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Mathematical finite group theory
group theory, the classical involution theorem of Aschbacher (1977a, 1977b, 1980) classifies simple groups with a classical involution and satisfying some
Classical_involution_theorem
Theorem classifying finite simple groups
of involution in "standard form" then it is a group of Lie type of odd characteristic. This was solved by Aschbacher's classical involution theorem. Quasi-standard
Classification of finite simple groups
Classification_of_finite_simple_groups
Function that is its own inverse
particular, every involution on an odd number of elements has at least one fixed point. This can be used to prove Fermat's two squares theorem. The graph of
Involution_(mathematics)
Noether–Castelnuovo theorem in this context. The Geiser involution and Bertini involution are two of the classical non-linear involutions of the plane Cremona
Cremona_group
Non-associative algebras with positive-definite quadratic form
Hurwitz's theorem has been applied in algebraic topology to problems on vector fields on spheres and the homotopy groups of the classical groups and
Hurwitz's theorem (composition algebras)
Hurwitz's_theorem_(composition_algebras)
Topics referred to by the same term
subgroup" can also mean an analogue of the Weyl group used in the classical involution theorem The infinite Thompson groups F, T and V studied by the logician
Thompson_group
Type of group in mathematics
nondegenerate Hermitian forms relative to an involution. Over C {\displaystyle \mathbb {C} } , the connected simple classical Lie groups are the families of types
Classical_group
Semigroup in abstract algebra
In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism
Semigroup_with_involution
Propositional logic theorem
to be a law of thought in classical logic, but it is disallowed by intuitionistic logic. The principle was stated as a theorem of propositional logic by
Double_negation
Representation theory
In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its
Plancherel theorem for spherical functions
Plancherel_theorem_for_spherical_functions
Identifies the commutant of a specific von Neumann algebra
In mathematics, a commutation theorem for traces explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the
Commutation theorem for traces
Commutation_theorem_for_traces
Method for producing composition algebras
construction takes any algebra with involution to another algebra with involution of twice the dimension. Hurwitz's theorem states that the reals, complex
Cayley–Dickson_construction
Mathematical transform that expresses a function of time as a function of frequency
Handbook of Fourier Theorems, Cambridge University Press, Bibcode:1987hft..book.....C Chandrasekharan, Komaravolu (1989), Classical Fourier Transforms
Fourier_transform
A branching law for subgroups fixed by an involution and a noncompact analogue of the Borel-Weil theorem, Progr. Math., vol. 220, Birkhäuser, pp. 291–353
Restricted_representation
Counts tuples of non-intersecting lattice paths
{\displaystyle f(P)} ). Construction of the involution: The idea behind the definition of the involution f {\displaystyle f} is to take choose two intersecting
Lindström–Gessel–Viennot lemma
Lindström–Gessel–Viennot_lemma
General concept and operation in mathematics
called primal). Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this
Duality_(mathematics)
Time reversal symmetry in physics
space. For a real (not complex) classical (unquantized) scalar field ϕ {\displaystyle \phi } , the time reversal involution can simply be written as T ϕ
T-symmetry
Technique for proving sets have equal size
giving a proof of a classical result on the number of certain integer partitions. Bijective proofs of the pentagonal number theorem. Bijective proofs of
Bijective_proof
Mathematical group
Wonenburger, Anna (1966). "Transformations which are products of two involutions". Journal of Mathematics and Mechanics. 16 (4): 327–338. Symplectic Group
Symplectic_group
Mathematical group
