# Operations on Algebras and their Elements ## Operations on Algebras ### `Centre(A): AlgAss -> AlgAss` The centre of the associative algebra $A$. ### `Centralizer(A, S): AlgAss, AlgAss -> AlgAss` ### `Centraliser(A, S): AlgAss, AlgAss -> AlgAss` The centralizer of the subalgebra $S$ of the associative algebra $A$, that is, the subalgebra of $A$ commuting elementwise with $S$. ### `Idealizer(A, B: parameters): AlgAss, AlgAss -> AlgAss` ### `Idealiser(A, B: parameters): AlgAss, AlgAss -> AlgAss` ```magma Side: MonStgElt Default: "Both" ``` Given an associative algebra $A$ and a subalgebra $B$ of $A$, compute the idealizer of $B$ in $A$, that is, the largest subalgebra of $A$ in which $B$ is an ideal. By default the two-sided idealizer, that is, the largest subalgebra in which $B$ is a two-sided ideal, is found; the left- or right-idealizer can be found by setting the parameter `Side` to `"Left"` or `"Right"` respectively. ### `LieAlgebra(A): AlgAss -> AlgGen, Map` For an associative structure constant algebra $A$, return the structure constant algebra $L$ with product given by the Lie bracket $(a,b) \mapsto a*b - b*a$. As a second value the map identifying the elements of $A$ and $L$ is returned. ### `CommutatorModule(A, B): AlgAss, AlgAss -> ModTupRng` Let $A$ and $B$ be subalgebras of an associative algebra with underlying module $M$. This function returns the submodule of $M$ which is spanned by the elements $[a,b] = a*b - b*a$, $a \in A, b \in B$. ### `CommutatorIdeal(A, B): AlgAss, AlgAss -> AlgAss` For two subalgebras $A$ and $B$ of an associative algebra, return the ideal generated by all $[a,b] = a*b - b*a$, $a \in A, b \in B$. ### `LeftAnnihilator(A, B): AlgAss, AlgAss -> AlgAss, AlgAss` For two subalgebras $A$ and $B$ of an associative algebra, return the left annihilator of $B$ in $A$; that is, the subalgebra of $A$ consisting of all elements $a$ such that $a*b = 0$ for all $b \in B$. ### `RightAnnihilator(A, B): AlgAss, AlgAss -> AlgAss, AlgAss` For two subalgebras $A$ and $B$ of an associative algebra, return the right annihilator of $B$ in $A$; that is, the subalgebra of $A$ consisting of all elements $a$ such that $b*a = 0$ for all $b \in B$. ### `Example: liealg (ex-28076e)` We create the Lie algebra $sl_3({\mathbb{Q}})$ as a structure constant algebra. First, we construct $gl_3({\mathbb{Q}})$ from the full matrix algebra $M_3({\mathbb{Q}})$ and get $sl_3({\mathbb{Q}})$ as the derived algebra of $gl_3({\mathbb{Q}})$. ```magma > gl3 := LieAlgebra(Algebra(MatrixRing(Rationals(), 3))); > sl3 := gl3 * gl3; > sl3; Lie Algebra of dimension 8 with base ring Rational Field ``` Let’s see how the first basis element acts. ```magma > for i in [1..8] do > print sl3.i * sl3.1; > end for; (0 0 0 0 0 0 0 0) ( 0 -1 0 0 0 0 0 0) ( 0 0 -2 0 0 0 0 0) (0 0 0 1 0 0 0 0) (0 0 0 0 0 0 0 0) ( 0 0 0 0 0 -1 0 0) (0 0 0 0 0 0 2 0) (0 0 0 0 0 0 0 1) ``` Since it acts diagonally, this element lies in a Cartan subalgebra. The next candidate seems to be the fifth basis element. ```magma > for i in [1..8] do > print sl3.i * sl3.5; > end for; (0 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) ( 0 0 -1 0 0 0 0 0) ( 0 0 0 -1 0 0 0 0) (0 0 0 0 0 0 0 0) ( 0 0 0 0 0 -2 0 0) (0 0 0 0 0 0 1 0) (0 0 0 0 0 0 0 2) ``` This also acts diagonally and commutes with `sl3.1`, hence we have luckily found a full Cartan algebra in $sl_3({\mathbb{Q}})$. We can now easily work out the root system. Obviously the root spaces correspond to the pairs `(sl3.2, sl3.4)`, `(sl3.3, sl3.7)` and `(sl3.6, sl3.8)`. The product of a positive root with its negative should lie in the Cartan algebra. ```magma > sl3.2*sl3.4; ( 1 0 0 0 -1 0 0 0) > sl3.3*sl3.7; (1 0 0 0 0 0 0 0) > sl3.6*sl3.8; (0 0 0 0 1 0 0 0) ``` Clearly some choices have to be made and we fix `sl3.3` as the element $e_\alpha$ corresponding to the first fundamental root $\alpha$, `sl3.7` as $e_{-\alpha}$ and get `sl3.1` as $h_\alpha = e_\alpha * e_{-\alpha}$. For the other fundamental root $\beta$ we have to find an element $e_\beta$ such that $e_\alpha * e_\beta$ is non-zero. ```magma > sl3.3*sl3.2; (0 0 0 0 0 0 0 0) > sl3.3*sl3.4; ( 0 0 0 0 0 -1 0 0) > sl3.3*sl3.6; (0 0 0 0 0 0 0 0) > sl3.3*sl3.8; (0 1 0 0 0 0 0 0) ``` We choose `sl3.8` as $e_\beta$, `sl3.6` as $e_{-\beta}$ and consequently `-sl3.5` as $h_\beta$. This now determines $e_{\alpha+\beta}$ to be `sl3.2` and $e_{-\alpha-\beta}$ to be `sl3.4`. ### `RestrictionOfScalars(A): AlgAss[FldAlg] -> AlgAss, Map` ### `RestrictionOfScalars(A): AlgAss[FldFun] -> AlgAss, Map` ### `RestrictionOfScalars(A, F): AlgAss[FldAlg], Fld -> AlgAss, Map` ### `RestrictionOfScalars(A, F): AlgAss[FldFun], FldFunG -> AlgAss, Map` Given an associative algebra $A$ over an algebraic number or function field $K$ and an optional coefficient field $F$ of $K$, return the algebra isomorphic to $A$ whose coefficient field is $F$ if given otherwise the coefficient field of $K$. ### `Example: restrict (ex-661f6a)` We illustrate some computations of restricting scalars, first over the coefficient field. ```magma > P := PolynomialRing(Rationals()); > F := NumberField(x^3-3*x-1); > Z_F := MaximalOrder(F); > A := QuaternionAlgebra; > A := AssociativeAlgebra(A); > A; Associative Algebra of dimension 4 with base ring F > RA, m := RestrictionOfScalars(A); > RA; Associative Algebra of dimension 12 with base ring Rational Field > m; Mapping from: AlgAss: A to AlgAss: RA given by a rule > m(A.1); (1 0 0 0 0 0 0 0 0 0 0 0) > $1 @@ m; (1 0 0 0) > RA.6 @@ m; ( 0 b^2 0 0) > m($1); (0 0 0 0 0 1 0 0 0 0 0 0) ``` We now show an example where the scalars we are restricting to are not necessarily the coefficient field but a coefficient field of the coefficient field. ```magma > P := PolynomialRing(QuadraticField(23)); > F := NumberField(x^3-3*x-1); > Z_F := MaximalOrder(F); > A := QuaternionAlgebra; > A := AssociativeAlgebra(A); > RA, m := RestrictionOfScalars(A); > RA; Associative Algebra of dimension 12 with base ring Quadratic Field with defining polynomial x^2 - 23 over the Rational Field > m; Mapping from: AlgAss: A to AlgAss: RA given by a rule > RA.6 @@ m; ( 0 b^2 0 0) > m($1); (0 0 0 0 0 1 0 0 0 0 0 0) > RAQ, mQ := RestrictionOfScalars(A, Rationals()); > RAQ, mQ; Associative Algebra of dimension 24 with base ring Rational Field Mapping from: AlgAss: A to AlgAss: RAQ given by a rule > mQ(A.1); (1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0) > $1 @@ mQ; (1 0 0 0) > RAQ.15 @@ mQ; ( 0 0 b^2 + 2*$.1*b + 23 0) > mQ($1); (0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0) ``` ## Operations on Elements ### `Centralizer(A, s): AlgAss, AlgAssElt -> AlgAss` ### `Centraliser(A, s): AlgAss, AlgAssElt -> AlgAss` The centralizer of the element $s$ of the associative algebra $A$, that is, the subalgebra of $A$ commuting with $s$. ### `LieBracket(a, b): AlgAssElt, AlgAssElt -> AlgAssElt` ### `(a, b): AlgAssElt, AlgAssElt -> AlgAssElt` The Lie bracket $a * b - b * a$ of $a$ and $b$, where $a$ and $b$ are elements of an associative algebra $A$. ### `IsScalar(a): AlgAssElt -> BoolElt, RngElt` Returns `true` (and $a$ coerced to $F$) iff a belongs to the base ring $F$ of its parent algebra. ### `RepresentationMatrix(a, M : parameters): AlgAssElt, AlgAss -> AlgMatElt` ```magma Side: MonStgElt Default: "Right" ``` Returns the matrix representation of `Side`-multiplication by the element $a$ in the associative algebra $A$ (which must have $1$) on the $A$-module $M$. ## Representations ### `MatrixAlgebra(A): AlgAss -> AlgMat` For an associative algebra $A$ of dimension $n$ return an isomorphic matrix algebra. If $A$ contains the identity-element, the matrix algebra will be of degree $n$, otherwise it will be of degree $n+1$. ### `MatrixAlgebra(A, M : parameters): AlgAss, AlgAss -> AlgMat, Map` ```magma Side: MonStgElt Default: "Right" ``` Given a finite-dimensional $R$-algebra $A$ and a `Side` $A$-module $M$ (both free as $R$-modules), return the matrix algebra of $A$-endomorphisms of $M$, and the $R$-algebra homomorphism from $A$ into this endomorphism ring. ### `RegularRepresentation(A : parameters): AlgAss -> AlgMat, Map` ```magma Side: MonStgElt Default: "Right" ``` For an associative algebra $A$ of dimension $n$ over $R$ return its regular representation. If $B = (e_1, e_2, \ldots, e_n)$ is the stored basis for $A$, an element $a \in A$ is mapped to the matrix in $R^{n \times n}$ which has as its $i$-th row the coordinates of $e_i * a$ with respect to $B$. As a second map, the homomorphism of $A$ onto the regular representation is returned. By default, the right-regular representation is computed. This can be changed to the left-regular representation (in which the $i$-th row of the image of $a$ contains the coordinates of $a * e_i$) by setting the parameter `Side` to `"Left"`. ## Decomposition of an Algebra This section describes a few functions that can be used to obtain information on the structure of a finite-dimensional associative algebra. ### `JacobsonRadical(A): AlgAssV -> AlgAssV` ```magma Al: MonStgElt Default: "Default" ``` This returns the largest nilpotent ideal of $A$. This function works for finite-dimensional associative algebras defined over a field of characteristic $0$, or over a finite field. The algorithm used by default is taken from [[Cohen *et al.*, 1997](../../references.md#cite-cohen-ivanyos-wales)]. The meataxe algorithm can be used by setting `Al := "Meataxe"`. ### `Example: Jac Rad (ex-564d9d)` We compute the Jacobson radical of the group algebra over the field of three elements of a $3$-group. In that case it is equal to the augmentation ideal. ```magma > G:= SmallGroup( 27, 5 ); > A:= GroupAlgebra( GF(3), G ); > JacobsonRadical( A ); Ideal of dimension 26 of the group algebra A ``` ### `DirectSumDecomposition(A): AlgAssV -> [ AlgAssV ], [ AlgAssVElt ]` ### `IndecomposableSummands(A): AlgAssV -> [ AlgAssV ], [ AlgAssVElt ]` Given an associative algebra $A$, return the direct sum decomposition of $L$ as a sequence of ideals of $L$ whose sum is $L$ and each of which cannot be further decomposed into a direct sum of ideals. The second sequence return contains the corresponding primitive central idempotents. For a description of the algorithm we refer to [[Eberly and Giesbrecht, 1996](../../references.md#cite-eberly-giesbrecht)]. ### `CentralIdempotents(A): AlgAssV -> SeqEnum, SeqEnum` Let $Z$ be the centre of the associative algebra $A$, and let $J(Z)$ denote its Jacobson radical. This function returns a sequence of primitive orthogonal idempotents in $Z$ such that their images in $Z/J(Z)$ span $J(Z)$. Each such idempotent generates a two-sided ideal in $A$. The second return value is the sequence of these ideals. In particular, if $A$ is a semisimple algebra, then this function returns a sequence of primitive orthogonal idempotents spanning $Z$. Furthermore, the ideals in the second sequence returned are simple algebras, and their direct sum equals $A$. For a description of the algorithm we refer to [[Eberly and Giesbrecht, 1996](../../references.md#cite-eberly-giesbrecht)]. ### `Example: Id Pots (ex-4239e7)` We compute the direct sum decomposition of a group algebra. ```magma > G:= SmallGroup( 10, 2 ); > A:= GroupAlgebra( Rationals(), G ); > ee, II:= CentralIdempotents( A ); > ee[1]; 1/10*Id(G) + 1/10*G.2 + 1/10*G.2^2 + 1/10*G.2^3 + 1/10*G.2^4 + 1/10*G.1 + 1/10*G.1 * G.2 + 1/10*G.1 * G.2^2 + 1/10*G.1 * G.2^3 + 1/10*G.1 * G.2^4 > II; [ Ideal of dimension 1 of the group algebra A Basis: Id(G) + G.2 + G.2^2 + G.2^3 + G.2^4 + G.1 + G.1 * G.2 + G.1 * G.2^2 + G.1 * G.2^3 + G.1 * G.2^4, Ideal of dimension 1 of the group algebra A Basis: Id(G) + G.2 + G.2^2 + G.2^3 + G.2^4 - G.1 - G.1 * G.2 - G.1 * G.2^2 - G.1 * G.2^3 - G.1 * G.2^4, Ideal of dimension 4 of the group algebra A Basis: Id(G) - G.2^4 + G.1 - G.1 * G.2^4 G.2 - G.2^4 + G.1 * G.2 - G.1 * G.2^4 G.2^2 - G.2^4 + G.1 * G.2^2 - G.1 * G.2^4 G.2^3 - G.2^4 + G.1 * G.2^3 - G.1 * G.2^4, Ideal of dimension 4 of the group algebra A Basis: Id(G) - G.2^4 - G.1 + G.1 * G.2^4 G.2 - G.2^4 - G.1 * G.2 + G.1 * G.2^4 G.2^2 - G.2^4 - G.1 * G.2^2 + G.1 * G.2^4 G.2^3 - G.2^4 - G.1 * G.2^3 + G.1 * G.2^4 ] ``` We see that here the group algebra is the direct sum of two $1$-dimensional and two $4$-dimensional ideals. The first idempotent is the sum over all group elements divided by the group order.