Allow multiplication with different algebras

[blegat]

Useful for https://github.com/kalmarek/SymbolicWedderburn.jl/issues/79

[kalmarek]

As expressed in our long call I think we need something better here, namely we need something (a struct)? that is going to combine Gramm (psd) constraint basis, Gramm matrix and (linear) constraint basis. What is the actual code that needs these? I bet (not too much though :D) it is not a simple * aka b'*Q*b, since the whole operation is treating Q as a quadratic form with the appropriate basis. Can you think of defining QuadraticForm struct that will do your bidding?

[blegat]

I don't know if we need to add the quadratic form in StarAlgebras. In SumOfSquares, I simply redirect operate!(::UnsafeAddMul, a::AlgebraElement, a::GramMatrix) to operate!(::UnsafeAddMul, a::AlgebraElement, b::AlgebraElement, c::AlgebraElement): https://github.com/jump-dev/SumOfSquares.jl/blob/cf146a6129adc388b64f4c77aac0c172353b6664/src/gram_matrix.jl#L147-L165 So a way to deal with algebra elements of different bases is all we need. At the moment, I use SparseCoefficient and FullBasis for a, b and c so that they have the same basis. If we have a multiplicative structure working with different basis, I could have a be in the constraint basis with a Vector or SparseVector coefficient and b/c be in the gram basis with SparseVector coefficient of one entry.

[kalmarek]

how about redirecting it to operate!(SA.UnsafeQuadratic(*), res::AlgebraElement, b::AlgebraElement, c::AlgebraElement) ? btw. what are args here https://github.com/jump-dev/SumOfSquares.jl/blob/cf146a6129adc388b64f4c77aac0c172353b6664/src/gram_matrix.jl#L151 ?

UnsafeQuadratic is explicit in what it does and unless we have a good reason, I'd avoid further overloading operate!(::UnsafeAddMul, ...) as it makes the code very fragile and hard to reason about. That is unless we change the algebraic model of our data.

So far we have assumed that Basis contains everything what is needed for algebra structure (both + and *). If this assumption does not hold for this particular case, I'd suggest that we need a new structure (QuadraticForm?) that encapsulates this use. It should contain both the input and output bases and propagate this information down the chain.

[blegat]

The args can be AlgebraElements. For instance, in Putinar's certificate, you have p = s0 + sum si fi so fi is in args.... For this reason, quadratic would not be enough but actually we also don't do symmetry reduction for Putinar's certificate yet anyway. But I was thinking of something that would also work without symmetry reduction so that's still relevant. So we would have

operate_to(::UnsafeAddMul, res::AlgebraElement, s::GramMatrix, f::AlgebraElement, b::AlgebraElement)

that becomes

operate_to(::UnsafeAddMul, res::AlgebraElement, a::AlgebraElement, b::AlgebraElement, f::AlgebraElement)

where each AlgebraElement would have basis SubBasis.

[kalmarek]

As I said, I'm against overloading the poor UnsafeAddMul with more meanings. If You're doing Putinar thing, we should be able to pass UnsafePutinarThing(*, inbasis, outbasis). If we're doing Gramm thing, lets pass UnsafeQuadraticThing(*, inbasis, outbasis). We could create some AbstractAddMul and supertype the three above if there's too much code overlap. But I honestly don't think there will be that much code overlap when it comes to performant code...

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[blegat]

This PR, in addition to allowing to deal with SymbolicWedderburn decomposition by defining a custom MultiplicativeStructure, also allow doing https://github.com/JuliaAlgebra/MultivariateBases.jl/pull/50, which suggest it might be a good direction :)