4.10 Setting Properties of an element in an algebra
4.10.1 SetInv[a, b, c, …]
Aliases: None
Description: SetInv[a,b,c,…] sets all the symbols a, b, c, … to be invertible (i.e. invQ[a],
invQ[b], invQ[c], … are set True).
Arguments: Symbols separated by commands
Comments / Limitations: If one does not set x to be invertible before the first use of invL[x]
or invR[x], then NCAlgebra may not make the substitution from invL[x] **x to 1 or
from x **invR[x] to 1 automatically.
4.10.2 SetSelfAdjoint[Symbols]
Aliases: None
Description: SetSelfAdjoint[a, b, …] will set a, b, … to be self-adjoint. The rules tp[a] := a,
tp[b] :=b, … and aj[a] := a, aj[b] := b, … will be automatically applied. See SelfAdjointQ.
Arguments: Symbols is one or more symbols separated by commas.
Comments / Limitations: If one does not set x to be self adjoint before the first use of
aj[x], then NCAlgebra may not make the substitution from aj[x] to x automatically.
Similary for tp.
4.10.3 SelfAdjointQ[aSymbol]
Aliases: None
Description: SelfAdjointQ[x] will return True if SetSelfAdjoint[x] was executed
previously. See SetSelfAdjoint.
Arguments: aSymbol is a symbol
Comments / Limitations: None
4.10.4 SetIsometry[Symbols]
Aliases: None
Description: SetIsometry[a,b,…] will set a, b, … to be isometries. If set the rules tp[a] ** a
:= Id, tp[b] ** b :=Id, … and aj[a] ** a := Id; aj[b] ** b := Id; … will be automatically
applied. See IsometryQ.
Arguments: Symbols is one or more symbols separated by commas.
Comments / Limitations: If one does not set x to be an isometry before the first use of aj[x],
then NCAlgebra may not make the substitution from aj[x] **x to 1 automatically.
Similarly for tp.
4.10.5 IsometryQ[aSymbol]
Aliases: None
Description: IsometryQ[x] will return True if SetIsometry[x] was executed previously. See
SetIsometry.
Arguments: aSymbol is a symbol.
Comments / Limitations: None
4.10.6 SetCoIsometry[Symbols]
Aliases: None
Description: SetCoIsometry[a,b,…] will set a, b, … to be co-isometries. The rules a ** tp[a]
:= Id, b ** tp[b] :=Id, … and a ** aj[a] := Id, b ** aj[b] := Id, … will be automatically
applied. See CoIsometryQ.
Arguments: Symbols is one or more symbols separated by commas.
Comments / Limitations: If one does not set x to be a coisometry before the first use of aj[x],
then NCAlgebra may not make the substitution from x **aj[x] to 1 automatically.
4.10.7 CoIsometryQ[aSymbol]
Aliases: None
Description: CoIsometryQ[x] will return True if SetCoIsometry[x] was executed
previously. See SetCoIsometry.
Arguments: aSymbol is a symbol.
Comments / Limitations: None
4.10.8 SetUnitary[Symbols]
Aliases: None
Description: SetUnitary[a,b,…] will set a, b, … to be isometries and co-isometries. Also
effects on UnitaryQ. See SetIsometry and SetCoIsometry.
Arguments: Symbols is one or more symbols separated by commas.
Comments / Limitations: If one does not set x to be a unitary before the first use of aj[x],
then NCAlgebra may not make the substitution from x**aj[x] to 1 or from aj[x] **x
to 1 automatically.
4.10.9 UnitaryQ[aSymbol]
Aliases: None
Description: UnitaryQ[x] will return True if SetUnitary[x] was executed previously.
Caution: If one executes SetIsometry[x]; SetCoIsometry[x]; then x is unitary, but
UnitaryQ remains uneffected. See SetUnitary.
Arguments: aSymbol is a symbol.
Comments / Limitations: None
4.10.10 SetProjection[Symbols]
Aliases: None
Description: SetProjection[a,b,…] will set a, b, … to be projections. The rules a ** a
:= a, b ** b :=b, … will be automatically applied. Caution: If one wants x to be a
self-adjoint projection, then one must execute SetSelfAdjoint[x]; SetProjection[x].
See ProjectionQ.
Arguments: Symbols is one or more symbols separated by commas.
Comments / Limitations: If one does not set x to be a projection before the first use of x,
then NCAlgebra may not make the substitution from x **x to x.
4.10.11 ProjectionQ[S]
Aliases: None
Description: ProjectionQ[x] will return true if SetProjection[x] was executed previously.
See SetProjection.
Arguments: S is a symbol.
Comments / Limitations: None
4.10.12 SetSignature[Symbols]
Aliases: None
Description: When SetSignature[a] and SetSelfAdjoint[a] are executed, the rule a ** a
:= -1 will be automatically applied. See SetSelfAdjoint and SignatureQ.
Arguments: Symbols is one or more symbols separated by commas.
Comments / Limitations: If one does not set x to be a signature matrix and self adjoing
before the first use of x, then NCAlgebra may not make the substitution from x **x
to -1.
4.10.13 SignatureQ[Symbol]
Aliases: None
Description: SignatureQ[x] will return True if SetSignature[x] was executed previously.
See SetSignature.