 22 + J 33 + J 23 + J 13 + J 12 

< , >: J × J → F, = tr R xy is a trace form, 

< , , >: J × J × J →J, = is trilinear. 

J ii = Fe i 

J ij = {a (k) |a € C} = C (k) 

Exercise: Let d €End J. 

Show that d is a derivaCon iff d leaves both < , > and 

< , , > invariant 

67

Der(Albert algebra)= f 4 

Jordan algebras 

Lie algebras 

DerivaCon algebras of Jordan algebras 

provides models for Lie algebras 

DerivaCon algebras of excepConal Jordan algebras 

provides models for excepConal Lie algebras 

68

J. E. Humphreys, Introduc=on to Lie algebras and 

representa=on theory, Springer-‐Verlag, New York,Heidelberg, 

Berlin 1972. 

ClassificaCon of irreducible root systems by using Dynkin diagrams 

69

PROOF: 

First step: dim Der(J)=52 

Der(Albert algebra)= f 4 

D 0 := {d €Der(J) | d(e i )=0 for all i} ≅ so(C,n) (≅d 4 , simple dim 28) 

If U€ so(C,n) , D U :J → J given by D U (e i ):=0, D U (a (i) ) := U i (a) (i) is a 

deriva=on, for U 1 = SUS -‐1 , U 2 =U’ and U 3 =U’’. These endomorphisms 

are given by the triality: U(xy)=U’(x)y+xU’’(y) and S(a)=ā denotes the 

conjuga=on in C. 

Di := {[Ra(i) , Rej-‐ek ]| a€C} ( Der(C,n)); (i,j,k cyclic permuta=on of 

1,2,3) (dim 8, since ≅ C as vector space, as {[R a(i) , R ej-‐ek ](e j )= -‐1/2a(i) ) 

Now check that Der (J)= D 0 + D 1 + D 2 + D 3 (for use that de i € C (j) +C (k) 

using the generalized Peirce decomposi=on) and that the sum is direct. 

70

f 4 is the algebra of deriva=ons of the 

Second step: Der(J) is simple 

Albert algebra (bis) 

Computa=ons show that [D i , D i ] D 0 and [D i , D j ] D k (i,j,k dis=nct) 

Lemma: A fin-‐dim arbitrary algebra A = S + I, dim S > dim I, S simple, SI=I IS. 

Then every proper ideal of A is contained in I. 

Finally if B J, B D 1 + D 2 + D 3 . Take d=d 1 +d 2 +d 3 € B. Fix i, there exists 

b € C with d i =[R b(i),R ej-‐ek ]. Thus [d i , [R a(i),R ej-‐ek ]](c(i) )=(-‐n(b,c)a+n(a,c)b) (i) 

=0, since [d i , [R a(i),R ej-‐ek ]] €D 0 (D 1 + D 2 + D 3) =0, so n(a,c)b=n(b,c)a for 

all a,c so that b=0. 

Corollary: Der J=[R J , R J ]=Inder J. 

Proof: [R J , R J ] is an ideal of Der J, which is simple. 

71

The structure algebras 

The structure algebra is the Lie algebra str(J):= der(J) + R J 

The inner structure algebra is the Lie algebra Instr(J):= Inder(J) + R J 

Example: If J=H(A,*) for A associa=ve with involu=on, then str(J)=A -‐ 

Example: If J=Albert algebra, str(J)=F+E6 ! 

now we 

check it 

The exceptional Lie algebras are related to 

the exceptional Jordan algebras 

72

PROOF: 

Str(Albert alg)= e 6 (+center) 

First step: dim R J0 +[R J , R J ]=78 

Direct sum as 1 € J 

Second step: it is simple 

If B is an ideal of RJ0 +[RJ , RJ ], by lemma, B RJ0 . Take Rc € B, 

then [Rc ,Rw ] € RJ0 [RJ , RJ ] =0, so c is in the center=F and in F J0 =0 

E6 B6 C6 73

PROOF: 

Str(Albert alg)= e 6 (+center) 

