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edit approved Mar 16, 2018 at 8:18
Alex Meiburg
edit approved Mar 16, 2018 at 8:18
Alex Meiburg
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Perhaps the following example can also be of some use. Consider the looped suspension of infinite-dimensional complex projective space, $\Omega\Sigma CP^\infty$$\Omega\Sigma \,\mathbb{CP}^\infty$. By Bott-Samelson its homology is the associative ring over the integers in countably many indeterminates, also known as the algebra of non-symmetric functions, NSymm. It is a Hopf algebra (but not primitively generated). It's dual is the cohomology of this space and is the so-called algebra of quasi-symmetric functions, QSymm; also a Hopf algebra of course. As an algebra over the integers QSymm is commutative free polynomial. See arXiv:math/0410366 for a proof and an explicit set of free polynomial generators.

The corresponding commutative situation is provided by of the homology and cohomology Hopf algebras of the classifying space BU. These are dual and isomorphic, and both equal to the free commutative algebra in countably many indeterminates, one in each double dimension, also known as Symm, the algebra of symmetric functions over the integers.

Perhaps the following example can also be of some use. Consider the looped suspension of infinite-dimensional complex projective space, $\Omega\Sigma CP^\infty$. By Bott-Samelson its homology is the associative ring over the integers in countably many indeterminates, also known as the algebra of non-symmetric functions, NSymm. It is a Hopf algebra (but not primitively generated). It's dual is the cohomology of this space and is the so-called algebra of quasi-symmetric functions, QSymm; also a Hopf algebra of course. As an algebra over the integers QSymm is commutative free polynomial. See arXiv:math/0410366 for a proof and an explicit set of free polynomial generators.

The corresponding commutative situation is provided by of the homology and cohomology Hopf algebras of the classifying space BU. These are dual and isomorphic, and both equal to the free commutative algebra in countably many indeterminates, one in each double dimension, also known as Symm, the algebra of symmetric functions over the integers.

Perhaps the following example can also be of some use. Consider the looped suspension of infinite-dimensional complex projective space, $\Omega\Sigma \,\mathbb{CP}^\infty$. By Bott-Samelson its homology is the associative ring over the integers in countably many indeterminates, also known as the algebra of non-symmetric functions, NSymm. It is a Hopf algebra (but not primitively generated). It's dual is the cohomology of this space and is the so-called algebra of quasi-symmetric functions, QSymm; also a Hopf algebra of course. As an algebra over the integers QSymm is commutative free polynomial. See arXiv:math/0410366 for a proof and an explicit set of free polynomial generators.

The corresponding commutative situation is provided by of the homology and cohomology Hopf algebras of the classifying space BU. These are dual and isomorphic, and both equal to the free commutative algebra in countably many indeterminates, one in each double dimension, also known as Symm, the algebra of symmetric functions over the integers.

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Michiel Hazewinkel
Michiel Hazewinkel
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Perhaps the following example can also be of some use. Consider the looped suspension of infinite-dimensional complex projective space, $\Omega\Sigma CP^\infty$. By Bott-Samelson its homology is the associative ring over the integers in countably many indeterminates, also known as the algebra of non-symmetric functions, NSymm. It is a Hopf algebra (but not primitively generated). It's dual is the cohomology of this space and is the so-called algebra of quasi-symmetric functions, QSymm; also a Hopf algebra of course. As an algebra over the integers QSymm is commutative free polynomial. See arXiv:math/0410366 for a proof and an explicit set of free polynomial generators.

The corresponding commutative situation is provided by of the homology and cohomology Hopf algebras of the classifying space BU. These are dual and isomorphic, and both equal to the free commutative algebra in countably many indeterminates, one in each double dimension, also known as Symm, the algebra of symmetric functions over the integers.