Tutorial Series Table of Contents Related Pages digitalcommons.usu. edu/dg Jordan Algebras and the Exceptional Lie algebra f Synopsis. The Compact Form of f References Release Notes Cartan Subalgebras. The Split form of f Author. f 6 6 References Synopsis Jordan algebras are a general class of commutative (but usually non-associative) algebras which satisfy a certain weak associativity identity. A Jordan algebra is called special if there is an underlying associative algebra A and the Jordan product is x + y = / xy C yx. Jordan algebras which are not special are called exceptional. Let O denote the octonions let O' denote the split octonions let pq be the diagonal matrix with its first p entries equal to + and its remaining q entries equal to -. Let M n A denote the algebra of n # n matrices over an algebra (possibly non-associative) A. Define the Jordan algebras J O = A M O A = A J O' = A M O' A = A J O = A M O A = A.
We view these as real algebras. Since the octonions are non-associative these Jordan algebras are exceptional. All are easily created in Maple. The derivations of an algebra A over a field k are the kklinear maps f : A / A such that f x $ y = f x $y C x$f y. The set of all derivations forms a Lie algebra. n 95 Chevalley and Schafer [] proved that the derivation algebra of J O is the compact real form of the exceptional Lie algebra f. n Section of this tutorial we shall verify this result of Chevalley and Schafer. n Section we show that the derivation algebra of J O' is the split real form of f. We shall calculate the Chevalley basis for the split form of f. n Section we show that the derivation algebra of J O' gives the sole non-compact non-split real form of f. We shall produce the Satake diagram for this form.. The Compact Form of f as the Derivation Algebra of the Jordan Algebra J O with(differentialgeometry): with(liealgebras): We use the AlgebraLibraryData command to obtain the multiplication rules for the exceptional Jordan algebra J O. We shall refer to this 7-dimensional algebra as J. AD := AlgebraLibraryData("Jordan(Octonions)" JO): We use the DGsetup command to initialize this algebra. DGsetup(AD); algebra name: JO (.) The algebra of derivations of J O is a matrix algebra of 7 # 7 matrices. A basis of matrices for the derivations is computed here. Because of the size of this Lie algebra some of the the following calculations may take a minute or so to execute. JO autj := Derivations(JO): The derivation algebra is 5-dimensional.
JO nops(autj); 5 (.) We use the command LieAlgebraData to find the structure equations for the Lie algebra of derivations and initialize it. The keyword argument check = "no" means that the code will not check that the matrices are actually closed with respect to the matrix commutator. JO JO LD := LieAlgebraData(autJ f check = "no"): DGsetup(LD); Lie algebra: f (.) Now we show that this 5-dimensional Lie algebra is the compact real form for f. First we find the Killing form and calculate its signature. f B := KillingForm(f): Next we use QuadraticFormSignature to find a list of subspaces on which the Killing form is positive-definite negative definite and null. f S d f S := Tensor:-QuadraticFormSignature(B); e e e e e5 e6 e7 e8 e9 e e e e e e5 e6 e7 e8 e9 e e e e e e5 e5 K e5 e5 K e5 C e5 e5 C e C e5 C e5 e6 e6 K e5 e6 C e C e5 e6 C e K e C e5 e7 e7 K e5 e7 K e9 C e5 e7 C e C e9 C e5 e8 e8 C e8 e8 C e K e8 e8 C e5 K e K e8 e9 e9 C e9 e9 K e K e9 e9 C e6 C e K e9 e e K e6 e K e C e6 e C e7 C e C e6 e e K e7 e C e C e7 e C e8 K e C e7 map(nops S); 5 (.) (.5) This shows that the Killing form is negative-definite and therefore our Lie algebra is the compact form of a semi-simple Lie algebra. We now classify this algebra using the structure theory of semi-simple Lie algebras. This involves three steps. First find a Cartan subalgebra. Second find the corresponding root space decomposition. Third identify the simple roots and calculate the Cartan matrix.