First step: dim R J0 +[R J , R J ]=78 

Direct sum as 1 € J 

Second step: it is simple 

If B is an ideal of R J0 +[R J , R J ], by lemma, B R J0 . Take R c € B, 

then [R c ,R w ] € R J0 [R J , R J ] =0, so c is in the center=F and in F J 0 =0 

One more thing: J is an irreducible module of dimension 27 

As J simple, J is RJ -‐irreducible and RJ0 -‐irreducible, so that alg< RJ0 > -‐irreducible 

E6 B6 C6 74

Tits-‐Kantor-‐Koecher construc=on 

If J is a Jordan algebra, we can build the Lie algebra (obviously Z-‐graded) 

L:= TKK (J) := L -‐1 ⊕ L 0 ⊕ L 1 

Where L -‐1 = J, L 1 = J, L 0 = Instr(J):= V J, J (inner structure algebra, which 

is a Lie algebra since [V x,y , V u,v ] = V {x,y,u},v – V u,{y,x,v} ) 

with an=commuta=ve product given by 

if T € Instr(J), x,y € J and the involu=on is V x,y *= V y,x . 

Thus TKK (J) is a Lie algebra with involu=on x 1 + V x,y + y -‐1 

y 1 + V y,x + x -‐1 

75

e 7 

TKK (Albert algebra) =The excep=onal Lie algebra e 7 

Proof: Simple, dim 133. 

But TKK-‐construc=on was not only important 

because of its connec=on to Lie algebras. 

76

A consequence of this Tits-‐Kantor-‐Koecher construc=on was the 

enlargement of the family of Jordan objects. 

77

Jordan triples 

Meyberg (in 1969) thought what was necessary to add to a Jordan algebra J in 

order to get that TKK(J) is a Lie algebra. 

DefiniCon: 

T is a (linear) Jordan triple system if there is a trilinear product 

{.,.,.}:T×T× T → T such that: 

{x,y,z} = {z,y,x} 

{x,y,{u,v,w}} = {{x,y,u},v,w}-{u,{y,x,v},w}+{u,v,{x,y,w}} 

(Remark: there is also another axioma=za=on in the pressence of ½ and 1/3) 

Example: If J is a Jordan algebra, also J is a Jordan triple system with 

{x, y, z} = 2 (x • (y • z) + (x • y) • z - (x • z) • y) 

(respectively {x, y, z} = U x,z (y)), as usual. That is, simply forget the bilinear 

product (respec=vely the U-‐quadra=c product). 

Example: M p×q (R) for p≠q with the product {x,y,z} = x y t z + z y t x is a Jordan 

triple system which does not come from a bilinear product. 

78

Jordan pairs 

Meyberg himself suggested that the en=re TKK construc=on would work for 

“pairs”, that is, for two independent spaces J + , J -‐ ac=ng on each other: 

DefiniCon: 

(V + ,V -‐ ) is a (linear) Jordan pair if there are two trilinear maps {.,.,.}:V + ×V -‐ × V + → V + 

such that for each s €{+,-‐}, x s ,u s ,w s €V s , y -s ,v -s €V -s 

Example: V + = M p×q (R) , V -‐ = M q×p (R) 

{.,.,.}:V -‐ ×V + × V -‐ → V -‐ 

{x s ,y -s ,z s } = {z s ,y -s ,x s } 

{x s ,y -s ,{u s ,v -s , w s }} = {{x s ,y -s ,u s },v -s , w s }-{u s ,{y -s ,x s ,v -s },w s }+{u s ,v -s ,{x s ,y -s ,w s }} 

Example: If T is a Jordan triple system, it can be doubled into a pair, 

V + = T , V -‐ = T, {x s ,y -s ,z s } = {x,y,z} 

Example: V + =Hom F (W,U), V -‐ = Hom F (U,W). Hence not any pair comes from a 

triple system (these two spaces have not neccesarily the same dimension). 

79

Jordan pairs (quadra=c version) 

DefiniCon: 

A pair V=(V + ,V -‐ ) of F-‐vector-‐spaces is a (quadra=c) Jordan pair if there are two 

quadra=c maps Q + : V + → Hom F (V -‐ ,V + ) and Q -‐ : V -‐ → Hom F (V + ,V -‐ ) which sa=sfy the 

following iden==es in all scalar extensions: 

(JP1) {x,y,Q s (x)z}= Q s (x){yxz} 

(JP2) {Q s (x)y,y,z}= {x,Q -‐s (y)x,z} 

(JP3) Q s (Q s (x)y)= Q s (x)Q -‐s (y) Q s (x) 

80

(Loos’ book) 

Mo=va=on for Jordan pairs 

DefiniCon: 

An element € J is inver=ble if it has a Jordan inverse, that is, y such that 

x • y = 1, x 2 • y = x. 

