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List Of Irreducible Tits Indices
In the mathematical theory of linear algebraic groups, a Tits index (or index) is an object used to classify semisimple algebraic groups defined over a base field ''k'', not assumed to be algebraically closed. The possible irreducible indices were classified by Jacques Tits, and this classification is reproduced below. (Because every index is a direct sum of irreducible indices, classifying ''all'' indices amounts to classifying irreducible indices.) Organization of the list An index can be represented as a Dynkin diagram with certain vertices drawn close to each other (the orbit of the vertices under the *-action of the Galois group of ''k'') and with certain sets of vertices circled (the orbits of the non-distinguished vertices under the *-action). This representation captures the full information of the index except when the underlying Dynkin diagram is D4, in which case one must distinguish between an action by the cyclic group ''C''3 or the permutation group ''S''3. Alterna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebraic Group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n where M^T is the transpose of M. Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(''n'',R).) The simple Lie groups were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include Maurer, Chevalley, and . In the 1950s, Armand Borel constructed much of the theory of algebraic groups as it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tits Index Bn
TIT, Tit, Tits, or tit may refer to: Birds * Tit (bird) or Paridae, a large family of passerine birds * Bearded tit, a small reed-bed passerine bird * Long-tailed tits or Aegithalidae, a family of passerine birds with long tails * Tit-babbler or ''Macronus'', a genus in the family Timaliidae * Tit berrypecker, a species of bird in the family Paramythiidae * Tit hylia, a species of bird in the family Cettiidae * Tomtit, a small passerine bird of the family Petroicidae * Wrentit, a small bird, the only species in the genus ''Chamaea'' Places * Tin Tsz stop, a Light Rail stop in Hong Kong * Tit, Adrar, a town in Adrar Province, central Algeria * Tit, Tamanrasset, a village in Tamanrasset Province, southern Algeria * Tit-e Olya, a village in West Azerbaijan Province, Iran * Tehran International Tower, a residential tower located in Tehran, Iran People * Jacques Tits (1930–2021), French/Belgian mathematician * Tit (name), Romanian male given name * Tit Linda Sou (born 1989) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tits Index 1E6,2;16
TIT, Tit, Tits, or tit may refer to: Birds * Tit (bird) or Paridae, a large family of passerine birds * Bearded tit, a small reed-bed passerine bird * Long-tailed tits or Aegithalidae, a family of passerine birds with long tails * Tit-babbler or ''Macronus'', a genus in the family Timaliidae * Tit berrypecker, a species of bird in the family Paramythiidae * Tit hylia, a species of bird in the family Cettiidae * Tomtit, a small passerine bird of the family Petroicidae * Wrentit, a small bird, the only species in the genus ''Chamaea'' Places * Tin Tsz stop, a Light Rail stop in Hong Kong * Tit, Adrar, a town in Adrar Province, central Algeria * Tit, Tamanrasset, a village in Tamanrasset Province, southern Algeria * Tit-e Olya, a village in West Azerbaijan Province, Iran * Tehran International Tower, a residential tower located in Tehran, Iran People * Jacques Tits (1930–2021), French/Belgian mathematician * Tit (name), Romanian male given name * Tit Linda Sou (born 1989) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tits Index 1E6,2;28
