Algebras

It’s been a long time since my last post about mathematics, which is kind of a shame, because I wanted math to be one of the main subjects of this blog.
So, I decided to start and write a little bit more about things I loved in mathematics (and hopefully I will follow).

In this post, I’m assuming some basic Linear Algebra knowledge. This post is a little boring, but it defines an important algebraic structure that is used in some very beautiful subjects and theorems, so stay tuned.

Algebras and Subalgebras

Definition: An algebra A is a vector space over field \mathbb{F}, endowed with a binary bilinear operation (*) s.t. \forall a,b,c\in A, \lambda,\mu\in\mathbb{F}:

    \[ a*(\lambda b+ \mu c)=\lambda a*b+\mu a*c \]

    \[ (\lambda b+ \mu c)*a=\lambda b*a+\mu c*a \]

For example, The polynomials with n variables, \mathbb{F}[x_1,\dots,x_n], with the multiplication as the operation * is an associative and commutative algebra. This algebra is usually called the Polynomial algebra.

But please note, NOT all algebras must be commutative! For example, we can look at the vector space of the n\times n matrices (M_n(\mathbb{F})), with matrix multiplication at the operation *. M_n(\mathbb{F}) is an associative, non-commutative algebra.

We can also define a subalgebra:

Definition: A subspace A'\subset A is called a subalgebra if \forall a,b\in A', a*b\in A'.

For example, the diagonal matrices are a subalgebra of the matrices algebra.

Homomorphism

And just like any other algebraic structure, we can define an homomorphism between two algebras as a linear map between the algebras the preserve the operation, that is:

Defenition: Let A,B be algebras, a linear map \phi:A\to B is called an homomorphism if:

    \[ \forall a,b\in A\;\phi(a+_Ab)=\phi(a)+_B\phi(b) \]

    \[ \forall a\in A, \lambda\in\mathbb{F}\;\phi(\lambda a)=\lambda\phi(a) \]

    \[ \forall a,b\in A\;\phi(a*_Ab)=\phi(a)*_B\phi(b) \]

Ideals

Definition: A subspace I of an algebra A is a left (resp. right, resp. two-sided) ideal if:

    \[ \forall a\in A, b\in I\;a*b\in I \]

(resp. b*a\in I, resp. a*b,b*a\in I).

It is clear, by the definition, that any ideal is a subalgebra.

Using the ideals, we can look at the quotient space A/I (a space where any vector in I means the zero vector, meaning that two vector that differ by only a vector in I are identical). The quotient space has the canonical algebra structure given by:

    \[ (a+I)$(b+I)=a*b+I \]

which called the quotient algebra A/I.
It’s easy to see that the canonical map A\to A/I that is given by a\mapsto a+I is an algebra homomorphism.

Moreover, if \phi:A\to B is an algebra homomorphism, the kernel \ker\phi is a two-sided ideal of A and the image \mathrm{Im}\phi is a subalgebra of B.

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