This happens iff U x is an inver=ble operator. 

DefiniCon: 

If u € J is inver=ble, we may consider the u-‐isotope J (u) : 

the same vector space J but with 

x • u y = x • (u • y) + (x • u) • y – u • (x • y) 

The descrip=on is simple using the quadra=c product: U x (u) =Ux U u 

DefiniCon: 

Two Jordan algebras J and J’ are isotopic if there is an automorphism between J 

and J’ (u) (called isotopism) 

Several important theorems in the theory 

of Jordan algebras hold only up to isotopy 

81

Mo=va=on for Jordan pairs 

The autotopism group of J, usually called the structure group and 

denoted by Str(J), plays a more important role than the 

automorphism group. 

There ought to be some algebraic object associated to J which 

somehow incorporates all homotopes of J and whose 

automorphism group is just the structure group of J. 

This object is the Jordan pair (J,J) associated with J . 

Automorphism group 

Structure group 

Lie algebra 

Deriva=on algebra 

Structure algebra 

82

Take T= J with the 

product {x,y,z} 

Rela=ons among these objects 

Jordan Triple System T 

T=V + ⊕V -‐ 

V=(T,T) 

Jordan Algebra J Jordan Pair V =(V + ,V -‐ ) 

And proper=es go up and down and conversely very smoothly 

83

Not only 

Jordan algebras Lie algebras 

but 

Jordan excep=onal algebras Lie excep=onal algebras

Freudenthal-‐Tits magic square 

It combines alterna=ve and Jordan algebras in a very exci=ng 

characteriza=on of the excep=onal simple Lie algebras (Tits). 

char F≠ 2,3 

U composiCon algebra 

J central simple Jordan algebra of degree 3 

L (U, J) = Der (U) ⊕ U 0 

J 0 ⊕ Der (J) 

• [Der(U), Der(J)] =0 

• Der (U) and Der (J) are Lie subalgebras 

• [a x, d] = d(a) x 

• [a x, D] = a D(x) 

• [a x, b y] = 1/12 D a,b + (a * b) (x * y) -‐ (a,b) [R x ,R y ] 

If a,b €U 0 , x,y€ J 0 , d € Der(U), D € Der (J) 

Where (a,b) is the bilinear form in U, a * b=ab-‐(a,b)1U is the projec=on of the 

product inU0 , =t(xy)=3/(dim J) trace Rxy , x * y=xy-‐1/31J is the 

projec=on of the product in J 0 , D a,b =[L a ,L b ]+[L a ,R b ]+[R a ,R b ] €Der (U) 

85

F algebraically closed, char F=0 

Magic Square (II) 

Recall that there are 4 composi=on (split) algebras: F, K=K(0) ,Q=Q(0,1), C =C(0,1,1) 

U 

F 

K 

Q 

C 

J F 

H3 (F) H3 (K) H3 (Q) H3 (C) 

86




U 

F 

K 

Q 

C 

J F H3 (F) H3 (K) H3 (Q) H3 (C) 

0 

0 

sl(2) 

sl(2) sl(3) sp(6) 

sl(3) sl(6) 

sp(6) 

sl(3)⊕sl(3) 

sl(6) 

o(12) 

g2 f e 

4 

6 e7 e8 f 4 

e 6 

e 7 

87




U 

F 

K 

Q 

C 

J F H3 (F) H3 (K) H3 (Q) H3 (C) 

0 

0 

sl(2) 

sl(2) sl(3) sp(6) 

sl(3) sl(6) 

sp(6) 

sl(3)⊕sl(3) 

sl(6) 

o(12) 

g2 f e 

4 

6 e7 e8 f 4 

e 6 

e 7 

88

(char F =0 for avoiding separability ques=ons) 

89

Aim: 

Any Jordan algebra is power-‐associa=ve 

Define powers: x 1 := x, x i+1 := x i • x 

(there is no ambiguity, x i (•)=x i if J is special) 

x i • x j = x i+j for all x€ J 

Hint: To prove this, first show that R xi belongs to the (associa=ve and 

commuta=ve) algebra spanned by R x and R x2, by using an induc=ve argument 

since 

Now apply R xi R xj = R xjR xi to x, and finish by induc=on on i. 

