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Mathematical coincidence
In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ai and bj of two families, usually
Exceptional_isomorphism
Construction in group theory
The isomorphism L2(9) ≅ A6 allows one to see the exotic outer automorphism of A6 in terms of field automorphism and matrix operations. The isomorphism L4(2)
Projective_linear_group
Low-rank isomorphisms in mathematics
In mathematics, the exceptional isomorphisms of classical groups (also: accidental isomorphism or sporadic isogenies) are unexpected coincidences between
Exceptional isomorphisms of classical groups
Exceptional_isomorphisms_of_classical_groups
of mathematics often relate to the exceptional objects in others. A related phenomenon is exceptional isomorphism, when two series are in general different
Exceptional_object
Group of even permutations of a finite set
(like (123)) with elements of shape 32 (like (123)(456)). There are some exceptional isomorphisms between some of the small alternating groups and small groups of
Alternating_group
Double cover Lie group of the special orthogonal group
dimensions, there are isomorphisms among the classical Lie groups called exceptional isomorphisms. For instance, there are isomorphisms between low-dimensional
Spin_group
Topics referred to by the same term
circumstances Exceptional objects, in mathematics Exceptional isomorphisms State of exception, a concept of extension of sovereign power Exceptionality (disambiguation)
Exception
Mathematical theory
{\displaystyle n=2,} the isomorphism s o 5 ≅ s p 2 {\displaystyle {\mathfrak {so}}_{5}\cong {\mathfrak {sp}}_{2}} corresponds to the isomorphisms of diagrams B 2
Compact_Lie_algebra
There are two isomorphism classes if n ≠ 6 and one isomorphism class if n = 6. The alternating group of degree n has one isomorphism class of Schur cover
Covering groups of the alternating and symmetric groups
Covering_groups_of_the_alternating_and_symmetric_groups
assumptions) that any birational regular map that is not an isomorphism has an exceptional divisor. A particularly important example is the blowup σ :
Exceptional_divisor
Nonabelian group of order 120
⋅ A 5 ≅ 2 I ; {\displaystyle 2\cdot A_{5}\cong 2I;} this isomorphism covers the isomorphism of the icosahedral group with the alternating group A 5 ≅
Binary_icosahedral_group
Sporadic simple group
with orbits of size 8 and 16. The linear group GL(4,2) has an exceptional isomorphism to the alternating group A8. The pointwise stabilizer O of an octad
Mathieu_group_M24
Pictorial representation of symmetry
D_{5}} These isomorphisms correspond to isomorphism of simple and semisimple Lie algebras, which also correspond to certain isomorphisms of Lie group
Dynkin_diagram
Mathematical classification
extends the families to include redundant terms, one obtains the exceptional isomorphisms D 3 ≅ A 3 , E 4 ≅ A 4 , E 5 ≅ D 5 , {\displaystyle D_{3}\cong A_{3}
ADE_classification
Invariant in projective geometry
only has three points, so this representation is an isomorphism, and is the exceptional isomorphism S 3 ≈ P G L ( 2 , Z 2 ) {\displaystyle \mathrm {S}
Cross-ratio
Finite simple group; sometimes classed as sporadic
of Lie type that is not a group of Lie type in any series from exceptional isomorphisms. It is sometimes considered a 27th sporadic group. The Ree groups
Tits_group
Type of non-Euclidean geometry
given explicit matrix realizations. In small dimensions, there are exceptional isomorphisms of Lie groups that yield additional ways to consider symmetries
Hyperbolic_geometry
Subspace defined by a polynomial of degree 2 over a field
Grassmannian for the symplectic group Sp(4,k). (This is related to the exceptional isomorphism of linear algebraic groups between SO(5,k) and Sp ( 4 , k ) /
Quadric_(algebraic_geometry)
Counterintuitive mathematical object
5). A similar but distinct phenomenon is that of exceptional objects (and exceptional isomorphisms), which occurs when there are a "small" number of
Pathological_(mathematics)
American mathematician
205–219, doi:10.5802/jtnb.354, MR 1925998 Baran, Burcu (2014), "An exceptional isomorphism between modular curves of level 13", Journal of Number Theory,
Jennifer_Balakrishnan
Non-commutative group with 6 elements
elements, the projective line has only 3 points, and this is thus the exceptional isomorphism S 3 ≈ P G L ( 2 , 2 ) . {\displaystyle S_{3}\approx \mathrm {PGL}
