ASKSAGE: Sage Q&A Forum - Latest question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sat, 16 Feb 2019 14:54:16 -0600Quotient of non-commutative ringhttps://ask.sagemath.org/question/45480/quotient-of-non-commutative-ring/ I'm trying to make a non-commutative ring, where X^2 = -1 ; XY = -YX and Y^2 = 0, something like this ;
R[X,Y] / (X^2 + 1 , XY + YX , Y^2)
To get the non-commutative part, I did :
A.<x,y> = FreeAlgebra(QQ,2)
P.<x,y> = A.g_algebra(relations={y*x:-x*y})
What is the next command to have the quotient by (X^2+1 , Y^2) ? Or maybe an other way to build the entire ring, if there exists a simpler method :) thanks !
franslagSat, 16 Feb 2019 14:54:16 -0600https://ask.sagemath.org/question/45480/derivative of non-commuting symbolic producthttps://ask.sagemath.org/question/37003/derivative-of-non-commuting-symbolic-product/Consider the product rule $\frac d{dt}[A(t)B(t)]=\dot{A}(t)B(t)+A(t)\dot{B}(t)$ with $A$ and $B$ not commuting, e.g, matrix valued. I'd like to replicate this in sage, however, I don't see how I can specify that $A$ and $B$ do not commute. So far I have
<pre>
var('t')
A=function('A')(t)
B=function('B')(t)
diff(A*B,t)
</pre>
which yields
`B(t)*diff(A(t), t) + A(t)*diff(B(t), t)`. But here sage has assumed that the operators and their derivatives do commute. Not what I want.
I did look into `sage.symbolic.function_factory.function` and the like, but could not find anything about products. Am I overlooking something or is this currently not possible?BjörnMon, 20 Mar 2017 09:57:40 -0500https://ask.sagemath.org/question/37003/how to sum up the function over all permutations of variables in associative non-commutative algebrahttps://ask.sagemath.org/question/34714/how-to-sum-up-the-function-over-all-permutations-of-variables-in-associative-non-commutative-algebra/ hello, i need to sum up $\lambda_{\sigma}(a_{\sigma(1)}a_{\sigma(2)}a_{\sigma(3)}-a_{\sigma(3)}a_{\sigma(4)}a_{\sigma(5)})$ over all $\sigma\in S_5$ where $a_i$ are elements of associative non-commutative algebra.
the result should be $E_1a_1a_2a_3a_4a_5+\cdots+E_{120}a_5a_4a_3a_2a_1$
and i need to express $E_i$ in terms of $\lambda_{\sigma}$.
actualy my tartget is to find non-zero solution of $E_i=0$ for all $i$.RadmirSun, 04 Sep 2016 08:19:15 -0500https://ask.sagemath.org/question/34714/Is there a way to solve differential equation with non-commutative variables?https://ask.sagemath.org/question/10687/is-there-a-way-to-solve-differential-equation-with-non-commutative-variables/I want to solve differential equation involving QM operators. For eg.
$i\frac{d}{dt}\rho=H\rho-\rho H$
The solution for this equation is $\rho(t)=exp(-iHt)\rho(0)exp(iHt)$, if $H$ is independent of $t$. Is there a way to solve the given differential equation using sage when the variables do not commute. ShashankSun, 03 Nov 2013 10:25:11 -0600https://ask.sagemath.org/question/10687/doubly indexed variables in a non-commutative ringhttps://ask.sagemath.org/question/9823/doubly-indexed-variables-in-a-non-commutative-ring/I'm trying to do some basic arithmetic in a non-commutative polynomial ring with variables indexed by two indices i and j (say 1 \le i < j \le 6). Does anyone know how to ask for a doubly indexed variable like this?
