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##
#W qea.gd QuaGroup Willem de Graaf
##
##
## Declarations for elements of quantized enveloping algebras, and highest
## weight modules
##
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##
#C IsQEAElement( <obj> )
#C IsQEAElementCollection( <obj> )
#C IsQEAElementFamily( <fam> )
##
## This is the category of elements of a quantized enveloping algebra.
##
DeclareCategory( "IsQEAElement", IsVector and IsRingElement and
IsMultiplicativeElementWithOne );
DeclareCategoryCollections( "IsQEAElement" );
DeclareCategoryFamily( "IsQEAElement" );
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#F CollectQEAElement
##
##
DeclareGlobalFunction( "CollectQEAElement" );
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##
#C IsQuantumUEA( <A> )
##
## The category of quantized enveloping algebras.
##
DeclareCategory( "IsQuantumUEA", IsAlgebra );
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##
#A QuantizedUEA( <R> )
##
## For a root system <R> this function returns the corresponding quantized
## enveloping algebra <U>. The attribute `GeneratorsOfAlgebra' of <U>
## is a list of generators of a PBW-type basis of <U>. These are computed
## relative to the reduced expression for the longest element of the
## Weyl group contained in `LongestWeylWord( <R> )'. If you would like to
## have a PBW-type basis relative to a different reduced expression than
## you must set the value of this attribute by hand (preferably directly
## after creating the root system).
##
DeclareAttribute( "QuantizedUEA", IsRootSystem );
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##
#O LeadingQEAMonomial( <novar>, <f> )
##
## For an element of the negative part of a qea, this gives its leading
## monomial with respect to the reverse lexicographical ordering.
## <novar> is the number of generators of that negative part (i.e.,
## the number of positive roots).
##
DeclareOperation( "LeadingQEAMonomial", [ IsInt, IsQEAElement ] );
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##
#F LeftReduceQEAElement( <novar>, <G>, <lms>, <lmtab>, <p> )
##
## Left reduction in the negative part of a qea.
##
DeclareGlobalFunction( "LeftReduceQEAElement" );
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##
#A QuantumParameter( <U> )
##
## The is the quantum parameter of the quantized universal enveloping
## algebra <U>.
##
DeclareAttribute( "QuantumParameter", IsQuantumUEA );
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##
#C IsGenericQUEA( <U> )
##
## A quantized uea has this category if it is defined over
## the QuantumField, i.e., uring the generic function for constructing
## quantized enveloping algebras.
##
DeclareCategory( "IsGenericQUEA", IsQuantumUEA );
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##
#A IrreducibleQuotient( <V> )
##
## Here <V> is a highest-weight module over a quantized enveloping
## algebra, possibly defined with a root of 1 as quantum parameter.
## This function returns the unique irreducible quotient of <V>,
## that contains a highest weight vector.
##
DeclareAttribute( "IrreducibleQuotient", IsAlgebraModule );
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##
#A CanonicalMapping( <A> )
##
DeclareAttribute( "CanonicalMapping", IsObject );
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#A GenericModule( <V> )
##
DeclareAttribute( "GenericModule", IsAlgebraModule);
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##
#O HWModuleByGenerator( <W>, <w>, <hw> )
##
DeclareOperation( "HWModuleByGenerator",
[ IsAlgebraModule, IsObject, IsList ] );
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##
#O InducedQEAModule( <U>, <V> )
##
DeclareOperation( "InducedQEAModule", [ IsQuantumUEA, IsAlgebraModule ] );
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##
#A FundamentalModules( <U> )
##
DeclareAttribute( "FundamentalModules", IsQuantumUEA );
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##
#O HWModuleByTensorProduct( <U>, <list> )
##
DeclareOperation( "HWModuleByTensorProduct", [ IsQuantumUEA, IsList ] );