Dihedral_group_of_order_6
Twisted spin group
{\displaystyle \operatorname {Spin} ^{\mathbb {C} }(n)} . Using the exceptional isomorphism Spin ( 2 ) ≅ U ( 1 ) {\displaystyle \operatorname {Spin} (2)\cong
Spinc_group
Algebraic structure used in analysis
definition of Lie algebras, two braiding isomorphisms are needed. If A is a vector space, the interchange isomorphism τ : A ⊗ A → A ⊗ A {\displaystyle \tau
Lie_algebra
State space for internal degrees of freedom of a subatomic particle
2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} due to an exceptional isomorphism. Examples include scalar fields, commonly denoted ϕ {\displaystyle
Multiplet
Coincidence in mathematics
just 0.0001%. Almost integer Anthropic principle Birthday problem Exceptional isomorphism Experimental mathematics Koide formula Narcissistic number Sophomore's
Mathematical_coincidence
3-regular graph with 30 vertices and 45 edges
( F 2 ) {\displaystyle Sp_{4}(\mathbb {F} _{2})} (there is an exceptional isomorphism between this group and the symmetric group S 6 {\displaystyle S_{6}}
Tutte–Coxeter_graph
Type of mathematical group
contained in the center Z(G) of G. Then Fischer (1971) proved that up to isomorphism G/Z(G) is one of the following groups and D is the image of the given
3-transposition_group
Direct sum of simple Lie algebras
enumeration is redundant, and one has exceptional isomorphisms between simple Lie algebras, which are reflected in isomorphisms of Dynkin diagrams; the En can
Semisimple_Lie_algebra
{\text{D}}^{\text{b}}(X)} , up to isomorphism. That helps to explain the name.) The triangulated subcategory generated by an exceptional object E is equivalent to
Semiorthogonal_decomposition
Group that is also a differentiable manifold with group operations that are smooth
Lie algebras; it is then reasonable to ask how isomorphism classes of Lie groups relate to isomorphism classes of Lie algebras. The first result in this
Lie_group
Twisted spin group
{\displaystyle \operatorname {Spin} ^{\mathbb {H} }(n)} . Using the exceptional isomorphism Spin ( 3 ) ≅ Sp ( 1 ) {\displaystyle \operatorname {Spin} (3)\cong
Spinh_group
Type of geometric transformation
{O}}(1)} for π. π is an isomorphism away from the exceptional divisor, but the exceptional divisor need not be in the exceptional locus of π. That is, π
Blowing_up
Graph representing edges of another graph
line graph. However, all such exceptional cases have at most four vertices. A strengthened version of the Whitney isomorphism theorem states that, for connected
Line_graph
Moduli space of the Seiberg–Witten equations
M\rightarrow \operatorname {BSpin} ^{\mathrm {c} }(4)} . Because of the exceptional isomorphism: Spin c ( 4 ) ≅ U ( 2 ) × U ( 1 ) U ( 2 ) = { U ± ∈ U
Seiberg–Witten_moduli_space
numbers. Over the real numbers, there are three such Jordan algebras up to isomorphism. One of them, which was first mentioned by Pascual Jordan, John von Neumann
Albert_algebra
Aspect of mathematical group theory
{\displaystyle \mathrm {S} _{n}\to \operatorname {Aut} (\mathrm {S} _{n})} is an isomorphism. n ≠ 1 , 2 , 6 {\displaystyle n\neq 1,2,6} : Out ( A n ) = S n / A
Automorphisms of the symmetric and alternating groups
Automorphisms_of_the_symmetric_and_alternating_groups
Isomorphism of an object to itself
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping
Automorphism
Geometric configuration of ten points and lines
Bernhild; Stroppel, Markus (2013), "Desargues, doily, dualities and exceptional isomorphisms" (PDF), Australasian Journal of Combinatorics, 57: 257 Wikimedia
Desargues_configuration
On the existence of zeros of homogeneous polynomials over the p-adic numbers
the exceptional set of primes p. If the degree d is 1, 2, or 3 the exceptional set is empty. Heath-Brown (2010) showed that if d = 5 the exceptional set
Ax–Kochen_theorem
the correspondence of isomorphism classes of octonion F-algebras with isomorphism classes of G2-torsors over F. These isomorphism classes form the non-abelian
Octonion_algebra
Concept in algebraic geometry
is up to isomorphism only one such surface, given by blowing up the projective plane in 2 distinct points. Degree 8: they have 2 isomorphism types. One
Del_Pezzo_surface
3D symmetry group