90

Idempotents and Peirce decomposi=on 

DefiniCon: e € J is said to be an idempotent if e 2 =e. 

Denote: J i ={x € J | xe=ix } 

J = J 0 ⊕ J 1/2 ⊕ J 1 is the Peirce decomposiCon associated to e. 

Proof: Check that 2R e 3 -‐3Re 2 +Re =0 to conclude that R e is diagonalizable 

with eigenvalues € {0, 1/2, 1}. 

ObservaCon: trace R e ≠0 = dim J 1 +1/2 dim J 1/2 

Behaviour 

Remark: There is a generalized Peirce decomposi=on when 1=e 1 +…+e t 

is a decomposi=on into pairwise orthogonal idempotents. 

91

The nilpotent Radical 

DefiniCons: 

A power-‐associa=ve algebra A is nil if every element is nilpotent (for each x€A, 

there is n€ N such that x n =0). A is nilpotent if there is n€ N such that any 

product x 1 …x n of n elements in A is 0, no maker how we associate these 

elements. A is solvable if A (n) =0 for some n€ N. (where A (1) =A, A (r+1) =(A (r) ) 2 ). 

nilpotent solvable nil 

If J is finite-‐dimensional, nil implies nilpotent: 

z Solvable implies J* nilpotent (equivalent to J nilpotent by an easy 

nonassocia=ve result) 

It can be proved by induc=on on dim J (clue: J=Fw+C J 2 with C solvable) 

z Nil implies solvable by induc=on on dim J. Take 0≠B≤ J maximal soluble 

and apply a technical lemma: if xB B,x 2 B B then x n B B, in par=cular it 

happens if x€u 2 B with uB B. 

Hence we can define Rad J= the maximal nilpotent ideal of J 

92

A counterexample 

Exercise: find a solvable not nilpotent Jordan algebra 

(obviously infinite-‐dimensional) 

Hint: (Shestakov’s example) (char F≠2) 

Take V a vector space with a countable basis. 

Consider the exterior algebra Λ(V), and let Λ 0 (V) be the subalgebra generated 

by V. 

Then J= Λ 0 (V) ⊕V with product (u+x)(v+y)=uy+vx € Λ 0 (V) is the algebra we are 

looking for. 

93

Semisimple =sum of simple ideals 

DefiniCon: J is semisimple if Rad J=0. (Always J/Rad J is semisimple) 

DefiniCon: A trace-‐form ( , ):A×Aà F is a symmetric bilinear form such that 

(xy,z)=(x,yz). 

ObservaCon: If B A then B⊥ ={y∈A| (B,y)=0} A. 

Example: (x,y)= trace R xy is a trace form in a Jordan algebra J 

Theorem: If J is finite-‐dimensional (char F=0 for the proof), Rad J = J ⊥ 

Proof: for ⊆, see that J ⊥ is nilideal, checking that R b is nilpotent when b∈J is 

nilpotent. (It also uses that no nil implies with idempotents). 

Corollary: If J is fin-‐dim semisimple (char F=0), then it is sum of simple 

ideals. 

Proof: use Diedonné’s Theorem, since Rad J=0=J ⊥ implies nondegeneracy of 

the trace form. 

94

Semisimple implies unital 

DefiniCon: e∈ J an idempotent is called principal if J 0 is nil (it does not have 

idempotents). 

Remark: Any finite-‐dimensional Jordan algebra not nil contains a principal 

idempotent. 

Proof: Take e idempotent. If it is not principal, take u idempotent in J 0 (e) and 

check that e’=e+u is an idempotent with J 1 (e) ⊆J 1 (e’) (but ≠). 

Theorem: J semisimple (char F=0) ⇒ 1∈J 

Proof: Take e a principal idempotent and see J 1 (e)=J. For if x∈J 0 ⊕J 1/2 then 

(x,y)=tr R xy =(computa=ons with the trace)=0 for all y so that x=0 (by 

nondegeneracy). 