sporadic group. They have involution centralizers of the form Z/2Z × PSL(2, q) for q = 3n, and by investigating groups with an involution centralizer of the
Group_of_Lie_type
German mathematician (1832–1903)
differential geometry, as well as number theory, algebras with involution and classical mechanics. Rudolf Lipschitz was born on 14 May 1832 in Königsberg
Rudolf_Lipschitz
(pseudo-)Riemannian manifold whose geodesics are reversible
subgroup H that is (a connected component of) the invariant group of an involution of G. This definition includes more than the Riemannian definition, and
Symmetric_space
Type of algebras, possibly non associative
N(xy)=N(x)N(y)} for all x and y in A. A composition algebra includes an involution called a conjugation: x ↦ x ∗ . {\displaystyle x\mapsto x^{*}.} The quadratic
Composition_algebra
Element mapped to itself by a mathematical function
fixed-point theorem is a result saying that at least one fixed point exists, under some general condition. For example, the Banach fixed-point theorem (1922)
Fixed_point_(mathematics)
Universal construction of a complex Lie group from a real Lie group
}} Then Sp(n,C) is the fixed point subgroup of the involution θ(g) = A (gt)−1 A−1 of SL(2n,C). It leaves the subgroups N±, TC and B
Complexification_(Lie_group)
Manifold with inversion symmetry
\displaystyle {g(Z)=(AZ+B)(CZ+D)^{-1}.}} The polydisk theorem takes the following concrete form in the classical cases: Type Ipq (p ≤ q): for every p × q matrix
Hermitian_symmetric_space
Type of physical or mathematical property
self-symmetrical or have symmetrical images under the involution π. In physics, the laws of motion of classical mechanics exhibit time reversibility, as long
Time_reversibility
geometry. An inversion of the Möbius plane with respect to any circle is an involution which fixes the points on the circle and exchanges the points in the interior
Möbius_plane
Algebra of 4D spacetime
spacetime. All Clifford or geometric algebras have three main involutions: grade involution, reversion, and Clifford conjugation. If g ∈ G 3 {\displaystyle
Algebra_of_physical_space
American mathematician (born 1937)
Cambridge University Press, pp. 1–48 Seiler, Werner M. (26 October 2009), Involution: The Formal Theory of Differential Equations and its Applications in Computer
David_Mumford
Logical operation
important difference between classical and intuitionistic negation. Algebraically, classical negation is called an involution of period two. However, in
Negation
True when either but not both inputs are true
The function is linear. Involution: Exclusive or with one specified input, as a function of the other input, is an involution or self-inverse function;
Exclusive_or
Algebraic structure used in theoretical physics
canonical involutive automorphism on any superalgebra called the grade involution. It is given on homogeneous elements by x ^ = ( − 1 ) | x | x {\displaystyle
Superalgebra
Concept in mathematics
Theorem 6.4. Farb & Margalit 2012, Theorem 6.15 and Theorem 6.12. Farb & Margalit 2012, Theorem 6.11. Ivanov 1992, Theorem 4. Ivanov 1992, Theorem 1
Mapping class group of a surface
Mapping_class_group_of_a_surface
Operation in Hamiltonian mechanics
must be in mutual involution, where n {\displaystyle n} is the number of degrees of freedom. Furthermore, according to Poisson's Theorem, if two quantities
Poisson_bracket
Construction in algebra
Hopf algebra is said to be involutive (and the underlying algebra with involution is a *-algebra). If H is finite-dimensional semisimple over a field of
Hopf_algebra
Normed vector space that is complete
C*-algebra is a complex Banach algebra A {\displaystyle A} with an antilinear involution a ↦ a ∗ {\displaystyle a\mapsto a^{*}} such that ‖ a ∗ a ‖ = ‖ a ‖ 2
Banach_space
278). involution 1. A transformation whose square is the identity. Cremona transformations that are involutions include Bertini involutions, Geiser
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
Transformations induced by a mathematical group
Similarly, an action of Z / 2Z on X is equivalent to the data of an involution of X. The symmetric group Sn and its subgroups act on the set {1, ...