A5, the alternating group of even permutations of five objects. This isomorphism can be realized by I acting on various compounds, notably the compound
Icosahedral_symmetry
Constructs the minimal model of a given smooth algebraic surface
to one point P {\displaystyle P} , and moreover this morphism is an isomorphism outside C {\displaystyle C} (i.e., X ∖ C {\displaystyle X\setminus C}
Castelnuovo's contraction theorem
Castelnuovo's_contraction_theorem
Mathematics study in geometry
dualizing sheaf on Y {\displaystyle Y} . The isomorphism between these two functors gives an isomorphism of the set of underlying points of the derived
Derived noncommutative algebraic geometry
Derived_noncommutative_algebraic_geometry
Generalizations of codimension-1 subvarieties of algebraic varieties
{\mathcal {O}}_{X}(D)} defines a monoid isomorphism from the Weil divisor class group of X to the monoid of isomorphism classes of rank-one reflexive sheaves
Divisor_(algebraic_geometry)
Not-necessarily-associative commutative algebra satisfying (xy)(xx) = x(y(xx))
group is the exceptional Lie group F4. Since over the complex numbers this is the only simple exceptional Jordan algebra up to isomorphism, it is often
Jordan_algebra
Algebraic structure with addition, multiplication, and division
norm residue isomorphism theorem, proved around 2000 by Vladimir Voevodsky, relates this to Galois cohomology by means of an isomorphism K n M ( F ) /
Field_(mathematics)
Geometric space whose points represent algebro-geometric objects of some fixed kind
whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to
Moduli_space
Magnitude or dimension of a thing
set" for well-ordered sets: ordinal number (equal if there is an order-isomorphism) In statistics (hypothesis testing), the "size" of the test refers to
Size
Nonassociative algebra over the real numbers
whereas the octonions have a positive-definite signature (8,0). Up to isomorphism, the octonions and the split-octonions are the only two 8-dimensional
Split-octonion
Mathematical group
{\displaystyle \operatorname {Sp} (2)\cong \operatorname {Spin} (5)} . This isomorphism is exhibited by identifying Sp ( 2 ) {\displaystyle \operatorname {Sp}
Symplectic_group
Counterexample in algebraic geometry
map V × B → B are given by a suitable isomorphism from V×C to itself, where C = B×AB = B × Z/2Z. Such an isomorphism is given by the action of Z/2Z on V
Hironaka's_example
Algebra based on a vector space with a quadratic form
2 then there is a natural isomorphism between ⋀V and Cl(V, Q) considered as vector spaces (and there exists an isomorphism in characteristic two, which
Clifford_algebra
Group of unitary complex matrices with determinant of 1
be associated with a subring of 2-by-2 complex matrices by the ring isomorphism mapping a 1 ^ + b i ^ + c j ^ + d k ^ {\displaystyle a\,{\hat {1}}+b\
Special_unitary_group
Non-singular cubic surface in mathematics
coordinates (in P4). Up to isomorphism, the Clebsch surface is the only cubic surface with this automorphism group. The 27 exceptional lines are: The 15 images
Clebsch_surface
Nilpotent subalgebra of a Lie algebra
unitary SU(n) Symplectic Sp(n) Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8 Other Lie groups Circle Lorentz Poincaré Conformal group
Cartan_subalgebra
Lie group of complex numbers of unit modulus; topologically a circle
denotes group quotient and ≅ {\displaystyle \cong } the existence of an isomorphism between the groups. A 1 × 1 {\displaystyle 1\times 1} matrix has
Circle_group
Geometric arrangements of points, foundational to Lie theory
algebra, that is, that the root system determines the Lie algebra up to isomorphism. Finally, we must show that for each irreducible root system, there is
Root_system
(categorical) isomorphism; for example, "The tensor product in a weak monoidal category is associative and unital up to a natural isomorphism." vanish To
Glossary of mathematical jargon
Glossary_of_mathematical_jargon
Algebra associated to any vector space
{\displaystyle V} defines an isomorphism of V {\displaystyle V} with V ∗ {\displaystyle V^{*}} , and so also an isomorphism of ⋀ k ( V ) {\displaystyle
Exterior_algebra
Concept in algebraic geometry
singularities of X). The map from the strict transform of X′ to X is an isomorphism away from the singular points of X. W′ is constructed by repeatedly blowing
Resolution_of_singularities
