Added the definition of “basic triples” of octonions, and the statement that they form a torsor over $Aut(\mathbb{O}) = G_2$.

]]>I have expanded slightly at *coalgebra – Properties – As filtered colimits of finite dimensional pieces*.

And I have added and cross-linked with corresponding remarks at *dg-coalgebra*, at *pro-object*, at *L-infinity algebra* and at *model structure for L-infinity algebras*.

added to homotopy groups of spheres the table

$k =$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | $\cdots$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\pi_k(\mathbb{S}) =$ | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{24}$ | $0$ | $0$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{240}$ | $(\mathbb{Z}_2)^2$ | $(\mathbb{Z}_2)^3$ | $\mathbb{Z}_6$ | $\mathbb{Z}_{504}$ | $0$ | $\mathbb{Z}_3$ | $(\mathbb{Z}_2)^2$ | $\mathbb{Z}_{480} \oplus \mathbb{Z}_2$ |

I discovered that there was an ancient stub entry *canonical commutation relation*. Have given it a bit more content now.

added references to *essentially algebraic theory*. Also equipped the text with a few more hyperlinks.

To support mentioning weak wreath product in a parallel discussion with Urs, I created a stub for weak bialgebra with redirect weak Hopf algebra.

]]>The Idea-section at *quasi-Hopf algebra* had been confused and wrong. I have removed it and written a new one.

added to *Cachy real number* a pointer to

- Egbert Rijke, Bas Spitters,
*Cauchy reals in the univalent foundations*(talk notes, May 2013) (pdf)

I looked at *real number* and thought I could maybe try to improve the way the Idea section flows. Now it reads as follows:

]]>A

real numberis something that may be approximated by rational numbers. Equipped with the operations of addition and multiplication induced from the rational numbers, real numbers form anumber field, denoted $\mathbb{R}$. The underlying set is thecompletionof the ordered field $\mathbb{Q}$ of rational numbers: the result of adjoining to $\mathbb{Q}$ suprema for every bounded subset with respect to the natural ordering of rational numbers.The set of real numbers also carries naturally the structure of a topological space and as such $\mathbb{R}$ is called the

real linealso known asthe continuum. Equipped with both the topology and the field structure, $\mathbb{R}$ is a topological field and as such is the uniform completion of $\mathbb{Q}$ equipped with the absolute value metric.Together with its cartesian products – the Cartesian spaces $\mathbb{R}^n$ for natural numbers $n \in \mathbb{N}$ – the real line $\mathbb{R}$ is a standard formalization of the idea of

continuous space. The more general concept of (smooth)manifoldis modeled on these Cartesian spaces. These, in turnm are standard models for the notion of space in particular in physics (seespacetime), or at least in classical physics. See atgeometry of physicsfor more on this.

the entry *[[group algebra]]* had been full of notation mismatch and also of typos. I have reworked it now.

made Ben Webster’s recent observation the Idea-section at Hall algebra (see the link given there)

]]>I finally gave this statement its own entry, in order to be able to conveniently point to it:

*embedding of smooth manifolds into formal duals of R-algebras*

I am trying to imrpove the complex of entries revolving around the *Hurwitz theorem*. I am not done yet at all, but since in the process I am touching a lot of entries, I thought I’d drop a note now for those anxiously following the RecentlyRevised notifications.

So I gave *Hurwitz theorem* its own entry, first of all, cross linking to the details (a proof,in fact), that may be found at *composition algebra*, but which previously could not be found from *normed division alegbra*. Now there are cross-links.

I also tried to add more references, but this needs work. It seems that Wikipedia says both that the source is

- Adolf Hurwitz,
*Über die Composition der quadratischen Formen von beliebig vielen Variabeln*, Nachr. Ges. Wiss. Göttingen (1898) 309–316

as well as that “was published posthumously in 1923”.

But I haven’t really spent much effort yet to check.

I also added cross-links with *Hopf invariant one*, but this is plain stubby for the moment.

I gave *Drinfel’d double* an Idea-section.

Also moved a paragraph on module categories from the References to a Properties-section.

]]>started some minimum at *Bost-Connes system*.

Hm, it seems that the statement is that that partition function of the BC-system

$\beta \mapsto Tr(\exp(- \beta H_{BostConnes}))$is the Riemann zeta function. But by the pertinent analogies the zeta functions are not supposed to *equal* partition functions, but to be related to them by the transformation

Hm.

]]>I gave the stub-entry *Hopf algebroid* a paragraph in the Idea-section that points out that already in commutative geometry there are two different kinds of Hopf algebroids associated with a groupoid (just as there are two versions of Hopf algebras associated with a group):

The commutative but non-co-commutative structure obtained by forming ordinary function algebras on objects and morphisms;

The non-commutative but co-commutative structure obtained by forming the groupoid convolution algebra.

For the moment I left the rest of the entry (which vaguely mentions commutative and non-commutative versions without putting them in relation) untouched, but I labelled the whole entry “under constructions”, since I think this issue needs to be discussed more for the entry not to be misleading.

I may find time to get back to this later…

]]>I started to greatly expand the entry [[module]]

The new toc now looks like this:

Idea

Basic idea

More general perspectives

Enriched presheaves

Stabilized overcategories

Details

Ordinary concept

In enriched category theory

Examples

- Modules over rings

Related concepts

- Vector bundles and sheaves of modules

I have split of *super 2-algebra* from super algebra. It’sa stub. Currently the only content is to provide the pointers into the video of Kapranov’s talk (minutes:seconds.)

I have expanded vertex operator algebra (more references, more items in the Properties-section) in partial support to a TP.SE answer that I posted here

]]>I gave this its own page, in order to have a convenient way to point to it: *Lazard’s criterion*.

I have expanded the Idea section at *state on a star-algebra* and added a bunch of references.

The entry used to be called “state on an operator algebra”, but I renamed it (keeping the redirect) because part of the whole point of the definition is that it makes sense without necessarily having represented the “abstract” star-algebra as a C*-algebra of linear operators.

]]>Stub Frobenius reciprocity.

]]>I had started to expand Eilenberg-Watts theorem a little. Stated it in more modern form as an equivalence of categories. Also started adding pointers to Hoyes homotopy-theoretical versions, but then I ran out of steam for the moment. I should come back to this later.

Is there more than Hovey’s article on the higher-algebra/homotopy-theoretic versions?

]]>I noticed only now that the entry *bimodule* is in bad shape and needs some attention. For the moment I have added here a mentioning of the 2-category of algebras, bimodules and intertwiners and a pointer to the Eilenberg-Watts theorem.

at *normed division algebra* it used to say that “A normed field is either $\mathbb{R}$ or $\mathbb{C}$. ” I have changed that to “a normed field over $\mathbb{R}$ is…” and changed *normed field* from being a redirect to “normed division algebra” to instead being an entry on its own.