# Operations on Structure Constant Algebras and Elements ## Operations on Structure Constant Algebras ### `IsCommutative(A): AlgGen -> BoolElt` Returns `true` if the algebra $A$ is commutative; otherwise `false`. ### `IsAssociative(A): AlgGen -> BoolElt` Returns `true` if the algebra $A$ is associative; otherwise `false`. Note that for a structure constant algebra of dimension $n$ this requires up to $n^3$ tests. ### `IsLie(A): AlgGen -> BoolElt` Returns `true` if the algebra $A$ is a Lie algebra; otherwise `false`. Note that for a structure constant algebra of dimension $n$ this requires about $n^3/3$ tests of the Jacobi identity. ### `DirectSum(A, B): AlgGen, AlgGen -> AlgGen` Construct a structure constant algebra of dimension $n+m$ where $n$ and $m$ are the dimensions of the algebras $A$ and $B$, respectively. The basis of the new algebra is the concatenation of the bases of $A$ and $B$ and the products $a * b$ where $a \in A$ and $b \in B$ are defined to be 0. ### `Example: jordan (ex-381ebc)` We define a structure constant algebra which is a Jordan algebra. ```magma > M := MatrixAlgebra( GF(3), 2 ); > B := Basis(M); > C := &cat[Coordinates(M,(B[i]*B[j]+B[j]*B[i])/2) : j in [1..#B], i in [1..#B]]; > A := Algebra< GF(3), #B | C >; > #A; 81 > IsAssociative(A); false > IsLie(A); false > IsCommutative(A); true ``` This is a good start, as one of the defining properties of Jordan algebras is that they are commutative. The other property is that the identity $(x^2 * y) * x = x^2 * (y * x)$ holds for all $x,y \in A$. We check this on a random pair. ```magma > x := Random(A); y := Random(A); print (x^2*y)*x - x^2*(y*x); (0 0 0 0) ``` The algebra is small enough to check this identity on all elements. ```magma > forall{: x, y in A | (x^2*y)*x eq x^2*(y*x)}; true ``` So the algebra is in fact a Jordan algebra (which was clear by construction). We finally have a look at the structure constants. ```magma > BasisProducts(A); [ [ (1 0 0 0), (0 2 0 0), (0 0 2 0), (0 0 0 0) ], [ (0 2 0 0), (0 0 0 0), (2 0 0 2), (0 2 0 0) ], [ (0 0 2 0), (2 0 0 2), (0 0 0 0), (0 0 2 0) ], [ (0 0 0 0), (0 2 0 0), (0 0 2 0), (0 0 0 1) ] ] ``` ## Indexing Elements ### `a[i]: AlgGenElt, RngIntElt -> RngElt` If $a$ is an element of a structure constant algebra $A$ of dimension $n$ and $1 \leq i\leq n$ is a positive integer, then the $i$-th component of the element $a$ is returned (as an element of the base ring $R$ of $A$). ### `a[i] := r: AlgGenElt, RngIntElt, RngElt -> AlgGenElt` Given an element $a$ belonging to a structure constant algebra of dimension $n$ over $R$, a positive integer $1 \leq i\leq n$ and an element $r \in R$, the $i$-th component of the element $a$ is redefined to be $r$. ## The Module Structure of a Structure Constant Algebra ### `Module(A): AlgGen -> ModTupRng` The module $R^n$ underlying the structure constant algebra $A$. ### `Degree(A): AlgGen -> RngIntElt` The degree (= dimension) of the module underlying the algebra $A$. ### `Degree(a): AlgGenElt -> RngIntElt` Given an element belonging to the structure constant algebra $A$ of dimension $n$, return $n$. ### `ElementToSequence(a): AlgGenElt -> SeqEnum` ### `Eltseq(a): AlgGenElt -> SeqEnum` The sequence of coefficients of the structure constant algebra element $a$. ### `Coordinates(S, a): AlgGen, AlgGenElt -> SeqEnum` Let $a$ be an element of a structure constant algebra $A$ and let $S$ be a subalgebra of $A$ containing $a$. This function returns the coefficients of $a$ with respect to the basis of $S$. ### `InnerProduct(a, b): AlgGenElt, AlgGenElt -> RngElt` The (Euclidean) inner product of the coefficient vectors of $a$ and $b$, where $a$ and $b$ are elements of some structure constant algebra $A$. ### `Support(a): AlgGenElt -> SetEnum` The support of the structure constant algebra element $a$; i.e. the set of indices of the non-zero components of $a$. ## Homomorphisms ### `hom< A -> B | Q >: AlgGen, AlgGen, [ AlgGenElt ] -> Map` ### `hom< A -> B | Q >: AlgGen, TupMod, [ TupModElt ] -> Map` Given a structure constant algebra $A$ of dimension $n$ over $R$ and either a structure