Group that admits a formal description in terms of reflections
{\displaystyle A_{n}} . Further arguments show that this quotient map is an isomorphism. Coxeter groups are an abstraction of reflection groups. Coxeter groups
Coxeter_group
American mathematician
Congress of Mathematicians, on her work with Philipp Kunde proving that isomorphism of smooth ergodic flows forms a binary relation that cannot be described
Marlies_Gerber
finite group the Galois group of a Galois extension of the rationals? Isomorphism problem of Coxeter groups Are there an infinite number of Leinster groups
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Tool to track locally defined data attached to the open sets of a topological space
X} is an isomorphism (respectively monomorphism) if and only if for every open set U ⊆ X {\displaystyle U\subseteq X} , we have an isomorphism F ( U )
Sheaf_(mathematics)
Type of group in abstract algebra
× S2, S5 is one of the three non-solvable groups of order 120, up to isomorphism. S5 is the Galois group of the general quintic equation, and the fact
Symmetric_group
mathematicians mean when they use the term 'object'. Abstract and concrete Exceptional object Impossible object List of mathematical objects List of mathematical
Mathematical_object
Relation between Lie algebras depicted as a square
proposed by Rozenfeld (1956) that the remaining exceptional Lie groups E6, E7, and E8 are isomorphism groups of projective planes over certain algebras
Freudenthal_magic_square
Unitary representations of a Lie group
exactly one representation, up to an isomorphism, with the specified central and infinitesimal characters. In the exceptional cases there are two or three representations
Representation theory of SL2(R)
Representation_theory_of_SL2(R)
Tree graph with one central node and leaves of length 1
as an induced subgraph. They are also one of the exceptional cases of the Whitney graph isomorphism theorem: in general, graphs with isomorphic line graphs
Star_(graph_theory)
Nonabelian group in algebraic group theory
over the finite field F3 with unit determinant, with this isomorphism covering the isomorphism of the projective special linear group PSL(2,3) with the
Binary_tetrahedral_group
Sporadic simple group
46 conjugacy classes of maximal subgroups. Non-abelian simple groups of some 60 isomorphism types are found as subgroups or as quotients of subgroups. The largest
Monster_group
and for n = 6, it has order 4 (elementary abelian). Other names: Altn. Isomorphisms: A1 and A2 are trivial. A3 is cyclic of order 3. A4 is isomorphic to
List_of_finite_simple_groups
Theorem classifying finite simple groups
revised version of the proof. Theorem—Every finite simple group is, up to isomorphism, one of the following groups: a member of 18 specific infinite families
Classification of finite simple groups
Classification_of_finite_simple_groups
Shape with six sides
up the dihedral group D6. There are 16 subgroups. There are 8 up to isomorphism: itself (D6), 2 dihedral: (D3, D2), 4 cyclic: (Z6, Z3, Z2, Z1) and the
Hexagon
Theorem in algebraic geometry
can be stated as follows: The isomorphism ⋆ ¯ E {\displaystyle {\bar {\star }}_{E}} induces a complex linear isomorphism: H p , q ( X , E ) ≅ H n − p
Serre_duality
Branch of mathematics that studies abstract algebraic structures
If α {\displaystyle \alpha } is invertible, then it is said to be an isomorphism, in which case V {\displaystyle V} and W {\displaystyle W} (or, more
Representation_theory
Representation of the symmetry group of spacetime in special relativity
{\text{SO}}(3;1)^{+}} is surjective. One way to show this is to make use of the isomorphism SO ( 3 ; 1 ) + ≅ PGL ( 2 , C ) , {\displaystyle {\text{SO}}(3;1)^{+}\cong
Representation theory of the Lorentz group
Representation_theory_of_the_Lorentz_group
Type of smooth complex surface of kodaira dimension 0
derivative of the map is an isomorphism at some point). Define a marking of a complex analytic K3 surface X to be an isomorphism of lattices from H 2 ( X
K3_surface
Study of vector bundles, principal bundles, and fibre bundles
α } {\displaystyle \{U_{\alpha }\}} is such a cover, then under the isomorphism φ α : E U α → U α × K r {\displaystyle \varphi _{\alpha }:E_{U_{\alpha
Gauge_theory_(mathematics)
\operatorname {Pic} \mathbf {P} _{\mathbf {k} }^{n}=\mathbb {Z} } , and the isomorphism is given by the degree of divisors. The invertible sheaves, or line bundles
