Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorsvennik
    • CommentTimeNov 13th 2020
    • (edited Nov 13th 2020)

    In this article it says that the groupoid cardinality of the 2D hypercomplex number systems is 32\frac 3 2 because there are three 2D hypercomplex number systems (up to isomorphism) each equipped with one non-trivial algebra automorphism. But surely, the dual numbers have infinitely many automorphisms satisfying 11,εkε1 \mapsto 1, \epsilon \mapsto k\epsilon (where kk is an arbitrary non-zero real number), so the value of the groupoid cardinality is actually 12+12+1=1\frac 1 2 + \frac 1 2 + \frac 1 \infty = 1.

    Who is right?

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 14th 2020

    Are you sure that’s an automorphism, or just an isomorphism to a different algebra?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 14th 2020

    I haven’t thought about what the entry claims (or even looked at it until a minute ago), but, in response to David:

    The operation ϕ k\phi_k : 111 \mapsto 1 and εkε\epsilon \mapsto k \epsilon for k{0}k \in \mathbb{R} \setminus \{0\} is certainly an automorphism of the ring of dual numbers [ε]/(ε 2)\mathbb{R}[\epsilon]/(\epsilon^2):

    Explicitly, check that

    ϕ k(1+aε)ϕ k(1+bε) =(1+kaε)(1+kbε) =(1+k(a+b)ε) =ϕ k(1+(a+b)ε) =ϕ k((1+aε)(1+bε)) \begin{aligned} \phi_k(1 + a \epsilon) \cdot \phi_k(1 + b \epsilon) & =\; (1 + k a \epsilon) (1 + k b \epsilon) \\ & =\; (1 + k (a + b) \epsilon) \\ & =\; \phi_k\big( 1 + (a + b) \epsilon \big) \\ & =\; \phi_k\big( (1 + a \epsilon) \cdot (1 + b \epsilon) \big) \end{aligned}

    More conceptually, notice that (Hadamard’s lemma)

    [ε]/(ε 2)C ()/(ε 2) \mathbb{R}[\epsilon]/(\epsilon^2) \;\simeq\; C^\infty(\mathbb{R})/(\epsilon^2)

    so that every diffeomorphism of \mathbb{R} which fixes the origin induces an automorphism of [ε]/(ε 2)\mathbb{R}[\epsilon]/(\epsilon^2) (given by the diffeomorphism’s first Taylor coefficient, which is the kk from above).

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 14th 2020

    These cardinality do behave oddly. Sometimes it makes better ’sense’ to take the cardinality of the open interval as 1-1, so that the nonzero reals have cardinality 2-2. See, e.g., here.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 14th 2020
    • (edited Nov 14th 2020)

    [ removed ]

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 14th 2020

    Ah, ok. I was thinking it was something like the function [ε]/(ε 2)[kε]/(k 2ε 2)\mathbb{R}[\varepsilon]/(\varepsilon^2) \to \mathbb{R}[k\varepsilon]/(k^2\varepsilon^2), but now I see what was intended!

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 15th 2020

    I have edited the entry, in an attempt to fix the issue. See the edit log in the entry’s thread here.

    • CommentRowNumber8.
    • CommentAuthorsvennik
    • CommentTimeNov 15th 2020
    • (edited Nov 15th 2020)

    Regarding the claim that the perplex numbers have precisely two automorphisms, this is true. First, it’s easy to see that ϕ 1(a+be)=a+be\phi_1(a + be) = a + be and ϕ 2(a+be)=abe\phi_2(a + be) = a - be are two distinct automorphisms. It remains to prove that all automorphisms over the perplex numbers are of this form. Let ϕ\phi be an automorphism over the perplex numbers. Observe that ϕ(e) 2=ϕ(e 2)=ϕ(1)=1\phi(e)^2 = \phi(e^2) = \phi(1) = 1. It follows that ϕ(e){1,1,e,e}\phi(e) \in \{1,-1,e,-e\}. But if ϕ(e){+1,1}\phi(e) \in \{+1, -1\} then ϕ\phi is not injective. So therefore ϕ(e){e,e}\phi(e) \in \{e, -e\}.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 15th 2020

    Thanks, that’s easy enough. I have edited that into the entry (logs here).

    So thanks for bringing up this issue and helping to fix it. I’d like to bow out now. Please feel invited to edit the entry further, as need be.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)