constant algebra $B$ over $R$ or a module $B$ over $R$, construct the homomorphism from $A$ to $B$ specified by $Q$. The sequence $Q$ may be of the form $[b_1, \ldots, b_n]$, $b_i \in B$, indicating that the $i$-th basis element of $A$ is mapped to $b_1$ or of the form $[, \ldots, ]$ indicating that $a_i$ maps to $b_i$, where the $a_i (1 \le i \le n)$ must form a basis of $A$. Note that this is in general only a module homomorphism, it is not checked whether it is an algebra homomorphism. ### `Example: cayley (ex-fafa2a)` We construct the real Cayley algebra, which is a non-associative algebra of dimension 8, containing 7 quaternion algebras. If the basis elements are labelled $1, \ldots, 8$ and 1 corresponds to the identity, these quaternion algebras are spanned by $\{ 1, (n+1) \bmod 7 + 2, (n+2) \bmod 7 + 2, (n+4) \bmod 7 + 4 \}$, where $0 \leq n \leq 6$. We first define a function, which, given three indices $i,j,k$ constructs a sequence with the structure constants for the quaternion algebra spanned by $1,i,j,k$ in the quadruple notation. ```magma > quat := func, , , , > <1,i,i, 1>, , <1,j,j, 1>, , <1,k,k, 1>, , > , , , , , ]>; ``` We now define the sequence of non-zero structure constants for the Cayley algebra using the function `quat`. Some structure constants are defined more than once and we have to get rid of these when defining the algebra. ```magma > con := &cat[quat((n+1) mod 7 +2, (n+2) mod 7 +2, (n+4) mod 7 +2):n in [0..6]]; > C := Algebra< Rationals(), 8 | Setseq(Set(con)) >; > C; Algebra of dimension 8 with base ring Rational Field > IsAssociative(C); false > IsAssociative( sub< C | C.1, C.2, C.3, C.5 > ); true ``` The integral elements in this algebra are those where either all coefficients are integral or exactly 4 coefficients lie in $1/2 + {\mathbb{Z}}$ in positions $i_1, i_2, i_3, i_4$, such that $i_1, i_2, i_3, i_4$ are a basis of one of the 7 quaternion algebras or a complement of such a basis. These elements are called the integral Cayley numbers and form a ${\mathbb{Z}}$-algebra. The units in this algebra are the elements with either one entry $\pm1$ and the others 0 or with 4 entries $\pm 1/2$ and 4 entries 0, where the non-zero entries are in the positions as described above. This gives 240 units and they form (after rescaling with $\sqrt{2}$) the roots in the root lattice of type $E_8$. ```magma > a := (C.1 - C.2 + C.3 - C.5) / 2; > MinimalPolynomial(a); $.1^2 - $.1 + 1 > MinimalPolynomial(a^-1); $.1^2 - $.1 + 1 > MinimalPolynomial(C.2+C.3); $.1^2 + 2 > MinimalPolynomial((C.2+C.3)^-1); $.1^2 + 1/2 ``` Tensoring the integral Cayley algebra with a finite field gives a finite Cayley algebra. As the ${\mathbb{Z}}$-algebra generated by the chosen basis for $C$ has index $2^4$ in the full integral Cayley algebra, we can get the finite Cayley algebras by applying the `ChangeRing` function for finite fields of odd characteristic. The Cayley algebra over $GF(q)$ has the simple group $G_2(q)$ as its automorphism group. Since the identity has to be fixed, every automorphism is determined by its image on the remaining 7 basis elements. Each of these has minimal polynomial $x^2 + 1$, hence one obtains a permutation representation of $G_2(q)$ on the elements with this minimal polynomial. As $\pm$-pairs have to be preserved, this number can be divided by 2. ```magma > C3 := ChangeRing( C, GF(3) ); > f := MinimalPolynomial(C3.2); > f; $.1^2 + 1 > #C3; 6561 > time Im := [ c : c in C3 | MinimalPolynomial(c) eq f ]; Time: 3.099 > #Im; 702 > C5 := ChangeRing( C, GF(5) ); > f := MinimalPolynomial(C5.2); > f; $.1^2 + 1 > #C5; 390625 > time Im := [ c : c in C5 | MinimalPolynomial(c) eq f ]; Time: 238.620 > #Im; 15750 ``` In the case of the Cayley algebra over $GF(3)$ we obtain a permutation representation of degree 351, which is in fact the smallest possible degree (corresponding to the representation on the cosets of the largest maximal subgroup $U_3(3):2$). Over $GF(5)$, the permutation representation is of degree 7875, corresponding to the maximal subgroup $L_3(5):2$, the smallest possible degree being 3906.