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I'm reading a book, where they claim that the free algebra $A=k\langle X_1,...,X_n\rangle$ is an algebra. I've never seen this notation and I've never heard of the free algebra, so I wonder how this is defined and why it is an algebra.

90intuition
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2 Answers2

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The free algebra $k\langle X_1, \ldots, X_n\rangle$ is, according to wikipedia, the non-commutative analogue of the polynomial ring in $n$ variables. That means that the basis elements in the free algebra are words $$ X_{i_1}X_{i_2}\cdots X_{i_m} $$ and a general element in the algebra is a sum of finitely many such words, each with a coefficient from $k$. So, for instance, one element of $\Bbb R\langle x, y, z\rangle$ looks like this $$ 3yzy - 2xzyx + 5 $$ Multiplication is carried out like you usually would multiply two polynomials, expanding parentheses ("distributing") and so on, just with the added detail that the variables don't commute, so you cannot rearrange $xyx$ into $x^2y$, and you'll have to keep track of which element is the left side and which is the right side of a multiplication. You can still rearrange terms, though, so $x + y$ is the same as $y + x$.

Example multiplication: Take the two elements $x -y$ and $x+y$ in $\Bbb R\langle x, y\rangle$. Their product is $$ (x-y)(x+y)=x(x+y) - y(x + y)\\ = xx + xy - yx - yy\\ = x^2 + xy - yx - y^2 $$ and since it's non-commutative, $xy$ and $yx$ are completely different terms, and thus this cannot be simplified any further.

Arthur
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  • @arthur Should I think of a basis like a basis of a vector space ? That is the only "basis" that I learned, but I thought you maybe mean something different with it. – 90intuition Jun 08 '14 at 19:30
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    @90intuition It's not wrong to think of it as the basis of a vector space. Algebras are modules with a multiplication in them, and modules are generalized vector spaces in the first place. Over a field there's even not much generalization going on. So if you think of this as a vector space where you can multiply two basis vectors to get a new basis vector, then that is a common way to view both the free algebra and the polynomial algebra over a ring. – Arthur Jun 08 '14 at 19:34
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It's by definition an algebra, as you define it to be the algebra generated by $k$ and $n$ algebraically-independent transcendental variables $X_1,\ldots,X_n$.

oxeimon
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  • I've never heard about "algebra generated by" kind of construction, could you expand on that, or point to some place where there is more information about such kind of constructions ? – 90intuition Jun 08 '14 at 19:11
  • And what is a transcedental variable ? – 90intuition Jun 08 '14 at 19:13
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    Transcendental variables are just the technical term for variables like $x$ and $y$ that you find in say the polynomial ring $k[x,y]$. The definition is that a variable $x$ is transcendental over $k$ if $x$ is not the root of any polynomial with coefficients in $k$. – oxeimon Jun 08 '14 at 19:20