Group_action
Direct sum of simple Lie algebras
Killing form is not all negative). Suppose, moreover, it has a Cartan involution θ {\displaystyle \theta } and let g = k ⊕ p {\displaystyle {\mathfrak
Semisimple_Lie_algebra
Formulation of classical mechanics using momenta
quantities Gi which are in involution (i.e., {Gi, Gj} = 0), then the Hamiltonian is Liouville integrable. The Liouville–Arnold theorem says that, locally, any
Hamiltonian_mechanics
Integral yoga Interpellation Intrinsic and extrinsic properties Intuition Involution Irrationality Is–ought problem Ius indigenatus Judgement Jus sanguinis
List of philosophical concepts
List_of_philosophical_concepts
support on G. These functions form a * algebra under convolution with involution F ∗ ( g ) = F ( g − 1 ) ¯ , {\displaystyle F^{*}(g)={\overline {F(g^{-1})}}
Zonal_spherical_function
Term in mathematics
Satake diagrams are a generalization of Dynkin diagrams that classify involutions of root systems that are relevant in several contexts. They were introduced
Satake_diagram
Russian mathematician (1937–2010)
Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to several
Vladimir_Arnold
Geometric concept of a 2D space with "points at infinity" adjoined
such embeddability is a consequence of a property known as Desargues' theorem, not shared by all projective planes. A projective plane is a rank 2 incidence
Projective_plane
Integrable classical system
physics, the Garnier integrable system, also known as the classical Gaudin model is a classical mechanical system discovered by René Garnier in 1917, and
Garnier_integrable_system
Topological space
1 } {\displaystyle x\in \{0,1\}} , every element of the group is an involution, i.e., ( x i ) + ( x i ) = ( 0 ) i ∈ Z {\displaystyle (x_{i})+(x_{i})=(0)_{i\in
Cantor_space
System of resource-aware logic
proposition A in CLL has a dual A⊥, defined as follows: Observe that (-)⊥ is an involution, i.e., A⊥⊥ = A for all propositions. A⊥ is also called the linear negation
Linear_logic
Integral transform and linear operator
{\displaystyle v\in L^{q}(\mathbb {R} )} . The Hilbert transform is an anti-involution, meaning that H ( H ( u ) ) = − u {\displaystyle \operatorname {H}
Hilbert_transform
Group of unitary matrices
{\displaystyle K} a ↦ a ¯ {\displaystyle a\mapsto {\bar {a}}} which is an involution and fixes exactly k {\displaystyle k} ( a = a ¯ {\displaystyle a={\bar
Unitary_group
A) and ¬ (A → ¬A) in propositional logic; they are theorems in connexive logic but not in classical logic. See also Boethius' theses. arity The number
Glossary_of_logic
non-classical logics with Hilbert-style calculi, algebraic semantics, and metamathematical properties known from other logics (completeness theorems, deduction
T-norm_fuzzy_logics
\mathbb {P} ^{3}} . The quotient of a K3 surface under a fixpointfree involution. Horrocks–Mumford surfaces, surfaces of degree 10 in projective 4-space
List of complex and algebraic surfaces
List_of_complex_and_algebraic_surfaces
Isometry group of Euclidean space
Chasles' theorem asserts that any element of E+(3) is a screw displacement. See also 3D isometries that leave the origin fixed, space group, involution. For
Euclidean_group
Branch of functional analysis
Hilbert space, the Hermitian adjoint map on operators gives a natural involution, which provides an additional algebraic structure that can be imposed
Operator_algebra
Classification in abstract algebra
projection operators. Since ω is odd, these algebras are exchanged by the involution α induced by v ↦ −v on the generating space: α ( C l n ± ( C ) ) = C l
Classification of Clifford algebras
Classification_of_Clifford_algebras
Four-dimensional number system
by q∗, qt, q ~ {\displaystyle {\tilde {q}}} , or q. Conjugation is an involution, meaning that it is its own inverse, so conjugating an element twice returns
Quaternion
Physical quantity conserved throughout a motion
integrable system. Such a collection of constants of motion are said to be in involution with each other. For a closed system (Lagrangian not explicitly dependent
Constant_of_motion
Algebraic manipulation of "true" and "false"
hence in both algebras it satisfies the double negation law (also called involution law) Double negation ¬ ( ¬ x ) = x {\displaystyle {\begin{aligned}&{\text{Double
Boolean_algebra
Problem in finite group theory
that map to the identity under the natural map from the free monoid with involution on A {\displaystyle A} to the group G {\displaystyle G} . If B {\displaystyle
Word_problem_for_groups
fixed by N. But then, by the first result, ξ must be fixed by G. The classical theorems of Gustav Hedlund from the early 1930s assert the ergodicity of the
Ergodic_flow
Generalized manifold
Orbifolds that arise in this way are called developable or good. A classical theorem of Henri Poincaré constructs Fuchsian groups as hyperbolic reflection
Orbifold
Mathematical objects that generalise the notion of Hilbert spaces
{\displaystyle A} be a C*-algebra (not assumed to be commutative or unital), its involution denoted by ∗ {\displaystyle {}^{*}} . An inner-product A {\displaystyle
Hilbert_C*-module
G must be simple and K of maximal rank. From the theorem of Borel and de Siebenthal, the involution σ is inner and K is the centralizer of a torus S.
