# symmetric algebra

Let $M$ be a module over a commutative ring $R$. Form the tensor algebra $T(M)$ over $R$. Let $I$ be the ideal of $T(M)$ generated by elements of the form

$$u\otimes v-v\otimes u$$ |

where $u,v\in M$.
Then the quotient algebra^{} defined by

$$S(M):=T(M)/I$$ |

is called the *symmetric algebra* over the ring $R$.

Remark. Let $R$ be a field, and $M$ a finite dimensional vector space^{} over $R$. Suppose $\{{e}_{1},{e}_{2},\mathrm{\dots},{e}_{n}\}$ is a basis of $M$ over $R$. Then $T(M)$ is nothing more than a free algebra^{} on the basis elements ${e}_{i}$. Alternatively, the basis elements ${e}_{i}$ can be viewed as non-commuting indeterminates in the non-commutative polynomial ring^{} $R\u27e8{e}_{1},{e}_{2},\mathrm{\dots},{e}_{n}\u27e9$. This then implies that $S(M)$ is isomorphic^{} to the “commutative^{}” polynomial ring $R[{e}_{1},{e}_{2},\mathrm{\dots},{e}_{n}]$, where ${e}_{i}{e}_{j}={e}_{j}{e}_{i}$.

Title | symmetric algebra |
---|---|

Canonical name | SymmetricAlgebra |

Date of creation | 2013-03-22 15:46:23 |

Last modified on | 2013-03-22 15:46:23 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 4 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 15A78 |