Symmetric algebra

In mathematics, symmetric algebras are used to define polynomials over arbitrary vector spaces. They play an important role as in the theory of Lie groups and in the theory of characteristic classes.

Formal definition

It is a vector space over a field. Next was

The -fold tensor product of the conventions and. The direct sum

Is the tensor algebra of.

The two-sided homogeneous ideal is generated by differences of Elementartensoren with " wrong order ":

The symmetric algebra is then defined as the quotient space

-Th power of the symmetric is defined as the image of In, it is designated. One has a decomposition

The product in the symmetric algebra is traditionally written as.

Analogously, one can define the symmetric algebra of modules over commutative rings.

Examples

For is isomorphic to the polynomial ring.

Generally, one can interpret the elements of as polynomials in the elements of a pre - selected based on.

Especially for the vector space of matrices over, one can interpret the elements of as polynomials in the entries of the matrices:

Polynomials on vector spaces

Are polynomials of degree over a vector space - by definition - the elements, with the dual space designated. These polynomials are linear maps

Which are invariant under the action of the symmetric group. (Note that such a polynomial by its values ​​for all is already clearly defined. )

The product

Is defined by