Borel–de_Siebenthal_theory
System for reasoning about vagueness
stems from Łukasziewicz fuzzy logic. A generalization of the classical Gödel completeness theorem is provable in EVŁ. Similar to the way predicate logic is
Fuzzy_logic
Concept in topology
{\displaystyle {\mathfrak {g}}} be a real semisimple Lie algebra with Cartan involution σ. Thus the fixed point subgroup of σ is the maximal compact subgroup
Maximal_compact_subgroup
Vector space equipped with a bilinear product
underlying Banach space, which turns them into Banach algebras. If an involution is given as well, we obtain B*-algebras and C*-algebras. These are studied
Algebra_over_a_field
Concept in projective geometry
these are: Desargues' theorem ⇔ Converse of Desargues' theorem Pascal's theorem ⇔ Brianchon's theorem Menelaus' theorem ⇔ Ceva's theorem Not only statements
Duality_(projective_geometry)
Set of the elements not in a given subset
follows from the equivalence of a conditional with its contrapositive). Involution or double complement law: ( A c ) c = A . {\displaystyle \left(A^{c}\right)^{c}=A
Complement_(set_theory)
Generalization of quaternions to other fields
Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998). The book of involutions. Colloquium Publications. Vol. 44. With a preface by J. Tits. Providence
Quaternion_algebra
Mathematical concept
if the composition f ∘ f is equal to idX. Such a function is called an involution. If f is invertible, then the graph of the function y = f − 1 ( x ) {\displaystyle
Inverse_function
Algebraic structure used in theoretical physics
condition that the Poincaré–Birkhoff–Witt theorem holds (and, in general, they are necessary conditions for the theorem to hold). Just as for Lie algebras,
Lie_superalgebra
pair of type II. For type I, one starts with a division algebra D with involution τ, a hermitian form on U, and a skew-hermitian form on V (both of them
Reductive_dual_pair
Flat-sided three-dimensional shape
polar reciprocation, it is possible that a self-duality might not be an involution, meaning that repeating the self-duality twice might not return each vertex
Polyhedron
Algebra based on a vector space with a quadratic form
of Clifford algebras. The automorphism α is called the main involution or grade involution. Elements that are pure in this Z2-grading are simply said to
Clifford_algebra
Pictorial representation of symmetry
{\displaystyle D_{4}\to B_{3}} in 3 different ways, if quotienting by an involution) E 6 → F 4 {\displaystyle E_{6}\to F_{4}} Similar foldings exist for affine
Dynkin_diagram
Mathematical transformation
double derivatives are all positive. The Legendre transformation is an involution, i.e., f ∗ ∗ = f {\displaystyle f^{**}=f~} . Proof. By using the above
Legendre_transformation
Topological invariant in knot theory
K\to \mathbb {Q} } be any linear function which is invariant under the involution t ⟼ t − 1 {\displaystyle t\longmapsto t^{-1}} , then composing it with
Signature_of_a_knot
Set of lines described by homogeneous polynomial equations
∧ 2 R n {\displaystyle V,W\subset \wedge ^{2}\mathbb {R} ^{n}} are in involution, or in Klein polarity, if V , W {\displaystyle V,W} are orthogonal complements
Line_complex
Algebra describing 2D conformal symmetry
cluster model. For any c , h ∈ C {\displaystyle c,h\in \mathbb {C} } , the involution L n ↦ L ∗ = L − n {\displaystyle L_{n}\mapsto L^{*}=L_{-n}} defines an
Virasoro_algebra
Polynomial sequence
singletons and n − k/2 (unordered) pairs. Equivalently, it is the number of involutions of an n-element set with precisely k fixed points, or in other words
Hermite_polynomials
Study of angle-preserving transformations
generated by inversion are the only conformal mappings. Liouville's theorem is a classical theorem of conformal geometry. The addition of a point at infinity to
Inversive_geometry
Mathematical functions
