Quelle CHAP003.htm Sprache: HTML

products/sources/formale Sprachen/GAP/pkg/liepring/htm/CHAP003.htm

<html><head><title>[LiePRing] 3 LiePRings in GAP</title></head>
<body text="#000000" bgcolor="#ffffff">
[<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Previous</a>] [<a href ="CHAP004.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
<h1>3 LiePRings in GAP</h1>

<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP003.htm#SECT001">Ordinary Lie p-rings</a>
<li> <A HREF="CHAP003.htm#SECT002">Generic Lie p-rings</a>
<li> <A HREF="CHAP003.htm#SECT003">Specialising Lie $p$-rings</a>
<li> <A HREF="CHAP003.htm#SECT004">Subrings of Lie p-rings</a>
<li> <A HREF="CHAP003.htm#SECT005">Elementary functions</a>
<li> <A HREF="CHAP003.htm#SECT006">Series of subrings</a>
<li> <A HREF="CHAP003.htm#SECT007">The Lazard correspondence</a>
</ol>

This package introduces a new datastructure that allows to define and
compute with Lie p-rings in GAP. We first describe this datastructure
in the case of ordinary Lie p-rings; that is, Lie p-rings for a fixed
prime p with given structure constants. Then we show how this datastructure
can also be used to define so-called 'generic' Lie p-rings; that is,
Lie p-rings with indeterminate prime p.


<h2><a name="SECT001">3.1 Ordinary Lie p-rings</a></h2>

Let p be a prime and let L be a Lie p-ring of order pn. Let
(l1, …, ln) be a basis for L. Then there exist coefficients
ci,j,k ∈ {0, …, p−1} so that the following relations hold
in L for 1 ≤ i,j ≤ n with i ≠ j:

 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> li ·lj = </td><td nowrap="nowrap" align="center">n ∑ k=i+1 </td><td nowrap="nowrap" align="center">ci,j,k lk, </td></tr></table></td></tr></table>
 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> p li = </td><td nowrap="nowrap" align="center">n ∑ k=i+1 </td><td nowrap="nowrap" align="center">ci,i,k lk·</td></tr></table></td></tr></table>

These structure constants define the Lie p-ring L. As the multiplication
in a Lie p-ring is anticommutative, it follows that ci,j,k = −cj,i,k
holds for each k and each i ≠ j. Thus the structure constants
ci,j,k for i ≥ j are sufficient to define the Lie p-ring L.

This package contains the new datastructure <var>LiePRing</var> that allows to
define Lie p-rings via their structure constants ci,j,k. To use
this datastructure, we first collect all relevant information into a
record as follows:

<dl compact>
<dt><var>dim</var> <dd>
 the dimension n of L;

<dt><var>prime</var> <dd>
 the prime p of L;

<dt><var>tab</var> <dd>
 a list with structure constants
 [c1,1, c2,1, c2,2, c3,1, c3,2, c3,3, …].
</dl>

Each entry ci,j in the list <var>tab</var> is a list [k1, ci,j,k1, k2, ci,j,k2, …] so that k1 < k2 < … and the entries
ci,j,k1, ci,j,k2, … are the non-zero structure contants
in the product li ·lj. Thus if li ·lj = 0, then ci,j
is the empty list. If an entry in the list <var>tab</var> is not bound, then it is
assumed to be the empty list.

<a name = "SSEC001.1"></a>
<li><code>LiePRingBySCTable( SC )</code>
<a name = "SSEC001.1"></a>
<li><code>LiePRingBySCTableNC( SC )</code>

These functions create a <var>LiePRing</var> from the structure constants table
record SC. The first version checks that the multiplication defined by
<var>tab</var> is alternating and satisfies the Jacobi-identity, the second version
assumes that this is the case and omits these checks. These checks can
also be carried out independently via the following function.

<a name = "SSEC001.2"></a>
<li><code>CheckIsLiePRing( L )</code>

This function takes as input an object <var>L</var> created via <var>LiePRingBySCTableNC</var>
and checks that the Jacobi identity holds in this ring.

The following example creates the Lie 2-ring of order 8 with trivial
multiplication.

<pre>
gap> SC := rec( dim := 3, prime := 2, tab := [] );;
gap> L := LiePRingBySCTable(SC);
<LiePRing of dimension 3 over prime 2>
gap> l := BasisOfLiePRing(L);
[ l1, l2, l3 ]
gap> l[1]*l[2];
0
gap> 2*l[1];
0
gap> l[1] + l[2];
l1 + l2
</pre>

The next example creates a LiePRing of order 54 with non-trivial
multiplication.