Step. We find a Cartan subalgebra. t has dimension so that the derivation algebra f has rank. f CSA := CartanSubalgebra(); CSA d e e e K e e7 C e6 (.6) Since we know that the algebra is semi-simple we know that all Cartan subalgebras are abelian. Hence an alternative way to calculate a Cartan subalgebra (and one that is usually computationally faster) is to calculate a maximal abelian subalgebra MAS. f MAS := MaximalAbelianSubalgebra([e]); MAS d e e e K e e7 C e6 (.7) The generalized null space of this MAS is the MAS itself. This proves that our MAS is self-normalizing and therefore a Cartan subalgebra. f GeneralizedNullSpace(MAS); e e e K e e7 C e6 (.8) Step. We see that all the roots in the root space decomposition are pure imaginary numbers. This is consistent with the fact that the Lie algebra is of compact type. f f RSD := RootSpaceDecomposition(CSA): PosRts := PositiveRoots(RSD); PosRts d K K K K K K K K K K (.9)
K K K K K K K K Step. Finally we find the simple roots and the Cartan matrix to conclude that the derivation algebra of the octonions is the compact real form of f. f SimRts := SimpleRoots(PosRts); SimRts d K K K K K (.) f CM := CartanMatrix(SimRts RSD); K K CM d K K K (.) K To classify the Lie algebra from its Cartan matrix we transform it to standard form. f CartanMatrixToStandardForm(CM); (.)
K K K K K "F" (.) K. The Split Real Form of f as the Derivation Algebra of the Jordan Algebra J O ' We repeat the calculations of the previous section using the split octonions in place of the octonions. We use the AlgebraLibraryData command to obtain the multiplication rules for the exceptional Jordan algebra J O'. We shall refer to this 7-dimensional algebra as JS. This section may be executed independently of Section. with(differentialgeometry): with(liealgebras): AD := AlgebraLibraryData("Jordan( Octonions)" JOS type = "Split"): We use the DGsetup command to initialize this algebra. DGsetup(AD); algebra name: JOS (.) The algebra of derivations is a matrix algebra of 7 # 7 matrices. We do not display the matrices. JOS autjos:= Derivations(JOS): The derivation algebra is 5-dimensional. JOS nops(autjos); 5 (.)
We find the structure equations for the Lie algebra and initialize. The keyword argument check = "no" means that the code will not check that the matrices are actually closed with respect to the matrix commutator. JOS JOS LD := LieAlgebraData(autJOS f check = "no"): DGsetup(LD); Lie algebra: f (.) Now we show that this 5-dimensional Lie algebra is the split real form for f. We calculate the Killing form and find its signature to be [8 ]. This shows that our algebra f is semi-simple. f f f B := KillingForm(f): S := Tensor:-QuadraticFormSignature(B): map(nops S); 8 (.) The steps to finding the Chevalley basis for f are: [] Find any Cartan subalgebra. [] Find the root space decomposition. f the roots are real then skip to step 5. Otherwise steps [] and [] construct maximally non-compact Cartan subalgebra. [] Find a Cartan decomposition. [] Find a maximal abelian sub-algebra in the positive part of the Cartan decomposition. At this point we now have a Cartan subalgebra giving a real root space decomposition [5] Recalculate the root space decomposition [6] Calculate the Chevalley basis. Step. Find a Cartan subalgebra for f. f CSA := CartanSubalgebra(); CSA d e e e K e e7 K e6 (.5) Step. Find the root space decomposition. We see that the roots are not real and therefore this choice of Cartan subalgebra will not lead to a real change of basis for the Chevalley basis. f f RSD := RootSpaceDecomposition(CSA): PosRts := PositiveRoots(RSD);
PosRts d K K K K K K K K K (.6) K K K K K K K K K Step. Find the Cartan decomposition. f T P := CartanDecomposition(CSA RSD PosRts); T P d e e C e5 e e K 5 e6 e5 e K e6 e K e e C e e7 C e e8 K e e9 (.7) K e e C e e5 e6 K 5 e8 C 5 e8 C e5 e7 e e e e5 C e8 C e5 e6 K e9 K e5 e8 C e7 C e5 e9 K 5 e C 5 e9 C e5 e e5 C 5 e7 C e5 e C e5 e K 5 e6 e7 e8 e9 e K e6 e K e e C e e7 C e e8 K e e9 K e e C e e6 K 5 e8 C 5 e8 C e5 e9 e e C e e K e K e7 K e5 e K e8 K e5 e K e9 K e5 e5 e6 e7 e8 K e e9 K 5 e C 5 e9 C e5 e e e5 C 5 e7 C e5 e6 Step. Find a maximal abelian ideal inside of the positive part P of the Cartan decomposition. This is the required maximally non-compact Cartan subalgebra.
