# example of algebras and coalgebras which cannot be turned into Hopf algebras

Let $H={\mathrm{\u0161\x9d\x95\x84}}_{n}\u0101\x81\xa2(k)$ be a matrix algebra over a field $k$ with standard multiplication^{} and assume that $n>1$. Assume that $H$ can be turned into a Hopf algebra. In particular, there is $\mathrm{\u012a\mu}:H\u0101\x86\x92k$ such that $\mathrm{\u012a\mu}$ is a morphism of algebras^{}. It can be shown that matrix algebra is simple, i.e. if $I\u0101\x8a\x86H$ is a two-sided ideal^{}, then $I=0$ or $I=H$. Thus we have that $\mathrm{ker}\u0101\x81\xa2\mathrm{\u012a\mu}=0$ (because $\mathrm{\u012a\mu}\u0101\x81\xa2(1)=1$). Contradiction^{}, because ${\mathrm{dim}}_{k}\u0101\x81\xa2H>1={\mathrm{dim}}_{k}\u0101\x81\xa2k$.

Now consider $H={\mathrm{\u0161\x9d\x95\x84}}^{c}\u0101\x81\xa2(n,k)$ a vector space^{} of all $n\u0106\x97n$ matrices over $k$. We introduce coalgebra structure^{} on $H$. Let ${E}_{i\u0101\x81\xa2j}$ be a matrix in $H$ with $1$ in $(i,j)$ place and $0$ everywhere else. Of course $\{{E}_{i\u0101\x81\xa2j}\}$ forms a basis of $H$ and it is sufficient to define comultiplication and counit on it. Define

$$\mathrm{\u012a\x94}\u0101\x81\xa2({E}_{i\u0101\x81\xa2j})=\underset{p=1}{\overset{n}{\u0101\x88\x91}}{E}_{i\u0101\x81\xa2p}\u0101\x8a\x97{E}_{p\u0101\x81\xa2j};$$ |

$$\mathrm{\u012a\mu}\u0101\x81\xa2({E}_{i\u0101\x81\xa2j})={\mathrm{\u012a\u201c}}_{i\u0101\x81\xa2j},$$ |

where ${\mathrm{\u012a\u201c}}_{i\u0101\x81\xa2j}$ denotes Kronecker delta. It can be easily checked, that $({\mathrm{\u0161\x9d\x95\x84}}^{c}\u0101\x81\xa2(n,k),\mathrm{\u012a\x94},\mathrm{\u012a\mu})$ is a coalgebra known as the matrix coalgebra. Also, is well known that the dual algebra ${\mathrm{\u0161\x9d\x95\x84}}^{c}\u0101\x81\xa2{(n,k)}^{*}$ is isomorphic^{} to the standard matrix algebra.

Now assume that matrix coalgebra $H={\mathrm{\u0161\x9d\x95\x84}}^{c}\u0101\x81\xa2(n,k)$ (where $n>1$) can be turned into a Hopf algebra. Since $H$ is finite dimensional, then we can take dual Hopf algebra ${H}^{*}$. But the underlaying algebra structure of ${H}^{*}$ is isomorphic to a matrix algebra (as we remarked earlier), which weāve already shown to be impossible. Thus matrix coalgebra cannot be turned into a Hopf algebra.

Title | example of algebras and coalgebras which cannot be turned into Hopf algebras |
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Canonical name | ExampleOfAlgebrasAndCoalgebrasWhichCannotBeTurnedIntoHopfAlgebras |

Date of creation | 2013-03-22 18:58:45 |

Last modified on | 2013-03-22 18:58:45 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Example |

Classification | msc 16W30 |