TIT, Tit, Tits, or tit may refer to: Birds * Tit (bird) or Paridae, a large family of passerine birds * Bearded tit, a small reed-bed passerine bird * Long-tailed tits or Aegithalidae, a family of passerine birds with long tails * Tit-babbler or ''Macronus'', a genus in the family Timaliidae * Tit berrypecker, a species of bird in the family Paramythiidae * Tit hylia, a species of bird in the family Cettiidae * Tomtit, a small passerine bird of the family Petroicidae * Wrentit, a small bird, the only species in the genus ''Chamaea'' Places * Tin Tsz stop, a Light Rail stop in Hong Kong * Tit, Adrar, a town in Adrar Province, central Algeria * Tit, Tamanrasset, a village in Tamanrasset Province, southern Algeria * Tit-e Olya, a village in West Azerbaijan Province, Iran * Tehran International Tower, a residential tower located in Tehran, Iran People * Jacques Tits (1930–2021), French/Belgian mathematician * Tit (name), Romanian male given name * Tit Linda Sou (born 1989) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tits Index 1E6,0;78
TIT, Tit, Tits, or tit may refer to: Birds * Tit (bird) or Paridae, a large family of passerine birds * Bearded tit, a small reed-bed passerine bird * Long-tailed tits or Aegithalidae, a family of passerine birds with long tails * Tit-babbler or ''Macronus'', a genus in the family Timaliidae * Tit berrypecker, a species of bird in the family Paramythiidae * Tit hylia, a species of bird in the family Cettiidae * Tomtit, a small passerine bird of the family Petroicidae * Wrentit, a small bird, the only species in the genus ''Chamaea'' Places * Tin Tsz stop, a Light Rail stop in Hong Kong * Tit, Adrar, a town in Adrar Province, central Algeria * Tit, Tamanrasset, a village in Tamanrasset Province, southern Algeria * Tit-e Olya, a village in West Azerbaijan Province, Iran * Tehran International Tower, a residential tower located in Tehran, Iran People * Jacques Tits (1930–2021), French/Belgian mathematician * Tit (name), Romanian male given name * Tit Linda Sou (born 1989) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tits Index 3D4,2;2
TIT, Tit, Tits, or tit may refer to: Birds * Tit (bird) or Paridae, a large family of passerine birds * Bearded tit, a small reed-bed passerine bird * Long-tailed tits or Aegithalidae, a family of passerine birds with long tails * Tit-babbler or ''Macronus'', a genus in the family Timaliidae * Tit berrypecker, a species of bird in the family Paramythiidae * Tit hylia, a species of bird in the family Cettiidae * Tomtit, a small passerine bird of the family Petroicidae * Wrentit, a small bird, the only species in the genus ''Chamaea'' Places * Tin Tsz stop, a Light Rail stop in Hong Kong * Tit, Adrar, a town in Adrar Province, central Algeria * Tit, Tamanrasset, a village in Tamanrasset Province, southern Algeria * Tit-e Olya, a village in West Azerbaijan Province, Iran * Tehran International Tower, a residential tower located in Tehran, Iran People * Jacques Tits (1930–2021), French/Belgian mathematician * Tit (name), Romanian male given name * Tit Linda Sou (born 1989) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tits Index 3D4,1;9
TIT, Tit, Tits, or tit may refer to: Birds * Tit (bird) or Paridae, a large family of passerine birds * Bearded tit, a small reed-bed passerine bird * Long-tailed tits or Aegithalidae, a family of passerine birds with long tails * Tit-babbler or ''Macronus'', a genus in the family Timaliidae * Tit berrypecker, a species of bird in the family Paramythiidae * Tit hylia, a species of bird in the family Cettiidae * Tomtit, a small passerine bird of the family Petroicidae * Wrentit, a small bird, the only species in the genus ''Chamaea'' Places * Tin Tsz stop, a Light Rail stop in Hong Kong * Tit, Adrar, a town in Adrar Province, central Algeria * Tit, Tamanrasset, a village in Tamanrasset Province, southern Algeria * Tit-e Olya, a village in West Azerbaijan Province, Iran * Tehran International Tower, a residential tower located in Tehran, Iran People * Jacques Tits (1930–2021), French/Belgian mathematician * Tit (name), Romanian male given name * Tit Linda Sou (born 1989) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tits Index 3D4,0;28