<pre>
gap> SC := rec( dim := 4, prime := 5, tab := [ [], [3, 1], [], [4, 1]]);;
gap> L := LiePRingBySCTableNC(SC);;
gap> ViewPCPresentation(L);
[l2,l1] = l3
[l3,l1] = l4
</pre>


<h2><a name="SECT002">3.2 Generic Lie p-rings</a></h2>

In a generic Lie p-ring, p is allowed to be an indeterminate and
the structure constants are allowed to be rational functions over a
polynomial ring in a finite set of
commuting indeterminates. It is generally assumed that the indeterminate
with name <var>p</var> represents the prime, the indeterminate with name <var>w</var>
represents the smallest primitive root modulo the prime and there are
further predefined indeterminates with the names <var>x</var>, <var>y</var>, <var>z</var>, <var>t</var>,
<var>j</var>, <var>k</var>, <var>m</var>, <var>n</var>, <var>r</var>, <var>s</var>, <var>u</var> and <var>v</var>. These indeterminates are used
in the database of Lie p-rings and they can be obtained via

<a name = "SSEC002.1"></a>
<li><code>IndeterminateByName( string )</code>

The structure constants records for generic Lie p-rings are similar to
those for ordinary Lie p-rings, but have the additional entry <var>param</var>
which is a list containing all indeterminates used in the considered Lie
p-ring. We exhibit an example.

<pre>
gap> p := IndeterminateByName("p");;
gap> x := IndeterminateByName("x");;
gap> S := rec( dim := 5,
> param := [ x ],
> prime := p,
> tab := [ [ 4, 1 ], [ 3, 1 ], [ 5, x ], [ 4, 1 ], [ 5, 1 ] ] );;
gap> L := LiePRingBySCTable(S);
<LiePRing of dimension 5 over prime p with parameters [ x ]>
gap> ViewPCPresentation(L);
p*l1 = l4
p*l2 = x*l5
[l2,l1] = l3
[l3,l1] = l4
[l3,l2] = l5
gap> l := BasisOfLiePRing(L);
[ l1, l2, l3, l4, l5 ]
gap> p*l[1];
l4
gap> l[1]+l[2];
l1 + l2
gap> l[1]*l[2];
-1*l3
</pre>


<h2><a name="SECT003">3.3 Specialising Lie $p$-rings</a></h2>

A generic Lie p-ring defines a family of ordinary Lie p-rings by
evaluating the parameters contained in its presentation. It is generally
assumed that the indeterminate p is evaluated to a rational prime P
and the indeterminate w is evaluated to the smallest primitive root
modulo P (this can be determined via <var>PrimitiveRootMod(P)</var>). All other
indeterminates can take arbitrary integer values (usually these values are
in {0, …, P−1}, but other choices are possible as well). The
following functions allow to evaluate the indeterminates.

<a name = "SSEC003.1"></a>
<li><code>SpecialiseLiePRing(L, P, para, vals)</code>

takes as input a generic Lie p-ring <var>L</var>, a rational prime <var>P</var>,
a list of indeterminates <var>para</var> and a corresponding list of values
<var>vals</var>. The function returns a new Lie p-ring in which the prime
<var>p</var> is evaluated to <var>P</var>, the parameter <var>w</var> is evaluated to
<var>PrimitiveRootMod(P)</var> and the parameters in <var>para</var> are evaluated
to <var>vals</var>.

<a name = "SSEC003.2"></a>
<li><code>SpecialisePrimeOfLiePRing(L, P)</code>

this is a shortcut for <var>SpecialiseLiePRing(L, P, [], [])</var>. We exhibit
a some example applications.

<pre>
gap> p := IndeterminateByName("p");;
gap> w := IndeterminateByName("w");;
gap> x := IndeterminateByName("x");;
gap> y := IndeterminateByName("y");;
gap> S := rec( dim := 7,
> param := [ w, x, y ],
> prime := p,
> tab := [ [ ], [ 6, 1 ], [ 6, 1 ], [ 7, 1 ], [ ],
> [ 6, x, 7, y ], [ ], [ 7, 1 ], [ 6, w ] ] );;
gap> L := LiePRingBySCTable(S);
<LiePRing of dimension 7 over prime p with parameters [ w, x, y ]>
gap> ViewPCPresentation(L);
p*l2 = l6
p*l3 = x*l6 + y*l7
[l2,l1] = l6
[l3,l1] = l7
[l4,l2] = l7
[l4,l3] = w*l6
gap> SpecialiseLiePRing(L, 7, [x, y], [0,0]);
<LiePRing of dimension 7 over prime 7>
gap> ViewPCPresentation(last);
7*l2 = l6
[l2,l1] = l6
[l3,l1] = l7
[l4,l2] = l7
[l4,l3] = 3*l6
gap> SpecialiseLiePRing(L, 11, [x, y], [0,10]);
<LiePRing of dimension 7 over prime 11>
gap> ViewPCPresentation(last);
11*l2 = l6
11*l3 = 10*l7
[l2,l1] = l6
[l3,l1] = l7
[l4,l2] = l7
[l4,l3] = 2*l6
gap> Cartesian([0,1],[0,1]);
[ [ 0, 0 ], [ 0, 1 ], [ 1, 0 ], [ 1, 1 ] ]
gap> List(last, v -> SpecialiseLiePRing(L, 2, [x,y], v));
[ <LiePRing of dimension 7 over prime 2>,
 <LiePRing of dimension 7 over prime 2>,
 <LiePRing of dimension 7 over prime 2>,
 <LiePRing of dimension 7 over prime 2> ]
</pre>