95

Degree 

DefiniCon: an idempotent e ∈ J is primiCve if e cannot be wriken as the 

sum of two orthogonal idempotents (iff e is the unique idempotent in J 1 ). 

DefiniCon: e is absolutely primiCve if it is primi=ve in J K =J⊗ F K for any scalar 

extension K|F. 

DefiniCon: a fininite-‐dimensional central simple Jordan algebra J is reduced 

if 1=e 1 +…+e t for pairwise orthogonal absolutely primi=ve idempotents e i ∈ J. 

In this case J ii =Fe i and the number t is unique and is called the degree of J 

(or the capacity of J). 

96

CLASSIFICATION (III) 

Simple-‐central finite dimensional reduced Jordan 

algebras 

Consider an arbitrary field F with char F≠2, t=degree 

∃ A central simple ass. alg. s.t J= A + 

∃ A simple ass. alg. with 

involu=on * s.t J= H(A, *) 

(≤ A + ) 

∃ ( , ): V × Và F nondegenerate bilinear form (dim V >1) s. t. J= F1+V 

x•y=(x,y)1 

J= H(M 3 (C), *); (a ij )*=(ā ji ) 

A I 

Second kind A II 

First kind 

E t=3 

J K ≅ Mt (K) + 

K=alg.closure of F 

JK =H(Mt (K), * ); x*=xt JK =H(M2t (K), * ); x*=c -‐1xtc c skew 

A 

97 

B 

C 

D t=2

RECENT RESEARCH 

DERIVATIONS. 

ALGEBRAS OF QUOTIENTS OF LIE ALGEBRAS 

98

THE ASSOCIATIVE CASE 

Y. Utumi, On quo=ents rings, Osaka J. Math. 8 (1956), 1-‐18. 

Equivalently:

Examples: 

1.-‐ 

2.-‐ 

3.-‐ 

4.-‐ 

THE ASSOCIATIVE CASE

THE NON-ASSOCIATIVE CASE

C. MARTÍNEZ, The ring of fractions of a Jordan algebra, J. Algebra 

237 (2001), 798-‐‑812. 

Associative seKing 

algebra of quotients 

UTUMI’s ring of quotients

MSM, Algebras of quotients of Lie algebras, 

J. Pure Appl. Algebra 188 (2004), 175-‐‑188. 

L ⊆ Q Lie algebras. Q is an algebra of quotients of L 

if the following equivalent conditions are satisfied: 

∀ p, q ∈ Q, p ≠ 0, there exists x ∈ L such that [x, p] ≠ 0 

and [x, q], [x, ad y 1 , … ad y n q], ∈ L, with y 1 , …, y n ∈ L. 

∀ 0 ≠ q ∈ Q there exists an ideal I of L with zero annihilator 

such that 0 ≠ [I, q] ⊆ L. 

Qm (L) : = lim Der (I, L) 

I ∈ Ie (L)

Nonassociative quotients 

E. GARCÍA, M. GÓMEZ LOZANO, Centers, centroid, extended centroid 

and quotiens of Jordan systems, Comm. Algebra 34 (2006), 4311-‐‑4326. 

J. A. ANQUELA, T. CORTÉS, E. GARCÍA, M. GÓMEZ LOZANO, Jordan 

centers and Martindale-‐‑like covers, J. Algebra 305 (2006), 615-‐‑628. 

F. PERERA, MSM, Associative and Lie algebras of quotients, 

Pub. Mat. 52 (2008), 129-‐‑149. 

F. MONTANER, Algebras of quotients of Jordan algebras 

J. Algebra. 52 (2008), 129-‐‑149. 

M. CABRERA, J. SÁNCHEZ ORTEGA, Lie quotients for skew Lie 

algebras, Algebra colloq. 16 (2009), 267-‐‑274. 

M. BRESAR, F. PERERA, J. SÁNCHEZ ORTEGA, MSM, Computing the 

maximal algebra of quotients of a Lie algebra, Forum Math. 21(2009), 601-‐‑620.




and [x, q], [x, ad y 1 , … ad y n q], ∈ L, with y 1 , …, y n ∈ L. 



Qm (L) : = lim Der (I, L) 

I ∈ Ie (L)

L = Lσ , 

σ ∈ G 

⊕ Q = Qσ σ ∈ G 

⊕ 

gr-‐‑ gr-‐‑ 



h(Q) 

h(L) 


and [x, q], h(Q) 

[x, ad y1 , … ad yn q], ∈ L, with y1 , …, yn ∈ L. 

gr-‐‑ 



Q m (L) : = lim 

gr-‐‑ 

J. SÁNCHEZ ORTEGA, MSM, Algebras of 

quotients of graded Lie algebras, J. Algebra 

323 (2010), 2002-‐‑2015. 

I ∈ Ie (L) 

gr-‐‑ 

Der (I, L) 

gr-‐‑

Abstract properties 

F. PERERA, MSM, Associative and Lie algebras of quotients, 

Pub. Mat. 52 (2008), 129-‐‑149. 

M. CABRERA, J. SÁNCHEZ ORTEGA, Lie quotients for skew Lie 

algebras, Algebra colloq. 16 (2009), 267-‐‑274. 

Computation of maximal algebras of quotients 


maximal algebra of quotients of a Lie algebra Forum Math. 21 (2009), 601-‐‑620. 

Is Q m (L) = Q m (I) for every essential ideal I of L? 

Computation of Q m (A -‐‑ /Z A ) 

Is Q m (L) = Q m (Q m (L))?

NOT in general 

Computation of maximal algebras of quotients 


maximal algebra of quotients of a Lie algebra Forum Math. 21 (2009), 601-‐‑620. 

Is Q m (L) = Q m (I) for every essential ideal I of L? 


Is Q m (L) = Q m (Q m (L))? 

YES when L is ssp

Der m (A) : = lim 

ϕ : Der m (A) Q m (A -‐‑ /Z A ) 

δ I 

I ∈ I e (A) 

⟝ 

Der (I, A) 


δ I 

Homomorphism of Lie algebras 

Lie ideals, Lie derivations of A 

Ker (ϕ) = {δ I∣δ(I) ⊆ Z(A)} 

-‐‑ /ZA Associative ideals, associative derivationsof A

H. BIERWIRTH, MSM, Lie ideals of graded associative algebras, 

Israel J. Math 191, 111-‐‑136 (2012). 

Theorem 

A gr-‐‑prime Lie algebra 

ϕ : Der gr-‐‑m (A) Q gr-‐‑m (A -‐‑ /Z A ) 

δ I 

⟝ 

δ I 

Isomorphism of graded Lie algebras if A does not satisfy 

[[x 2 , y], [x, y]] = 0 for x, y ∈ h(A).

F. PERERA, MSM, Strongly nondegenerate Lie algebras, 

Proceedings of the Amer Math. Soc 136, 4115-‐‑4124 (2008). 

Theorem 

Application 

A gr-‐‑semiprime Lie algebra 2 and 3 -‐‑ torsion free 

Der gr (A) / I is a gr-‐‑strongly nondegenerate Lie algebra 

I = {δ ∈ Dergr (A) ∣δ(A) ⊆ Z(A)} = AnnDer (A) (Inn(A)) 

gr

Theorem 

Application 



I = {δ ∈ Der gr (A) ∣δ(A) ⊆ Z(A)} = Ann Der (A) (Inn(A)) 

gr 

Corollary 

A gr-‐‑prime Lie algebra , non commutative, 

2 and 3 -‐‑ torsion free 

Der gr (A) is a gr-‐‑strongly nondegenerate Lie algebra

Theorem 



I = {δ ∈ Dergr (A) ∣δ(I) Z(A)} = AnnDergr(A) (Inn(A)) 

L Lie algebra 

for x homogeneous 

Corollary 

Application 

L is gr-‐‑strongly nondegenerate if [x, [x, L]] = 0 

implies x = 0. 


2 and 3-‐‑ torsion free 


Theorem 



I = {δ ∈ Dergr (A) ∣δ(I) Z(A)} = AnnDergr(A) (Inn(A)) 

A graded semiprime associative algebra 

Corollary A -‐‑ /ZA is graded strongly nondegenerate 

Applications 


2 and 3-‐‑ torsion free 


L graded semiprime Lie algebra 

Q gr-‐‑m (L) 

≈ Q m (L) 

?