f CSA := MaximalAbelianSubalgebra([e + *e5] P); CSA d e C e5 e6 K 5 e8 C 5 e8 C e5 e K e6 e (.8) All steps []--[] can be executed using the keyword argument maximallynoncompact = "yes". f CartanSubalgebra(maximallynoncompact = "yes"); e C e5 e6 K 5 e8 C 5 e8 C e5 e K e6 e (.9) Step 5. Find the root space decomposition for CSA. Now the roots are all real. f f RSD := RootSpaceDecomposition(CSA): PosRts := PositiveRoots(RSD); PosRts d K K K K K K 8 K K (.) K K K K K K K K K K K K Step 6. Find the Chevalley basis. f CB := ChevalleyBasis(RSD PosRts): Finally let's examine the structure equations in this basis.
f LD := LieAlgebraData(CB fcb): n the Chevalley basis the Cartan subalgebra is given by the first vectors the next vectors are bases for the positive root spaces and the last vectors are bases for the negative root spaces. JOS VectorLabels := [h h h h seq(x i i =.. ) seq(y i i =.. )] ; VectorLabels d h h h h x x x x x5 x6 x7 x8 x9 x x x x x x5 x6 x7 x8 x9 x x x x x y y y y y5 y6 y7 y8 y9 y y y y y y5 y6 y7 y8 y9 y y y y y f DGsetup(LD VectorLabels ['o']); Lie algebra: fcb (.) (.) The entire multiplication table is too large to display so we consider just a block in the upper left-hand corner. fcb MultiplicationTable(fcb "LieTable" rows = [ ] columns = [ 5 6 7 8 9 ]); fcb h h h h x x x x x5 x6 x7 x8 x9 x h x K x K x5 x7 x8 K x9 K x h K x x K x x5 K x6 x7 x9 h K x x K x x6 K x7 K x9 x h K x x K x5 x6 K x8 x9 K x (.) We see that h h h h give the Cartan subalgebra (they are abelian and act by scaling on the remaining root spaces). We see that the coefficients in the multiplication table are all integers between - and. The Cartan matrix appears as the coefficients in the structure equations for hi xj.. The Remaining Real Form of f f 6 6 ) as the Derivation Algebra of the Jordan Algebra J O We repeat the calculations of the previous two sections using the exceptional Jordan algebra J O. This algebra is not stored in DifferentialGeometry and so must be constructed by hand.
