Symmetric algebra

In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is a special kind of commutative algebra that includes V and has a very important property. This property, called the universal property, means that for any linear map from V to another commutative algebra A, there is a unique way to extend this map to the whole symmetric algebra.

If we know a basis B for the vector space V, the symmetric algebra can be thought of as the polynomial ring K[B], where each basis element acts like a variable in a polynomial. This helps us see the symmetric algebra as a polynomial ring that works for any vector space, not just one with a chosen basis.

The symmetric algebra can also be constructed from the tensor algebra T(V) by dividing out by certain elements that measure how far two elements are from commuting. This construction shows how symmetric algebra captures the idea of commuting elements in a natural algebraic way. These ideas can be extended to more general settings, such as when V is a module over a commutative ring.

Construction

The symmetric algebra can be built using the tensor algebra. Think of it as a special kind of algebra made by taking the tensor algebra and simplifying it so that everything commutes. This means that when you multiply two elements, it doesn't matter in which order you multiply them.

Another way to build the symmetric algebra is by using polynomial rings. If you have a vector space with a basic set of building blocks, you can create a polynomial ring where these building blocks act like variables. This polynomial ring ends up being the same as the symmetric algebra, making the two ideas essentially identical.

Grading

The symmetric algebra is a graded algebra. This means it can be broken down into parts, where 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 something called symmetric tensors. A symmetric tensor of degree n is a special kind of tensor that stays the same no matter how you rearrange its parts. These symmetric tensors form a structure called a graded vector space.

Over fields of characteristic zero, symmetric tensors and the symmetric algebra can sometimes be matched up in a way that keeps their vector space properties the same. However, they behave differently when you multiply them together. This matching doesn't work for fields with positive characteristics or for certain rings.

Categorical properties

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

Because of this universal property, the symmetric algebra behaves nicely within category theory. It acts like a bridge between modules and commutative algebras, allowing module maps to be turned into algebra maps in a unique way. This makes the symmetric algebra a left adjoint to the process that forgets the algebra structure and just looks at the underlying module.

Symmetric algebra of an affine space

We can build something called the symmetric algebra for an affine space too. This is a bit different from the usual symmetric algebra. Instead of being organized by degrees like layers, it is called a filtered algebra. This means we can still figure out the overall degree of a polynomial, but we can’t always break it down into simpler, same-degree pieces.

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 ) behave somewhat like another mathematical idea called exterior powers. As ( k ) grows larger, the size of ( S^k ) also grows. Specifically, the size of ( S^k(V) ) is given by a special number called a binomial coefficient, written as ( \binom{n+k-1}{k} ), where ( n ) is the size of the original space ( V ). This number counts how many ways we can combine ( n ) variables in products of degree ( k ). Both the symmetric algebra and the exterior algebra are important parts when studying how symmetry groups act on tensor products.

As a Hopf algebra

The symmetric algebra can also be seen as a special kind of mathematical 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 closely related to something called a universal enveloping algebra. Specifically, it is the universal enveloping algebra of an abelian Lie algebra. This means the Lie bracket (a special operation in Lie algebras) is always zero. In simpler terms, the symmetric algebra works like a special kind of algebra that fits perfectly with certain mathematical structures.