Thanks,
AnnaannabWed, 20 Feb 2013 05:44:19 -0600https://ask.sagemath.org/question/9823/non-commutative algebra with formal functionshttps://ask.sagemath.org/question/9595/non-commutative-algebra-with-formal-functions/I'd like to look at the following: the set of formal functions with 2 variables f(x,y) and the real numbers a,b,c..., including an addition and a non-commutative multiplication, such that things like
expand((a+f(x,y))*(b+c*f(u,v))) = a*b + a*c*f(u,v) + b*f(x,y) + c*f(x,y)*f(u,v)
are possible (and vice versa), and with the multiplication of the functions
f(x,y)*f(u,v) != f(u,v)*f(x,y)
being non-commutative, however with the multiplication of the functions by the real scalars
a*f(x,y) == f(x,y)*a
still commutative. Can I construct something like that with sage?
MarkSat, 01 Dec 2012 11:37:19 -0600https://ask.sagemath.org/question/9595/Skew commuting variableshttps://ask.sagemath.org/question/9512/skew-commuting-variables/I want to work in the ring `QQ<x0, x1, x2> / (xi*xj = -xj*xi for i \neq j)`. (In particular, xi^2 \neq 0; this is not the exterior algebra.)
I seems like FreeAlgebraQuotient is the thing to use, but I am not sure how. In the documentation for FreeAlgebraQuotient, the algebras are 4-dimensional as modules over QQ. However in my application, the algebra is infinite-dimensional as a module, so I can't write down the matrices for the action of the generators.
Is there another way to obtain this ring?
Thanks.OliverPWed, 07 Nov 2012 11:22:17 -0600https://ask.sagemath.org/question/9512/Non-commutative ring with inverseshttps://ask.sagemath.org/question/8808/non-commutative-ring-with-inverses/Hello Sage,
I would like to make a ring over $\mathbb{Q}$ with $n$ variables which are non-commutative and also include their inverses.
So I want to generate the free algebra over $\mathbb{Q}$ with generates $x_1, x_2, ..., x_n$ and $x_1^{-1}, x_2^{-1}, ..., x_n^{-1}$. How can I do this?
Best regards,
NoudNoudWed, 21 Mar 2012 02:26:28 -0500https://ask.sagemath.org/question/8808/Free algebra over infinite alphabethttps://ask.sagemath.org/question/8599/free-algebra-over-infinite-alphabet/Is it possible to work with a free algebra over an infinite alphabet ? Or to define an Infinite ring of **non-commutative** polynomials ?
I need one of these structures to work with the quasi shuffle product.Matthieu DeneufchâtelThu, 29 Dec 2011 00:39:25 -0600https://ask.sagemath.org/question/8599/Documentation for Plural G-algebra support?https://ask.sagemath.org/question/8569/documentation-for-plural-g-algebra-support/Googling seems to show that Sage supports calculations in certain non-commutative rings (see e.g. http://trac.sagemath.org/sage_trac/ticket/4539).
However I'm having trouble finding enough documentation to get started with this. Can anyone point me to some, or failing this say how to build a g-algebra, add and multiply elements, and test elements for equality? Thank you!Steven Glenn JacksonFri, 16 Dec 2011 08:24:56 -0600https://ask.sagemath.org/question/8569/Factorization of non-commutative Laurent polynomialshttps://ask.sagemath.org/question/8417/factorization-of-non-commutative-laurent-polynomials/Hi, can Sage factorize non-commutative Laurent polynomials in several variables?
By those polynomials I mean elements in the group algebra Z[F(n)], where Z is the integers and F(n) is the free group on n letters.
(The case with Z/2- instead of Z-coefficients would also be interesting.)
Thank you!bmTue, 25 Oct 2011 19:37:53 -0500https://ask.sagemath.org/question/8417/Noncommuting variableshttps://ask.sagemath.org/question/7784/noncommuting-variables/I am extremely new to Sage, and even newer to this site, so I apologize if anything is not up to standards.
I am dealing with a multivariable polynomial ring over $\mathbb{Z}$ with noncommuting variables. Is there a way to implement this with Sage? The closest thing I have found is FreeAlgebra, where the variables are noncommutative, but I have not found any way to impose relations that I want.
As stated before, I am extremely new to all of this so don't assume that I know anything, and don't hesitate to give any and all suggestions.Eric A BunchMon, 06 Dec 2010 14:53:32 -0600https://ask.sagemath.org/question/7784/