Hey, if $\sigma$ is the permutation $(1~2~3)\in S_3$, can I write $\sigma(a,b,c)=(b,c,a)$? If not, what's the proper notation for applying a permutation to an ordered tuple?
And let's say $A\subseteq S_n$ is a permutation group, and $X,Y$ are $n$-tuples. Is there an accepted notation to signify that $\exists \sigma\in A:\sigma X=Y$? I would write $X\sim_A Y$ if I had to make something up, but I wonder if there's something standard.
@MikeMiller I made this mistake in teaching linear algebra before. (not including the 3rd axiom because I didn't think about the case where the subset is empty).
Hochschild homology happens to be really geometric when dealing with smooth algebras, and in this case there is a famous theorem by Konstant Rosenberg and Hochschild that says that Hochschild homology coincides with the homology of differential forms on $A$.
And you have Künneth-like theorems for HH, so you can interpret cohomology too.
@PedroTamaroff There are some special kind of groups called amenable groups. Inspired from their group algebras, amenability of Banach algebras was developed which extensively uses Hochschild cohomology.
@Semiclassical: Consider the algebra $L^\infty(G)$ of functions that are bounded except at a measure 0 set mod functions of measue zero, on $G$. A mean is a continuous nonnegative functional on this of norm 1. The norm should be inconsequential, since you can just scale.
A group is amenable if it has a left-invariant mean.
i created a math question can anyone tell me if it makes sense or not
A triangle $T$ is given in the plane with sides $3,4,5$. A second triangle $R(T)$ is created by reflecting $T$ about a line. Find $\min\{[R(T) \cup T]\}$ where $[\cdot]$ denotes area.
@GeorgeSmyridis Um, if $x\ll d$ just means that $x$ is much smaller than $d$… we could still have $x\ne0$, no? So we'd just have $(\frac d2)^2+x^2\approx(\frac d2)^2$.
Let $H = \{\phi\in A(S) : x_0\phi = x_0 ~\text{or}~ x_1 \phi = x_1\}$. Is $H$ a subgroup of $A(S)$? ($A(S)$ is the set of all one-to-one mappings from $S$ onto itself).