This section may be executed independently of Sections and. with(differentialgeometry): with(liealgebras): The first step in creating the exceptional Jordan algebra is to create the octonions. AD := AlgebraLibraryData("Octonions" H); AD d e e = e e e = e e e = e e e = e e e5 = e5 e e6 = e6 e e7 = e7 e e8 = e8 e e = e e e = K e e e = e e e = K e e e5 = e6 e e6 = K e5 e e7 = K e8 e e8 = e7 e e = e e e = K e e e = K e e e = e e e5 = e7 e e6 = e8 e e7 = K e5 e e8 = K e6 e e = e e e = e e e = K e e e = K e e e5 = e8 e e6 = K e7 e e7 = e6 e e8 = K e5 e5 e = e5 e5 e = K e6 e5 e = K e7 e5 e = K e8 e5 e5 = K e e5 e6 = e e5 e7 = e e5 e8 = e e6 e = e6 e6 e = e5 e6 e = K e8 e6 e = e7 e6 e5 = K e e6 e6 = K e e6 e7 = K e e6 e8 = e e7 e = e7 e7 e = e8 e7 e = e5 e7 e = K e6 e7 e5 = K e e7 e6 = e e7 e7 = K e e7 e8 = K e e8 e = e8 e8 e = K e7 e8 e = e6 e8 e = e5 e8 e5 = K e e8 e6 = K e e8 e7 = e e8 e8 = K e DGsetup(AD [x] [o]); algebra name: H (.) (.) Note that we are using x to denote algebra elements with o denoting elements of the dual. Here are the Jordan matrices over the octonions which are Hermitian with respect to the standard -dimensional Minkowski inner product. H JM := JordanMatrices( H signature = [ ]); x x x x x x x x x x x x x x x JM := x x x x x x x x x K x x x K x x x (.) x x x x x x x x x x x x x x x x x x x x5 x x x6 x x x7 x x x8 x K x x x K x5 x x K x6 x x K x7 x x K x8 x x x x x x x x x x x x x x x x x
x x x x x x x x x x x x x x x5 x x x x x x x x x x x x x x x K x x x x x x x x x x x x x5 x x x x x6 x x x7 x x x8 x x x x x x x x x x x x x x x x x x x x x x6 x x x7 x x x8 x x x K x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x5 x x x6 x x x7 x x x8 x x x x x x x x5 x x x6 x x x7 x x x8 x x x x x x x x x x Calculate the structure constants for these matrices. AD := AlgebraData(JM JordanProduct JO): We use the DGsetup command to initialize this algebra. DGsetup(AD); algebra name: JO (.) The algebra of derivations is a matrix algebra of 7 # 7 matrices. We do not display the matrices. JO autjo:= Derivations(JO): The derivation algebra is 5-dimensional. JO nops(autjo); 5 (.5)
We find the structure equations for the Lie algebra of derivations and initialize it. The keyword argument check = "no" means that the code will not check that the matrices are actually closed with respect to the matrix commutator. JO JO LD := LieAlgebraData(autJO f check = "no"): DGsetup(LD); Lie algebra: f (.6) Now we show that this 5-dimensional Lie algebra is the non-compact non-split real form for f. We calculate the Killing form and find its signature to be [6 6]. f f f B := KillingForm(f): S := Tensor:-QuadraticFormSignature(B): map(nops S); 6 6 (.7) The steps to finding the Satake diagram for f are: [] Find a maximally non-compact Cartan subalgebra. [] Calculate the root space decomposition. [] Examine the simple roots and determine which ones are Satake associates. Step. Find a maximally compact Cartan subalgebra for f. f CSA := CartanSubalgebra(maximallynoncompact = "yes"); CSA := e9 e5 C e7 C e5 e9 K e e8 C e5 (.8) Step. Find the root space decomposition and the simple roots. f f f RSD := RootSpaceDecomposition(CSA): PosRts := PositiveRoots(RSD): SimRts := SimpleRoots(PosRts); (.9)
SimRts := K K K K K (.9) f CM := CartanMatrix(SimRts RSD); K K CM := K K K K (.) f CM := CartanMatrixToStandardForm(CM SimRts); K CM := K K K K "F" (.) K Step. The first roots are purely imaginary roots and these are therefore colored. in the Satake diagram. The th root is white and is its own Satake associate. This gives the correct Satake diagram for f 6 6 or f K. References. John Baez The octonions Bull. Amer. Math. Soc. 9() 5-5. C. Chevalley and R. D. Schafer The exceptional simple Lie algebras F and E6 Proc. Nat. Acad. Sci. U.S. 6 7 (95). W. Fulton and J. Harris Representation Theory Graduate Texts in Mathematics 9 Springer 99.. http://en.wikipedia.org/wiki/octonion http://en.wikipedia.org/wiki/split-octonion 5. http://en.wikipedia.org/wiki/f_(mathematics)
Release Notes This worksheet was compiled with Maple 7 and DG release USU available by request from ian.anderson@usu.edu Author an M. Anderson Department of Mathematics Utah State University December