What does triality mean?

Definitions for triality
tri·al·i·ty

This dictionary definitions page includes all the possible meanings, example usage and translations of the word triality.


Did you actually mean treelet or trilled?

Wiktionary

  1. trialitynoun

    A relationship between three vector spaces.

Wikipedia

  1. Triality

    In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional rotation group SO(8), arising because the group has an outer automorphism of order three. There is a geometrical version of triality, analogous to duality in projective geometry. Of all simple Lie groups, Spin(8) has the most symmetrical Dynkin diagram, D4. The diagram has four nodes with one node located at the center, and the other three attached symmetrically. The symmetry group of the diagram is the symmetric group S3 which acts by permuting the three legs. This gives rise to an S3 group of outer automorphisms of Spin(8). This automorphism group permutes the three 8-dimensional irreducible representations of Spin(8); these being the vector representation and two chiral spin representations. These automorphisms do not project to automorphisms of SO(8). The vector representation—the natural action of SO(8) (hence Spin(8)) on F8—consists over the real numbers of Euclidean 8-vectors and is generally known as the "defining module", while the chiral spin representations are also known as "half-spin representations", and all three of these are fundamental representations. No other connected Dynkin diagram has an automorphism group of order greater than 2; for other Dn (corresponding to other even Spin groups, Spin(2n)), there is still the automorphism corresponding to switching the two half-spin representations, but these are not isomorphic to the vector representation. Roughly speaking, symmetries of the Dynkin diagram lead to automorphisms of the Tits building associated with the group. For special linear groups, one obtains projective duality. For Spin(8), one finds a curious phenomenon involving 1-, 2-, and 4-dimensional subspaces of 8-dimensional space, historically known as "geometric triality". The exceptional 3-fold symmetry of the D4 diagram also gives rise to the Steinberg group 3D4.

Webster Dictionary

  1. Trialitynoun

    three united; state of being three

Wikidata

  1. Triality

    In mathematics, triality is a relationship between three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin, the double cover of 8-dimensional rotation group SO, arising because the group has an outer automorphism of order three. There is a geometrical version of triality, analogous to duality in projective geometry. Of all simple Lie groups, Spin has the most symmetrical Dynkin diagram, D4. The diagram has four nodes with one node located at the center, and the other three attached symmetrically. The symmetry group of the diagram is the symmetric group S3 which acts by permuting the three legs. This gives rise to an S3 group of outer automorphisms of Spin. This automorphism group permutes the three 8-dimensional irreducible representations of Spin; these being the vector representation and two chiral spin representations. These automorphisms do not project to automorphisms of SO. The vector representation – the natural action of SO on – is also known as the "defining module", while the chiral spin representations are also known as "half-spin representations", and all three of these are fundamental representations.

How to pronounce triality?

How to say triality in sign language?

Numerology

  1. Chaldean Numerology

    The numerical value of triality in Chaldean Numerology is: 8

  2. Pythagorean Numerology

    The numerical value of triality in Pythagorean Numerology is: 6


Translations for triality

From our Multilingual Translation Dictionary

Get even more translations for triality »

Translation

Find a translation for the triality definition in other languages:

Select another language:

  • - Select -
  • 简体中文 (Chinese - Simplified)
  • 繁體中文 (Chinese - Traditional)
  • Español (Spanish)
  • Esperanto (Esperanto)
  • 日本語 (Japanese)
  • Português (Portuguese)
  • Deutsch (German)
  • العربية (Arabic)
  • Français (French)
  • Русский (Russian)
  • ಕನ್ನಡ (Kannada)
  • 한국어 (Korean)
  • עברית (Hebrew)
  • Gaeilge (Irish)
  • Українська (Ukrainian)
  • اردو (Urdu)
  • Magyar (Hungarian)
  • मानक हिन्दी (Hindi)
  • Indonesia (Indonesian)
  • Italiano (Italian)
  • தமிழ் (Tamil)
  • Türkçe (Turkish)
  • తెలుగు (Telugu)
  • ภาษาไทย (Thai)
  • Tiếng Việt (Vietnamese)
  • Čeština (Czech)
  • Polski (Polish)
  • Bahasa Indonesia (Indonesian)
  • Românește (Romanian)
  • Nederlands (Dutch)
  • Ελληνικά (Greek)
  • Latinum (Latin)
  • Svenska (Swedish)
  • Dansk (Danish)
  • Suomi (Finnish)
  • فارسی (Persian)
  • ייִדיש (Yiddish)
  • հայերեն (Armenian)
  • Norsk (Norwegian)
  • English (English)

Word of the Day

Would you like us to send you a FREE new word definition delivered to your inbox daily?

Please enter your email address:


Citation

Use the citation below to add this definition to your bibliography:

Style:MLAChicagoAPA

"triality." Definitions.net. STANDS4 LLC, 2024. Web. 19 Mar. 2024. <https://www.definitions.net/definition/triality>.

Discuss these triality definitions with the community:

0 Comments

    Are we missing a good definition for triality? Don't keep it to yourself...

    Image or illustration of

    triality

    Credit »

    Free, no signup required:

    Add to Chrome

    Get instant definitions for any word that hits you anywhere on the web!

    Free, no signup required:

    Add to Firefox

    Get instant definitions for any word that hits you anywhere on the web!

    Browse Definitions.net

    Quiz

    Are you a words master?

    »
    a light informal meal
    A collation
    B profaneness
    C viverrine
    D volubility

    Nearby & related entries:

    Alternative searches for triality: