Symmetric algebra

Main article: Symmetric algebra

Further information: Algebra, Commutative algebra

What is Symmetric Algebra?

In mathematics, symmetric algebra is a special kind of commutative algebra. It is built from a vector space and has an important property called the universal property.

How is it Built?

If we know a basis for the vector space, the symmetric algebra can be seen as a kind of polynomial ring. Each basis element acts like a variable in a polynomial.

Related Ideas

The symmetric algebra can also be made from the tensor algebra by focusing only on elements that commute with each other. These ideas can be used in more general settings as well.

Construction

The symmetric algebra can be built using the tensor algebra. Imagine it as a special kind of algebra made from the tensor algebra. In this algebra, everything commutes. This means when you multiply two elements, the order does not matter.

You can also build the symmetric algebra using polynomial rings. If you have a vector space with basic building blocks, you can make a polynomial ring where these blocks work like variables. This polynomial ring is the same as the symmetric algebra, so the two ideas are very similar.

Grading

The symmetric algebra is a graded algebra. This means it can be split into parts. Each part is related to how many elements from the original space are multiplied together.

For example, the part called the symmetric square of V, written as S²(V), deals with products of two elements from V. This structure helps organize the algebra into levels based on these products.

Relationship with symmetric tensors

The symmetric algebra of a vector space is closely related to symmetric tensors. A symmetric tensor of degree n is a special kind of tensor that stays the same when you rearrange its parts. These symmetric tensors form a structure called a graded vector space.

For fields of characteristic zero, symmetric tensors and the symmetric algebra can sometimes match up while keeping their vector space properties the same. However, they act differently when multiplied together. This matching does not work for fields with positive characteristics or for some rings.

Categorical properties

The symmetric algebra of a module ( V ) over a commutative ring ( K ) has a special rule called the universal property. This rule says that for any linear map from ( V ) to a commutative algebra ( A ), there is one and only one way to extend this map to the symmetric algebra of ( V ).

Because of this universal property, the symmetric algebra works well in category theory. It connects modules and commutative algebras, letting us change module maps into algebra maps in a unique way. This makes the symmetric algebra a left adjoint to the process that ignores the algebra structure and just looks at the module underneath.

Symmetric algebra of an affine space

We can build something called the symmetric algebra for an affine space. This is different from the usual symmetric algebra. It is called a filtered algebra. This means we can still find the overall degree of a polynomial, but we can’t always split it into simpler parts with the same degree.

For example, with a linear polynomial in a vector space, we can find its constant part by checking its value at zero. But in an affine space, there isn’t one special point to use, so this method doesn’t work directly. Choosing a point would change the affine space into a vector space.

Analogy with exterior algebra

The parts of the symmetric algebra called ( S^k ) are similar to another idea in math called exterior powers. As ( k ) gets bigger, the size of ( S^k ) also gets bigger. The size of ( S^k(V) ) is a special number called a binomial coefficient, written as ( \binom{n+k-1}{k} ). Here, ( n ) is the size of the original space ( V ). This number tells us how many ways we can combine ( n ) variables in products of degree ( k ). Both the symmetric algebra and the exterior algebra help us understand how symmetry groups work with tensor products.

As a Hopf algebra

The symmetric algebra can also be seen as a special kind of math structure called a Hopf algebra. For more information about related concepts, see the article on the Tensor algebra.

As a universal enveloping algebra

The symmetric algebra ( S(V) ) is related to a universal enveloping algebra. It is the universal enveloping algebra of an abelian Lie algebra. This means a special operation in Lie algebras, called the Lie bracket, is always zero. In simple words, the symmetric algebra is a special kind of algebra that fits well with certain mathematical structures.