} or ± i ϖ {\displaystyle \pm i\varpi } ), negates each function, an involution: cl ( z + ϖ ) = cl ( z + i ϖ ) = − cl z sl ( z + ϖ ) = sl (
Lemniscate_elliptic_functions
Setting of relativistic physics in geometric algebra
algebra containing pseudoscalars with a non-zero square. Grade involution (main involution, inversion) transforms every r {\displaystyle r} -vector
Spacetime_algebra
Distance from zero to a number
{\displaystyle \mathbb {R} ^{2}} . Every composition algebra A has an involution x → x* called its conjugation. The product in A of an element x and its
Absolute_value
Algebra of formal sums
of involutions of free abelian groups, the automorphisms that are their own inverse. Given a basis for a free abelian group, one can find involutions that
Free_abelian_group
Polynomial function of degree 4
{1}{2}}(x_{0}-x_{1}-x_{2}+x_{3}),\end{aligned}}} then since the transformation is an involution we may express the roots in terms of the four si in exactly the same way
Quartic_function
Branch of mathematics
posets with a unique bottom element 0, as well as an order-reversing involution ∗ {\displaystyle *} such that a ≤ a ∗ ⟹ a = 0. {\displaystyle a\leq a^{*}\implies
Order_theory
French mathematician (1869–1951)
Systems of PDEs, Cartan–Kähler theorem Theory of equivalence Integrable systems, theory of prolongation and systems in involution Infinite-dimensional groups
Élie_Cartan
Surface with constant mean curvature
{C} \setminus \{0\}} , ρ {\displaystyle \rho } is an antiholomorphic involution and L {\displaystyle L} is a line bundle on Σ {\displaystyle \Sigma }
Constant-mean-curvature surface
Constant-mean-curvature_surface
Algebraic structure designed for geometry
group, although Lundholm deprecates this usage). We denote the grade involution as S ^ {\displaystyle {\widehat {S}}} and reversion as S ~ {\displaystyle
Geometric_algebra
Element of a unital algebra over the field of real numbers
construction used involutions to generate complex numbers, quaternions, and octonions out of the real number system. Hurwitz and Frobenius proved theorems that put
Hypercomplex_number
Open convex self-dual cones
invertible operators that take the cone onto itself. By the Koecher–Vinberg theorem these correspond to the cone of squares in finite-dimensional real Euclidean
Symmetric_cone
Identities and relationships involving sets
B)^{\complement }=A^{\complement }\cup B^{\complement }} double complement or involution law ( A ∁ ) ∁ = A {\displaystyle (A^{\complement })^{\complement }=A}
Algebra_of_sets
Mathematical rule
rule, such as (Gasharov 1998), and (Stembridge 2002) using Bender-Knuth involutions. Littelmann (1994) used the Littelmann path model to generalize the
Littlewood–Richardson_rule
Non-self-adjoint compact operator used to solve boundary value problems for the Laplacian
transform and the Fredholm eigenvalues of bounded planar domains. Green's theorem for a bounded region Ω in the plane with smooth boundary ∂Ω states that
Neumann–Poincaré_operator
Mathematical object
conjugacy class. For G = GL(n), the transposition can serve as such an involution. In this case, there is the following criterion for the pair (G, K) to
Gelfand_pair
Rational numbers with root 5 added
{\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}} is an involution, ( α ¯ ) ¯ = α {\displaystyle {\overline {({\overline {\alpha }})}}=\alpha
Golden_field
Locus of the zeros of a polynomial of degree two
a) An exterior line, a tangent line or a secant line is mapped by the involution σ P {\displaystyle \sigma _{P}} on an exterior, tangent and secant line
Quadric
CLASSICAL INVOLUTION-THEOREM
CLASSICAL INVOLUTION-THEOREM
CLASSICAL INVOLUTION-THEOREM
CLASSICAL INVOLUTION-THEOREM
CLASSICAL INVOLUTION-THEOREM
CLASSICAL INVOLUTION-THEOREM
CLASSICAL INVOLUTION-THEOREM
CLASSICAL INVOLUTION-THEOREM
CLASSICAL INVOLUTION-THEOREM