Algebraic geometry of projective spaces
Algebraic_geometry_of_projective_spaces
Non-obvious mathematical equivalence
mathematics, but "cryptomorphism" is only very distantly related to "isomorphism", "homomorphism", or "morphisms". The equivalence may in a cryptomorphism
Cryptomorphism
Relate the direct image and the pull-back of sheaves
{F}}} is not usually an isomorphism. Instead the extension by zero functor f ! {\displaystyle f_{!}} satisfies an isomorphism g ∗ f ! F → f ! ′ g ∗ F
Base_change_theorems
Hypercomplex number system
numbers. Octonions are related to exceptional structures[clarification needed] in mathematics, among them the exceptional Lie groups. Octonions have applications
Octonion
Concept in algebraic geometry
bundles of X {\displaystyle X} and D {\displaystyle D} . It is a natural isomorphism ω D = i ∗ ( ω X ⊗ O ( D ) ) . {\displaystyle \omega _{D}=i^{*}(\omega
Canonical_bundle
Online database of integer sequences
short to do any analysis with", for example, A079243, the number of isomorphism classes of associative non-commutative non-anti-associative anti-commutative
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Surgery operation in minimal model program
the flip of X i {\displaystyle X_{i}} is a birational map (in fact an isomorphism in codimension 1) f : X i → X i + {\displaystyle f\colon X_{i}\rightarrow
Flip_(algebraic_geometry)
Category in mathematics
Z) is determined up to isomorphism by the morphism X → Y {\displaystyle X\to Y} , although not always up to a unique isomorphism. Every triangle isomorphic
Triangulated_category
Concept in mathematics
between compact connected Lie groups and complex reductive groups, up to isomorphism. For a compact Lie group K with complexification G, the inclusion from
Reductive_group
Graph of chess rook moves
Square rook's graphs are connected-homogeneous, meaning that every isomorphism between two connected induced subgraphs can be extended to an automorphism
Rook's_graph
Vector bundle existing over a Grassmannian
left the bracket means homotopy class and on the right is the set of isomorphism classes of real vector bundles of rank n. The inverse map is given as
Tautological_bundle
Hexagonal lattice made of carbon atoms
double bonds within the carbon structure. Graphene is known for its exceptionally high tensile strength, electrical conductivity, transparency, and being
Graphene
Group without normal subgroups other than the trivial group and itself
group have the same length and the same factors, up to permutation and isomorphism. In a huge collaborative effort, the classification of finite simple
Simple_group
Generalization of affine connections
intuitive idea is that ω(X) provides a vertical component of X, using the isomorphism of the fibers of π with H to identify vertical vectors with elements
Cartan_connection
Topological manifold whose homology coincides with that of a sphere
way to choose the b′s, and the homology sphere does not depend (up to isomorphism) on the choice of b′s.) If r is at most 2 this is just the usual 3-sphere;
Homology_sphere
Block design in combinatorial mathematics
(mod 6). The abbreviation SQS(n) is often used for these systems. Up to isomorphism, SQS(8) and SQS(10) are unique, there are 4 SQS(14)s and 1,054,163 SQS(16)s
Steiner_system
are maps that factor through isomorphisms with subschemes. Specifically, an open immersion factors through an isomorphism with an open subscheme and a
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
{\displaystyle {\bar {g}}:X\times _{Z}Y\rightarrow X} , there exists a canonical isomorphism R f ¯ ∗ R g ¯ ! ≅ R f ! R g ∗ {\displaystyle R{\bar {f}}_{*}R{\bar {g}}^{
Image_functors_for_sheaves
Concept in mathematics
of symbols, below, for a more precise statement of the nature of the isomorphism. It is useful, perhaps, to split the process into two steps. In the first
Universal_enveloping_algebra
graphs are isomorphic if there is an isomorphism between them; see isomorphism. isomorphism A graph isomorphism is a one-to-one incidence preserving correspondence
Glossary_of_graph_theory
EXCEPTIONAL ISOMORPHISM
EXCEPTIONAL ISOMORPHISM
EXCEPTIONAL ISOMORPHISM
EXCEPTIONAL ISOMORPHISM
EXCEPTIONAL ISOMORPHISM
EXCEPTIONAL ISOMORPHISM
EXCEPTIONAL ISOMORPHISM
EXCEPTIONAL ISOMORPHISM
EXCEPTIONAL ISOMORPHISM