GAP (Guarana) - Chapter 2: Computing the Mal'cev correspondence

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2 Computing the Mal'cev correspondence
2.1 The main functions
2.1-1 MalcevObjectByTGroup
2.1-2 UnderlyingGroup
2.1-3 UnderlyingLieAlgebra
2.1-4 Dimension
2.1-5 MalcevGrpElementByExponents
2.1-6 MalcevLieElementByCoefficients
2.1-7 RandomGrpElm
2.1-8 RandomLieElm
2.1-9 Log
2.1-10 Exp
2.1-11 *
2.1-12 Comm
2.1-13 MalcevSymbolicGrpElementByExponents
2.1-14 MalcevLieElementByCoefficients
2.2 An example application

### 2 Computing the Mal'cev correspondence

#### 2.1 The main functions

Let \(G\) be a \(T\)-group and \(G^\) its \(\Q\)-powered hull. In this chapter we describe functionality for setting up the Mal'cev correspondence between \(G^\) and the Lie algebra \(L(G)\). The data structures needed for computations with \(G^\) and \(L(G)\) are stored in a so-called Mal'cev object. Computational representations of elements of \(G^\), respectively \(L(G)\), will be called Mal'cev group elements, respectively Mal'cev Lie elements.

##### 2.1-1 MalcevObjectByTGroup

`‣ MalcevObjectByTGroup`( N ) | ( function )

If N is a a T-group (i.e. a finitely generated torsion-free nilpotent group), given by a polycyclic presentation with respect to a Mal'cev basis, then this function computes the Mal'cev correspondence for N and stores the result in a so-called Mal'cev object. Otherwise this function returns `fail'. In the moment this function is restricted to groups N of nilpotency class at most 9.

##### 2.1-2 UnderlyingGroup

`‣ UnderlyingGroup`( mo ) | ( function )

For a Mal'cev object mo this function returns the T-group, which was used to build mo.

##### 2.1-3 UnderlyingLieAlgebra

`‣ UnderlyingLieAlgebra`( mo ) | ( function )

For a Mal'cev object mo this function returns the Lie algebra, which underlies the correspondence described by mo.

##### 2.1-4 Dimension

`‣ Dimension`( mo ) | ( function )

Returns the dimension of the Lie algebra that underlies the Mal'cev object mo.

##### 2.1-5 MalcevGrpElementByExponents

`‣ MalcevGrpElementByExponents`( mo, exps ) | ( function )

For a Mal'cev object mo and an exponent vector exps with rational entries, this functions returns the Mal'cev group element, which has exponents exps with respect to the Mal'cev basis of the underlying group of mo.

##### 2.1-6 MalcevLieElementByCoefficients

`‣ MalcevLieElementByCoefficients`( mo, coeffs ) | ( function )

For a Mal'cev object mo and a coefficient vector coeffs with rational entries, this functions returns the Mal'cev Lie element, which has coefficients coeffs with respect to the basis of the underlying Lie algebra of mo.

##### 2.1-7 RandomGrpElm

`‣ RandomGrpElm`( mo, range ) | ( function )

For a Mal'cev object mo this function returns the output of MalcevGrpElementByExponents( mo, exps ), where exps is an exponent vector whose entries are randomly chosen integers between -range and range.

##### 2.1-8 RandomLieElm

`‣ RandomLieElm`( mo, range ) | ( function )

For a Mal'cev object mo this function returns the output of MalcevLieElementByExponents( mo, coeffs ), where coeffs is a coefficient vector whose entries are randomly chosen integers between -range and range.

##### 2.1-9 Log

`‣ Log`( g ) | ( function )

For Mal'cev group element g this function returns the corresponding Mal'cev Lie element.

##### 2.1-10 Exp

`‣ Exp`( x ) | ( function )

For Mal'cev Lie element x this function returns the corresponding Mal'cev group element.

