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I want to define Sweedler's four-dimensional Hopf algebra, which is freely generated by $x,y$ and subject to the relations \begin{align} x^2 = 1, \qquad y^2 = 0, \qquad x\cdot y = - y\cdot x~ , \end{align} but I don't see how to do it.

I have tried the following:

sage: A.<x,y> = FreeAlgebra(QQbar)
sage: I = A*[x*x - 1, y*y, x*y + y*x]*A
sage: H.<x,y> = A.quo(I)
sage: H
Quotient of Free Algebra on 2 generators (x, y) over Algebraic Field by the ideal (-1 + x^2, y^2, x*y + y*x)

But then I get

sage: H.one() == H(x*x)
False

So is this currently possibly using a different method?

Thanks

I want to define Sweedler's four-dimensional Hopf algebra, which is freely generated by $x,y$ and subject to the relations \begin{align} $$ x^2 = 1, \qquad y^2 = 0, \qquad x\cdot y = - y\cdot x~ , \end{align} $$ but I don't see how to do it.

I have tried the following:

sage: A.<x,y> = FreeAlgebra(QQbar)
sage: I = A*[x*x - 1, y*y, x*y + y*x]*A
sage: H.<x,y> = A.quo(I)
sage: H
Quotient of Free Algebra on 2 generators (x, y) over Algebraic Field by the ideal (-1 + x^2, y^2, x*y + y*x)

But then I get

sage: H.one() == H(x*x)
False

So is this currently possibly using a different method?

Thanks

I want to define (the algebra part of) Sweedler's four-dimensional Hopf algebra, which is freely generated by $x,y$ and subject to the relations $$ x^2 = 1, \qquad y^2 = 0, \qquad x\cdot y = - y\cdot x~ , $$ but I don't see how to do it.

I have tried the following:

sage: A.<x,y> = FreeAlgebra(QQbar)
sage: I = A*[x*x - 1, y*y, x*y + y*x]*A
sage: H.<x,y> = A.quo(I)
sage: H
Quotient of Free Algebra on 2 generators (x, y) over Algebraic Field by the ideal (-1 + x^2, y^2, x*y + y*x)

But then I get

sage: H.one() == H(x*x)
False

So is this currently possibly using a different method?

Thanks

I want to define (the algebra part of) Sweedler's four-dimensional Hopf algebra, which is freely generated by $x,y$ and subject to the relations $$ x^2 = 1, \qquad y^2 = 0, \qquad x\cdot y = - y\cdot x~ , $$ but I don't see how to do it.

I have tried the following:

sage: A.<x,y> = FreeAlgebra(QQbar)
sage: I = A*[x*x - 1, y*y, x*y + y*x]*A
sage: H.<x,y> = A.quo(I)
sage: H
Quotient of Free Algebra on 2 generators (x, y) over Algebraic Field by the ideal (-1 + x^2, y^2, x*y + y*x)

But then I get

sage: H.one() == H(x*x)
False

So is this currently possibly using a different method?

Thanks

I want to define (the algebra part of) Sweedler's four-dimensional Hopf algebra, which is freely generated by $x,y$ and subject to the relations $$ x^2 = 1, \qquad y^2 = 0, \qquad x\cdot y = - y\cdot x~ , $$ but I don't see how to do it.

I have tried the following:

sage: A.<x,y> = FreeAlgebra(QQbar)
sage: I = A*[x*x - 1, y*y, x*y + y*x]*A
sage: H.<x,y> = A.quo(I)
sage: H
Quotient of Free Algebra on 2 generators (x, y) over Algebraic Field by the ideal (-1 + x^2, y^2, x*y + y*x)

But then I get

sage: H.one() == H(x*x)
False

So is this currently possibly using a different method?

Thanks