L graded semiprime Lie algebra ? 

is a monomorphism 

ϕ : Q gr-‐‑m(L) Q m(L) 

δ I 

⟝ 

δ I 

L = A -‐‑ /Z A

A graded semiprime associative algebra 

ϕ : Q gr-‐‑m (A -‐‑ /Z A ) Q m (A -‐‑ /Z A ) 

δ I 

⟝ 

Is it an isomorphism? 

YES when… 

δ I 

A prime 

Grading given by an ordered group 

Grading finite 

is a monomorphism 

J. SÁNCHEZ ORTEGA, MSM, Finite gradings of Lie algebras, 

J. Algebra 372, 161-‐‑171 (2012).

Farnsteiner (1988) 

Proposition 

⊕ (nnassociative) 

R = R σ 

σ ∈ G 

finite grading 

Corollary 

σ ∈ G 

⊕ 

R = R σ 

finite grading 

Der (I, R) = Der gr (I, R) 

I graded ideal 

Der (R) = Der gr (R)

Theorem 

L = Lσ σ ∈ G 

⊕ 

G abelian ordered group 

is a strongly nondegenerate lie algebra 

Finite grading (τ the maximum) 

L τ ∩ I ≠ 0 for every nonzero ideal I of L 

Q m (L) = Q gr-‐‑m (L) and l (Supp(L))= l (Supp(Q m (L)))

Theorem 

A = Aσ σ ∈ G 

⊕ 

G abelian ordered group 

Finite grading 

is an associative prime algebra 

Q m (A -‐‑ /Z A ) = Q gr-‐‑m (A -‐‑/Z A ) = Der m(A) 

and l (Supp(L)) = l (Supp(Q m (L)))

USE OF ALGEBRAS OF 

QUOTIENTS 

121

H. BIERWIRTH, MSM, Structure of strongly nondegenerate 

Prime Lie algebras (preprint). 

Theorem 

122

JORDAN AND LIE QUOTIENTS 

123

JORDAN AND LIE QUOTIENTS 

Let V= (V + , V -‐ ) be a Jordan pair 

L:= TKK (V) := L -‐1 ⊕ L 0 ⊕ L 1 

where L -‐1 = V-‐, L 1 = V+, L 0 = Instr(V):= (inner structure 

algebra, which is a Lie algebra) is the TKK Lie algebra of V. 

Defini=on: 

V ≤ W. W is said to be a Jordan pair of quoCents of V if for every 

nonzero q € W σ there exists an ideal I of V such that 

being nonzero some of the two sets. 

124

Theorem 

The following are equivalent condi=ons: 

(i) W is a Jordan pair of quo=ents of V. 

(ii) TKK(W) is a Lie algebra of quo=ents of TKK(V). 

Theorem 

(i) Let V be a strongly-‐nondegenerate Jordan pair. Then: 

(ii) If L= L -‐1 ⊕ L 0 ⊕L 1 is a strongly nondegenerate Lie algebra such that 

Q m (L) is Jordan 3-‐graded then: 

where V= (L -‐1 , L 1 ) is the Jordan pair associated to L 

125

Analogue result for Jordan triple systems 

Theorem 

(i) Let V be a strongly-‐nondegenerate Jordan pair. Then: 

(ii) If L= L -‐1 ⊕ L 0 ⊕L 1 is a strongly nondegenerate Lie algebra such that 

Q m (L) is Jordan 3-‐graded then: 

where V= (L -‐1 , L 1 ) is the Jordan pair associated to L 

126

For Jordan algebras 

Theorem 

Let J be a strongly nondegenerate Jordan algebra over a ring with 1/6. 

Then: 

Where J T is the JTS associated to J and 

V(J T )= (J T , J T ) is the Jordan pair associated to J T . 

127


K. McCrimmon: “Of course, this can be turned around: nine Cmes 

out of ten, when you open up a Lie algebra you find a Jordan 

algebra inside which makes it thick.” 

There is a lot of work to do to proof that both worlds are the same.

GRACIAS

L - Cimat

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