##### 2.1-11 *

`‣ *`( g, h ) | ( function )

Returns the product of Mal'cev group elements.

##### 2.1-12 Comm

`‣ Comm`( x, y ) | ( function )

If x,y are Mal'cev group elements, then this function returns the group theoretic commutator of x and y. If x,y are Mal'cev Lie elements, then this function returns the Lie commutator of x and y.

##### 2.1-13 MalcevSymbolicGrpElementByExponents

`‣ MalcevSymbolicGrpElementByExponents`( mo, exps ) | ( function )

For a Mal'cev object mo and an exponent vector exps with rational indeterminates as entries, this functions returns the Mal'cev group element, which has exponents exps with respect to the Mal'cev basis of the underlying group of mo.

##### 2.1-14 MalcevLieElementByCoefficients

`‣ MalcevLieElementByCoefficients`( mo, coeffs ) | ( function )

For a Mal'cev object mo and a coefficient vector coeffs with rational indeterminates as entries, this functions returns the Mal'cev Lie element, which has coefficients coeffs with respect to the basis of the underlying Lie algebra of mo.

#### 2.2 An example application

    	gap> n := 2;
    	2
    	gap> F := FreeGroup( n );
    	<free group on the generators [ f1, f2 ]>
    	gap> c := 3;
    	3
    	gap> N := NilpotentQuotient( F, c );
    	Pcp-group with orders [ 0, 0, 0, 0, 0 ]

    	gap> mo := MalcevObjectByTGroup( N );
    	<<Malcev object of dimension 5>>
    	gap> dim := Dimension( mo );
    	5
    	gap> UnderlyingGroup( mo );
    	Pcp-group with orders [ 0, 0, 0, 0, 0 ]
    	gap> UnderlyingLieAlgebra( mo );
    	<Lie algebra of dimension 5 over Rationals>

    	gap> g := MalcevGrpElementByExponents( mo, [1,1,0,2,-1/2] );
    	[ 1, 1, 0, 2, -1/2 ]
    	gap> x := MalcevLieElementByCoefficients( mo, [1/2, 2, -1, 3, 5 ] );
    	[ 1/2, 2, -1, 3, 5 ]

    	gap> h := RandomGrpElm( mo );
    	[ 5, -3, 0, -2, 8 ]
    	gap> y := RandomLieElm( mo );
    	[ 3, 9, 5, 5, 2 ]

    	gap> z := Log( g );
    	[ 1, 1, -1/2, 7/3, -1/3 ]
    	gap> Exp( z ) = g;
    	true
    	gap> k := Exp( y );
    	[ 3, 9, 37/2, 77/4, 395/4 ]
    	gap> Log( k ) = y;
    	true

    	gap> g*h;
    	[ 6, -2, 5, 10, -15/2 ]
    	gap> Comm(g,h);
    	[ 0, 0, 8, 10, -18 ]
    	gap> Comm(x,y);
    	[ 0, 0, 3/2, -25/4, -79/4 ]

    	gap> indets := List( List( [1..dim], i->Concatenation( "a_", String(i) ) ),
    	>                   x->Indeterminate( Rationals, x : new ) );
    	[ a_1, a_2, a_3, a_4, a_5 ]
    	gap> g_sym := MalcevSymbolicGrpElementByExponents( mo, indets );
    	[ a_1, a_2, a_3, a_4, a_5 ]
    	gap> x_sym := Log( g_sym );
    	[ a_1, a_2, -1/2*a_1*a_2+a_3, 1/12*a_1^2*a_2+1/4*a_1*a_2-1/2*a_1*a_3+a_4,
    	-1/12*a_1*a_2^2+1/4*a_1*a_2-1/2*a_2*a_3+a_5 ]
    	gap> g_sym * g;
    	[ a_1+1, a_2+1, a_2+a_3, a_3+a_4+2, 1/2*a_2^2+1/2*a_2+a_3+a_5-1/2 ]

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