It is not necessary to specialise all parameters at once. In particular,
it is possible to leave the prime <var>p</var> as indeterminate and specialize only
some of the parameters. (Except for <var>w</var> which is linked to <var>p</var>.)

<pre>
gap> SpecialiseLiePRing(L, p, [x], [0]);
<LiePRing of dimension 7 over prime p with parameters [ y, w ]>
gap> ViewPCPresentation(last);
p*l2 = l6
p*l3 = y*l7
[l2,l1] = l6
[l3,l1] = l7
[l4,l2] = l7
[l4,l3] = w*l6
gap> SpecialiseLiePRing(L, p, [y], [3]);
<LiePRing of dimension 7 over prime p with parameters [ x, w ]>
gap> ViewPCPresentation(last);
p*l2 = l6
p*l3 = x*l6 + 3*l7
[l2,l1] = l6
[l3,l1] = l7
[l4,l2] = l7
[l4,l3] = w*l6
</pre>

It is also possible to specialise the prime only, but leave all or
some of the parameters indeterminate. Note that specialising <var>p</var>
also specialises <var>w</var>. Again, we continue to use the generic Lie
p-ring L as above.

<pre>
gap> SpecialisePrimeOfLiePRing(L, 29);
<LiePRing of dimension 7 over prime 29 with parameters [ y, x ]>
gap> ViewPCPresentation(last);
29*l2 = l6
29*l3 = x*l6 + y*l7
[l2,l1] = l6
[l3,l1] = l7
[l4,l2] = l7
[l4,l3] = 2*l6
</pre>

<a name = "SSEC003.3"></a>
<li><code>LiePValues(K)</code>

if K is obtained by specialising, then this attribute is set and
contains the parameters that have been specialised and their values.

<pre>
gap> L := LiePRingsByLibrary(6)[14];
<LiePRing of dimension 6 over prime p with parameters [ x ]>
gap> K := SpecialisePrimeOfLiePRing(L, 5);
<LiePRing of dimension 6 over prime 5 with parameters [ x ]>
gap> LiePValues(K);
[ [ p, w ], [ 5, 2 ] ]
</pre>


<h2><a name="SECT004">3.4 Subrings of Lie p-rings</a></h2>

Let L be a Lie p-ring with basis (l1, …, ln) and let U be a
subring of L. Then U is a Lie p-ring and thus also has a basis (u1, …, um). For 1 ≤ i ≤ m we define the coefficients ai,j ∈ {0, …, p−1} via
 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> ui = </td><td nowrap="nowrap" align="center">n ∑ j=1 </td><td nowrap="nowrap" align="center">ai,j li </td></tr></table></td></tr></table>
and we denote with A the matrix with entries ai,j. We say that the
basis (u1, …, um) is inducedif A is in upper triangular
form. Further, the basis (u1, …, um) is canonicalif A is
in upper echelon form; that is, it is upper triangular, each row in A has
leading entry 1 and there are 0's above the leading entry. Note that a
canonical basis is unique for the subring.

<a name = "SSEC004.1"></a>
<li><code>LiePSubring(L, gens)</code>

Let <var>L</var> be a (generic or ordinary) Lie p-ring and let <var>gens</var> be a set of
elements in <var>L</var>. This function determines a canonical basis for the subring
generated by <var>gens</var> in <var>L</var> and returns the LiePSubring of <var>L</var> generated by
<var>gens</var>. Note that this function may have strange effects for generic Lie
p-rings as the following example shows.

<pre>
gap> L := LiePRingsByLibrary(6)[100];
<LiePRing of dimension 6 over prime p>
gap> l := BasisOfLiePRing(L);
[ l1, l2, l3, l4, l5, l6 ]
gap> U := LiePSubring(L, [5*l[1]]);
<LiePRing of dimension 3 over prime p>
gap> BasisOfLiePRing(U);
[ l1, l4, l6 ]
gap> K := SpecialisePrimeOfLiePRing(L, 5);
<LiePRing of dimension 6 over prime 5>
gap> b := BasisOfLiePRing(K);
[ l1, l2, l3, l4, l5, l6 ]
gap> LiePSubring(K, [5*b[1]]);
<LiePRing of dimension 2 over prime 5>
gap> BasisOfLiePRing(last);
[ l4, l6 ]
gap> K := SpecialisePrimeOfLiePRing(L, 7);
<LiePRing of dimension 6 over prime 7>
gap> b := BasisOfLiePRing(K);
[ l1, l2, l3, l4, l5, l6 ]
gap> U := LiePSubring(K, [5*b[1]]);
<LiePRing of dimension 3 over prime 7>
gap> BasisOfLiePRing(U);
[ l1, l4, l6 ]
</pre>

<a name = "SSEC004.2"></a>
<li><code>LiePIdeal(L, gens)</code>

return the ideal of L generated by gens. This function computes a
an induced basis for the ideal.

<pre>
gap> LiePIdeal(L, [l[1]]);
<LiePRing of dimension 5 over prime p>
gap> BasisOfLiePRing(last);
[ l1, l3, l4, l5, l6 ]
</pre>

<a name = "SSEC004.3"></a>
<li><code>LiePQuotient(L, U)</code>

return a Lie p-ring isomorphic to L/U where U must be an ideal of
L. This function requires that L is an ordinary Lie p-ring.

<pre>
gap> LiePIdeal(K, [b[1]]);
<LiePRing of dimension 5 over prime 7>
gap> LiePIdeal(K, [b[2]]);
<LiePRing of dimension 4 over prime 7>
gap> LiePQuotient(K,last);
<LiePRing of dimension 2 over prime 7>
</pre>


<h2><a name="SECT005">3.5 Elementary functions</a></h2>

The functions described in this section work for ordinary and generic
Lie p-rings and their subrings.

<a name = "SSEC005.1"></a>
<li><code>PrimeOfLiePRing(L)</code>

returns the underlying prime. This can either be an integer or an
indeterminate.

<a name = "SSEC005.2"></a>
<li><code>BasisOfLiePRing(L)</code>

returns a basis for L.

<a name = "SSEC005.3"></a>
<li><code>DimensionOfLiePRing(L)</code>

returns the dimension of <var>L</var>.

<a name = "SSEC005.4"></a>
<li><code>ParametersOfLiePRing(L)</code>

returns the list of indeterminates involved in L. If L is a subring
of a Lie p-ring defined by structure constants, then the parameters of
the parent are returned.

<a name = "SSEC005.5"></a>
<li><code>ViewPCPresentation(L)</code>

prints the presentation for L with respect to its basis.


<h2><a name="SECT006">3.6 Series of subrings</a></h2>

Let <var>L</var> be a generic or ordinary Lie p-ring or a subring of such such
a Lie p-ring.

<a name = "SSEC006.1"></a>
<li><code>LiePLowerCentralSeries(L)</code>

returns the lower central series of L.

<a name = "SSEC006.2"></a>
<li><code>LiePLowerPCentralSeries(L)</code>

returns the lower exponent-p central series of L.

<a name = "SSEC006.3"></a>
<li><code>LiePDerivedSeries(L)</code>

returns the derived series of L.

<a name = "SSEC006.4"></a>
<li><code>LiePMinimalGeneratingSet(L)</code>

returns a minimal generating set of L; that is, a generating set of
smallest possible size.


<h2><a name="SECT007">3.7 The Lazard correspondence</a></h2>

The following function has been implemented by Willem de Graaf. It uses
the Baker-Campbell-Hausdorff formula as described in <a href="biblio.htm#CGV12"><cite>CGV12</cite></a> and it
is based on the Liering package <a href="biblio.htm#CdG10"><cite>CdG10</cite></a>.

<a name = "SSEC007.1"></a>
<li><code>PGroupByLiePRing(L)</code>

Let <var>L</var> be an ordinary Lie p-ring with cl(L) < p. Then this function
returns the p-group G obtained from L via the Lazard correspondence.


[<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Previous</a>] [<a href ="CHAP004.htm">Next</a>] [<a href = "theindex.htm">Index</a>]

<address>LiePRing manual June 2024
</address></body></html>

Messung V0.5

¤ Dauer der Verarbeitung: 0.7 Sekunden ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.