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#############################################################################
##
#W liefam.gd GAP library Thomas Breuer
##
#H @(#)$Id: liefam.gd,v 4.24 2002/04/15 10:04:54 sal Exp $
##
#Y Copyright (C) 1997, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St. Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains the definition of the family of Lie elements of a
## family of ring elements.
#1
## Let $x$ be a ring element, then `LieObject(x)' wraps $x$ up into an
## object that contains the same data (namely $x$). The multiplication
## `*' for Lie objects is formed by taking the commutator. More exactly,
## if $l_1$ and $l_2$ are the Lie objects corresponding to
## the ring elements $r_1$ and $r_2$, then $l_1 * l_2$ is equal to the
## Lie object corresponding to $r_1 * r_2 - r_2 * r_2$. Two rules
## for Lie objects are worth noting:
## \beginlist%unordered
## \item{--}
## An element is *not* equal to its Lie element.
##
## \item{--}
## If we take the Lie object of an ordinary (associative) matrix
## then this is again a matrix;
## it is therefore a collection (of its rows) and a list.
## But it is *not* a collection of collections of its entries,
## and its family is *not* a collections family.
## \endlist
##
## Given a family $F$ of ring elements, we can form its Lie family $L$.
## The elements of $F$ and $L$ are in bijection, only the multiplications
## via `\*' differ for both families.
## More exactly, if $l_1$ and $l_2$ are the Lie elements corresponding to
## the elements $f_1$ and $f_2$ in $F$, we have $l_1 * l_2$ equal to the
## Lie element corresponding to $f_1 * f_2 - f_2 * f_2$.
##
## The main reason to distinguish elements and Lie elements on the family
## level is that this helps to avoid forming domains that contain
## elements of both types.
## For example, if we could form vector spaces of matrices then at first
## glance it would be no problem to have both ordinary and Lie matrices
## in it, but as soon as we find out that the space is in fact an algebra
## (e.g., because its dimension is that of the full matrix algebra),
## we would run into strange problems.
##
## Note that the family situation with Lie families may be not familiar.
##
## \beginlist%unordered
## \item{--}
## An element is *not* equal to its Lie element.
##
## \item{--}
## If we take the Lie object of an ordinary (associative) matrix
## then this is again a matrix;
## it is therefore a collection (of its rows) and a list.
## But it is *not* a collection of collections of its entries,
## and its family is *not* a collections family.
##
## \item{--}
## We have to be careful when installing methods for certain types
## of domains that may involve Lie elements.
## For example, the zero element of a matrix space is either an ordinary
## matrix or its Lie element, depending on the space.
## So either the method must be aware of both cases, or the method
## selection must distinguish the two cases.
## In the latter situation, only one method may be applicable to each
## case; this means that it is not sufficient to treat the Lie case
## with the additional requirement `IsLieObjectCollection' but that
## we must explicitly require non-Lie elements for the non-Lie case.
##
## \item{--}
## Being a full matrix space is a property that may hold for a space
## of ordinary matrices or a space of Lie matrices.
## So methods for full matrix spaces must also be aware of Lie matrices.
## \endlist
##
Revision.liefam_gd :=
"@(#)$Id: liefam.gd,v 4.24 2002/04/15 10:04:54 sal Exp $";
#############################################################################
##
#C IsLieObject( <obj> )
#C IsLieObjectCollection( <obj> )
##
## An object lies in `IsLieObject' if and only if it lies in a family
## constructed by `LieFamily'.
##
DeclareCategory( "IsLieObject",
IsRingElement
and IsZeroSquaredElement
and IsJacobianElement );
DeclareCategoryCollections( "IsLieObject" );
#############################################################################
##
#A LieFamily( <Fam> )
##
## is a family $F$ in bijection with the family <Fam>,
## but with the Lie bracket as infix multiplication.
## That is, for $x$, $y$ in <Fam>, the product of the images in $F$ will be
## the image of $x \* y - y \* x$.
##
## The standard type of objects in a Lie family <F> is `<F>!.packedType'.
##
## \indextt{Embedding!for Lie algebras}
## The bijection from <Fam> to $F$ is given by `Embedding( <Fam>, $F$ )';
## this bijection respects addition and additive inverses.
##
DeclareAttribute( "LieFamily", IsFamily );
#############################################################################
##
#A UnderlyingFamily( <Fam> )
##
## If <Fam> is a Lie family then `UnderlyingFamily( <Fam> )'
## is a family $F$ such that `<Fam> = LieFamily( $F$ )'.
##
DeclareAttribute( "UnderlyingFamily", IsObject );
#############################################################################
##
#A LieObject( <obj> )
##
## Let <obj> be a ring element. Then `LieObject( <obj> )' is the
## corresponding Lie object. If <obj> lies in the family <F>,
## then `LieObject( <obj> )' lies in the family LieFamily( <F> )
## (see~"LieFamily").
##
DeclareAttribute( "LieObject", IsRingElement );
#############################################################################
##
#F IsLieObjectsModule( <V> )
##
## If a free $F$-module <V> is in the filter `IsLieObjectsModule' then
## this expresses that <V> consists of Lie objects (see~"..."),
## and that <V> is handled via the mechanism of nice bases (see~"...")
## in the following way.
## Let $K$ be the default field generated by the vector space generators of
## <V>.
## Then the `NiceFreeLeftModuleInfo' value of <V> is irrelevant,
## and the `NiceVector' value of $v \in <V>$ is defined as the underlying
## element for which <v> is obtained as `LieObject' value.
##
DeclareHandlingByNiceBasis( "IsLieObjectsModule",
"for free left modules of Lie objects" );
#############################################################################
##
#E
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