TIT, Tit, Tits, or tit may refer to: Birds * Tit (bird) or Paridae, a large family of passerine birds * Bearded tit, a small reed-bed passerine bird * Long-tailed tits or Aegithalidae, a family of passerine birds with long tails * Tit-babbler or ''Macronus'', a genus in the family Timaliidae * Tit berrypecker, a species of bird in the family Paramythiidae * Tit hylia, a species of bird in the family Cettiidae * Tomtit, a small passerine bird of the family Petroicidae * Wrentit, a small bird, the only species in the genus ''Chamaea'' Places * Tin Tsz stop, a Light Rail stop in Hong Kong * Tit, Adrar, a town in Adrar Province, central Algeria * Tit, Tamanrasset, a village in Tamanrasset Province, southern Algeria * Tit-e Olya, a village in West Azerbaijan Province, Iran * Tehran International Tower, a residential tower located in Tehran, Iran People * Jacques Tits (1930–2021), French/Belgian mathematician * Tit (name), Romanian male given name * Tit Linda Sou (born 1989) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tits Index 2Dn
TIT, Tit, Tits, or tit may refer to: Birds * Tit (bird) or Paridae, a large family of passerine birds * Bearded tit, a small reed-bed passerine bird * Long-tailed tits or Aegithalidae, a family of passerine birds with long tails * Tit-babbler or ''Macronus'', a genus in the family Timaliidae * Tit berrypecker, a species of bird in the family Paramythiidae * Tit hylia, a species of bird in the family Cettiidae * Tomtit, a small passerine bird of the family Petroicidae * Wrentit, a small bird, the only species in the genus ''Chamaea'' Places * Tin Tsz stop, a Light Rail stop in Hong Kong * Tit, Adrar, a town in Adrar Province, central Algeria * Tit, Tamanrasset, a village in Tamanrasset Province, southern Algeria * Tit-e Olya, a village in West Azerbaijan Province, Iran * Tehran International Tower, a residential tower located in Tehran, Iran People * Jacques Tits (1930–2021), French/Belgian mathematician * Tit (name), Romanian male given name * Tit Linda Sou (born 1989) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tits Index 1Dn
TIT, Tit, Tits, or tit may refer to: Birds * Tit (bird) or Paridae, a large family of passerine birds * Bearded tit, a small reed-bed passerine bird * Long-tailed tits or Aegithalidae, a family of passerine birds with long tails * Tit-babbler or ''Macronus'', a genus in the family Timaliidae * Tit berrypecker, a species of bird in the family Paramythiidae * Tit hylia, a species of bird in the family Cettiidae * Tomtit, a small passerine bird of the family Petroicidae * Wrentit, a small bird, the only species in the genus ''Chamaea'' Places * Tin Tsz stop, a Light Rail stop in Hong Kong * Tit, Adrar, a town in Adrar Province, central Algeria * Tit, Tamanrasset, a village in Tamanrasset Province, southern Algeria * Tit-e Olya, a village in West Azerbaijan Province, Iran * Tehran International Tower, a residential tower located in Tehran, Iran People * Jacques Tits (1930–2021), French/Belgian mathematician * Tit (name), Romanian male given name * Tit Linda Sou (born 1989) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic group and is also denoted by \mathrm(n). Many authors prefer slightly different notations, usually differing by factors of . The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group is denoted , and is the compact real form of . Note that when we refer to ''the'' (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension . The name "symplectic group" is due to Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex". The metaplect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed-point Subring
In algebra, the fixed-point subring R^f of an automorphism ''f'' of a ring ''R'' is the subring of the fixed points of ''f'', that is, :R^f = \. More generally, if ''G'' is a group acting on ''R'', then the subring of ''R'' :R^G = \ is called the fixed subring or, more traditionally, the ring of invariants under . If ''S'' is a set of automorphisms of ''R'', the elements of ''R'' that are fixed by the elements of ''S'' form the ring of invariants under the group generated by ''S''. In particular, the fixed-point subring of an automorphism ''f'' is the ring of invariants of the cyclic group generated by ''f''. In Galois theory, when ''R'' is a field and ''G'' is a group of field automorphisms, the fixed ring is a subfield called the fixed field of the automorphism group; see Fundamental theorem of Galois theory. Along with a module of covariants, the ring of invariants is a central object of study in invariant theory. Geometrically, the rings of invariants are the coordinate rin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |