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Algebra of meromorphic vector fields on the Riemann sphere
In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are
Witt_algebra
Algebra describing 2D conformal symmetry
mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional
Virasoro_algebra
Creating a "larger" Lie algebra from a smaller one, in one of several ways
algebra in two spacetime dimensions. The Virasoro algebra is the universal central extension of the Witt algebra. Central extensions are needed in physics, because
Lie_algebra_extension
Topics referred to by the same term
Witt ring may be A ring of Witt vectors The Witt ring (forms), a ring structure on the Witt group of symmetric bilinear forms See also Witt algebra,
Witt_ring
Explicitly describes the universal enveloping algebra of a Lie algebra
Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a
Poincaré–Birkhoff–Witt theorem
Poincaré–Birkhoff–Witt_theorem
German mathematician (1911–1991)
theorem is basic to the study of Lie algebras. In algebraic geometry, the Hasse–Witt matrix of an algebraic curve over a finite field determines the
Ernst_Witt
Shirshov–Witt theorem Witt algebra Witt decomposition Witt design (Witt geometry) Witt group Witt index Witt polynomial Witt ring Grothendieck-Witt ring Witt scheme
List of things named after Ernst Witt
List_of_things_named_after_Ernst_Witt
Group that is also a differentiable manifold with group operations that are smooth
circle. Its Lie algebra is (more or less) the Witt algebra, whose central extension the Virasoro algebra (see Virasoro algebra from Witt algebra for a derivation
Lie_group
Concept in mathematics
enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal
Universal_enveloping_algebra
Nonlinear differential operator used to study conformal mappings
result giving a 1-cocycle for Vect(S1), the Lie algebra of smooth vector fields, and hence for the Witt algebra, the subalgebra of trigonometric polynomial
Schwarzian_derivative
Basic result in the algebraic theory of quadratic forms, on extending isometries
theory. In mathematics, Witt's theorem, named after Ernst Witt, is a basic result in the algebraic theory of quadratic forms: any isometry between two subspaces
Witt's_theorem
American mathematician
was the first to discover the central extension of the Witt algebra that gives the Virasoro algebra, though his discovery went unnoticed for many years.
Richard_Earl_Block
universal enveloping algebra of a free Lie algebra on a set X is the free associative algebra generated by X. By the Poincaré–Birkhoff–Witt theorem it is the
Free_Lie_algebra
Study of angle-preserving transformations of a geometric space
satisfying 1. and 2. Hence the Lie algebra of infinitesimal symmetries of the conformal structure, the Witt algebra, is infinite-dimensional. The conformal
Conformal_geometry
Conformal field theory on a 2D spacetime
differential operators ℓ n {\displaystyle \ell _{n}} generate a Witt algebra. Unfortunately the Witt algebra on its own always generates a space of particle states
Two-dimensional conformal field theory
Two-dimensional_conformal_field_theory
Algebraic structure in mathematics
extension Koszul algebra Poincaré–Birkhoff–Witt theorem Alexander J. Hahn (1994) Quadratic algebras, Clifford algebras, and Arithmetic Witt Groups, page 5
Quadratic_algebra
Algebra term
forms. The Witt ring of C, and indeed any algebraically closed field or quadratically closed field, is Z/2Z. The Witt ring of R is Z. The Witt ring of a
Witt_group
Algebraic study of differential equations
and is essentially equivalent when K {\displaystyle K} is a field.) A Witt algebra is a differential ring that contains the field Q {\displaystyle \mathbb
Differential_algebra
Mathematical concept named for Ernst Witt
mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors
Witt_vector
Algebraic structure used in analysis
algebra is the quotient ring U ( g ) = T ( g ) / I {\displaystyle U({\mathfrak {g}})=T({\mathfrak {g}})/I} . It satisfies the Poincaré–Birkhoff–Witt theorem:
Lie_algebra
Transformation in quantum mechanics
invariant. The technique can be further extended to the Witt algebra, which is the centerless Virasoro algebra. Spin wave Jordan–Wigner transformation Jordan–Schwinger
Holstein–Primakoff transformation
Holstein–Primakoff_transformation
Unified Theory Supergroup Lie superalgebra Twistor theory Anyon Witt algebra Virasoro algebra Erlangen programme Homogeneous space Principal homogeneous space
List_of_Lie_groups_topics
Quantum field theory enjoying conformal symmetry
Witt algebra of infinitesimal conformal transformations has to be centrally extended. The quantum symmetry algebra is therefore the Virasoro algebra,
Conformal_field_theory
American mathematician
of Kansas. Her research involves commutative algebra, representation theory, and singularity theory. Witt is a 2005 graduate of the University of Chicago
Emily_E._Witt
Branch of Galois theory in mathematics
Schreier (1927) introduced Artin–Schreier theory for extensions of prime degree p, and Witt (1936) generalized it to extensions of prime power degree pn. If K is a field
Artin–Schreier_theory
Algebra associated to any vector space
In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle
Exterior_algebra
In mathematics, the Hasse–Witt matrix H of a non-singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping (p-th power mapping
Hasse–Witt_matrix
Concept in mathematical group theory
conformal symmetries of 2d Euclidean space is the infinite-dimensional Witt algebra. In 1908, Harry Bateman and Ebenezer Cunningham, two young researchers
Conformal_group
Mathematical homomorphism
identity homomorphism. For Witt vectors, the Verschiebung takes (a0, a1, a2, ...) to (0, a0, a1, ...). On the Hopf algebra of symmetric functions, the
Verschiebung_operator
Korean Canadian mathematician (1922–2005)
British Columbia in Vancouver, Canada. He completed his dissertation on Witt algebras in 1955. His thesis advisor was Stephen Arthur Jennings. Following the
Rimhak_Ree
African-American mathematician & academic
research on Sklyanin algebras in Poisson geometry, on the actions of Hopf algebras, and on the universal enveloping algebra of the Witt algebra. She was elected
Chelsea_Walton
Abstract mathematical group
In abstract algebra, the Virasoro group or Bott–Virasoro group (often denoted by Vir) is an infinite-dimensional Lie group defined as the universal central
Virasoro_group
invariant (or Hasse–Witt invariant) of a quadratic form Q over a field K takes values in the Brauer group Br(K). The name "Hasse–Witt" comes from Helmut
Hasse invariant of a quadratic form
Hasse_invariant_of_a_quadratic_form
Concept in mathematics
algebra. For example, Witt's decomposition theorem says that a nondegenerate quadratic form over a field is determined up to isomorphism by its Witt index
Reductive_group
German mathematician (1882–1935)
p-adic Theory in Noncommutative Algebras], Monatshefte für Mathematik (in German), 44 (1): 203–224, doi:10.1007/BF01699316 Witt, Ernst (1935), "Riemann-Rochscher
Emmy_Noether
Algebraic structure with addition and multiplication
commutative ring A returns the ring Wn(A) of p-isotypic Witt vectors of length n over A. In algebraic topology, a ring spectrum is a spectrum X together with
Ring_(mathematics)
Sporadic simple group
In the area of modern algebra known as group theory, the Mathieu group M22 is a sporadic simple group of order 443,520 = 27 · 32 · 5 · 7 · 11 ≈ 4×105
Mathieu_group_M22
In mathematics, Witt vector cohomology was an early p-adic cohomology theory for algebraic varieties introduced by Serre (1958). Serre constructed it by
Witt_vector_cohomology
result concerning commutators. It is a consequence of Philip Hall and Ernst Witt's eponymous identity. In what follows, the following notation will be employed:
Three_subgroups_lemma
Weil cohomology theory for schemes X over a base field k
because it produces modules over the ring of Witt vectors of the ground field. So if the ground field is an algebraic closure of Fp, its values are modules over
Crystalline_cohomology
American mathematician
generalized Witt algebras over algebraically closed fields of finite characteristic. From 1965 to the mid 1980s Sweeder worked on commutative algebra and related
Moss_Sweedler
as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of A and B. The quaternion algebras are linked if and only if
Biquaternion_algebra
Proof that every structure with certain properties is isomorphic to another structure
Boolean algebras and that of Stone spaces. The Poincaré–Birkhoff–Witt theorem states that every Lie algebra embeds into the commutator Lie algebra of its
Representation_theorem
French mathematician (born 1926)
1954–55 was one based on Witt vector coefficients. Around 1958, Serre suggested that isotrivial principal bundles on algebraic varieties—those that become
Jean-Pierre_Serre
Model in theoretical physics
Unruh–DeWitt detector model serves as a foundational theoretical tool within quantum field theory in curved spacetime, providing a localized, algebraically defined
Unruh–DeWitt_detector
Ring theory
\prod _{n\geq 0}(1{-}t^{n})^{-a_{n}}} . Witt vector Hazewinkel, Michiel (2009). "Witt vectors I". Handbook of Algebra. Vol. 6. Elsevier/North-Holland. pp
Necklace_ring
Swiss mathematician, working in algebra
1970) is a Swiss mathematician working in tensor triangular geometry, algebraic geometry, modular representation theory, and homotopy theory. He is a
Paul_Balmer
Operation measuring the failure of two entities to commute
every associative algebra can be turned into a Lie algebra. The anticommutator of two elements a and b of a ring or associative algebra is defined by {
Commutator
Virasoro algebra A central extension of the Witt algebra of polynomial vector fields on a circle. w A complex number W A W-boson W-algebra A sort of
Glossary_of_string_theory
In algebraic geometry, F-crystals are objects introduced by Mazur (1972) that capture some of the structure of crystalline cohomology groups. The letter
F-crystal
Poincaré–Birkhoff–Witt theorem (universal enveloping algebras) Shirshov–Cohn theorem (Jordan algebras) Shirshov–Witt theorem (Lie algebras) Beck's monadicity
List_of_theorems
Algebraic structure with addition, multiplication, and division
operations on rational numbers do. Fields are fundamental algebraic structures that are widely used in algebra, number theory, and many other areas of mathematics
Field_(mathematics)
Type of mathematical object
from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups
Group_scheme
In algebra, the Nichols algebra of a braided vector space (with the braiding often induced by a finite group) is a braided Hopf algebra which is denoted
Nichols_algebra
as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of A and B. The quaternion algebras are linked if and only if
Linked_field
Group in group theory and physics
enveloping algebra is an associative algebra into which h n {\displaystyle {\mathfrak {h}}_{n}} injectively imbeds. By the Poincaré–Birkhoff–Witt theorem
Heisenberg_group
Algebraic structure used in theoretical physics
Poincaré–Birkhoff–Witt theorem holds (and, in general, they are necessary conditions for the theorem to hold). Just as for Lie algebras, the universal enveloping
Lie_superalgebra
Representation theory of the symplectic group
\quad L_{1}={\begin{pmatrix}0&0\\-1&0\end{pmatrix}}.} They satisfy the Witt algebra [ L m , L n ] = ( m − n ) L m + n , m , n ∈ { 1 , 0 , − 1 } . {\displaystyle
Oscillator_representation
Manx mathematician (1944–2021)
sequences of Witt groups". Journal of Algebra. 74 (1): 206–210. doi:10.1016/0021-8693(82)90013-8. Lewis, D. W. (1977). "Forms over real algebras and the multisignature
David W. Lewis (mathematician)
David_W._Lewis_(mathematician)
Soviet mathematician (1921–1981)
free Lie algebras. He proved the Shirshov–Witt theorem, which states that any Lie subalgebra of a free Lie algebra is itself a free Lie algebra. Anatoly
Anatoly_Shirshov
Type of group in mathematics
matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension
Orthogonal_group
Ring without nonzero zero divisors
uses the standard filtration on the universal enveloping algebra and the Poincaré–Birkhoff–Witt theorem. Suppose that G is a group and K is a field. Is
Domain_(ring_theory)
Pictorial representation of symmetry
Dynkin diagrams arise in the classification of semisimple Lie algebras over algebraically closed fields, in the classification of Weyl groups and other
Dynkin_diagram
Objects in representation theory of Lie algebras
algebras, a branch of mathematics. Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra
Verma_module
Algebraic structure
In algebra, a field K {\displaystyle K} is perfect if any one of the following equivalent conditions holds: Every irreducible polynomial over K {\displaystyle
Perfect_field
In mathematics, a restricted Lie algebra (or p-Lie algebra) is a Lie algebra over a field of characteristic p>0 together with an additional "pth power"
Restricted_Lie_algebra
Algebraic structure also called skew field
finite fields. (Ernst Witt gave a simple proof.) Frobenius theorem: The only finite-dimensional associative division algebras over the reals are the
Division_ring
of an algebraic group G over a field is an invariant of forms of G taking values in a Galois cohomology group. Suppose that G is an algebraic group defined
Cohomological_invariant
enveloping algebra of a Lie algebra g {\displaystyle {\mathfrak {g}}} over a field k; it is filtered by degree. The Poincaré–Birkhoff–Witt theorem implies
Associated_graded_ring
Property of some binary operations
map sending each element to its adjoint action is a Lie algebra homomorphism. The Hall–Witt identity is the analogous identity for the commutator operation
Jacobi_identity
Anticommutating number
(linguist and mathematician) Superspace Exterior algebra DeWitt 1984, Chapter 1, page 1. DeWitt 1984, pp. 1–2. DeWitt 1984, p. 2. Rogers 2007a, Chapter 1 (available
Grassmann_number
Broad concept generalizing scalars in mathematics and physics
subspace of some Clifford algebra. Witt vector, an infinite sequence of elements of a commutative ring, which belongs to an algebra over this ring, and has
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Indian-American mathematician (born 1983)
Princeton University and works in arithmetic geometry and commutative algebra. Bhatt graduated with a B.S. in Applied Mathematics, summa cum laude from
Bhargav_Bhatt_(mathematician)
Mathematical formal infinite series
over a finite field K of characteristic p (or their algebraic closure), the field of Hahn–Witt series with value group Γ (containing the integers) would
Hahn_series
Sporadic simple group
In the area of modern algebra known as group theory, the Mathieu group M12 is a sporadic simple group of order 95,040 = 12 · 11 · 10 · 9 · 8 = 26 ·
Mathieu_group_M12
Concept in mathematics
intermediate between Lie groups (or algebraic groups) and Lie algebras. They are used in algebraic number theory and algebraic topology. A one-dimensional formal
Formal_group_law
German mathematician
Mathematical Society. Her research concerns abstract algebra including the theory of field extensions and algebraic groups. She is a professor emerita at the University
Ina_Kersten
of an algebraic curve over an algebraically closed field has u ≤ 2; this follows from Tsen's theorem that such a field is quasi-algebraically closed
U-invariant
German mathematician (born 1958)
Joachim; Deninger, Christopher (2015), "Witt vector rings and the relative de Rham Witt complex", Journal of Algebra, 440: 545–593, arXiv:1410.5249, doi:10
Christopher_Deninger
Real numbers adjoined with a nil-squaring element
In algebra, the dual numbers are a quadratic algebra first introduced in the 19th century. They are expressions of the form a + bε, where a and b are
Dual_number
Array of numbers
"two-by-three matrix", a 2 × 3 matrix, or a matrix of dimension 2 × 3. In linear algebra, matrices are used as linear maps. In geometry, matrices are used for geometric
Matrix_(mathematics)
Public school in New York City
broadcast from DeWitt Clinton High School from C-SPAN's American Writers Images: Algebra at DeWitt Clinton High School Stairwell at DeWitt Clinton High School
DeWitt_Clinton_High_School
French mathematician
cotangent complex and deformations, crystalline cohomology and the De Rham–Witt complex, and logarithmic geometry. In 2012, he was awarded the Émile Picard
Luc_Illusie
of algebra Glossary of field theory Glossary of ring theory List of abstract algebra topics List of algebraic structures List of Boolean algebra topics
Lists_of_mathematics_topics
24-dimensional repeating pattern of points
and the Leech lattice", Journal of Algebra, 322 (6): 2186–2190, doi:10.1016/j.jalgebra.2009.03.021, MR 2542837 Witt, Ernst (1941), "Eine Identität zwischen
Leech_lattice
Physical theory with fields invariant under the action of local "gauge" Lie groups
the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises
Gauge_theory
American mathematician (1911–1996)
conjecture Pierce–Birkhoff ring Poincaré–Birkhoff–Witt theorem Algebraic statistics Median algebra Staff. A COMMUNITY OF SCHOLARS: The Institute for Advanced
Garrett_Birkhoff
German mathematician (born 1943)
Library "Reduced forms and reduced Witt rings of higher level" by Eberhard Becker and Alex Rosenberg, Journal of Algebra, Vol 92, Issue 2, Feb 1985, pp 477-503
Eberhard_Becker
Commuting Lie algebra operator
with all other elements of the original group—often embedded within a Lie algebra. In some cases, such as two-dimensional conformal field theory, a central
Central_charge
Dutch mathematician and mayor
Schooten made of Descartes' La Géométrie, Hudde, together with Johan de Witt and Hendrik van Heuraet, published work of their own. Hudde's contribution
Johannes_Hudde
Tool in homological algebra
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral
Spectral_sequence
Function that is its own inverse
John Wiley & Sons Alexander J. Hahn (1994) Quadratic algebras, Clifford algebras, and Arithmetic Witt Groups, page 77, Universitext, Springer, ISBN 0-387-94110-X
Involution_(mathematics)
defined by mapping an algebra to the pair consisting of its grade and determinant. There is a map from the additive group of the Witt–Grothendieck ring to
Brauer–Wall_group
Module over the non-commutative Dieudonné ring
the non-commutative Dieudonné ring, which is generated over the ring of Witt vectors by two special endomorphisms F {\displaystyle F} and V {\displaystyle
Dieudonné_module
Polynomial with all terms of degree two
place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal groups), differential geometry (the Riemannian
Quadratic_form
Isomorphism of commutative rings constructed in the theory of Lie algebras
Lie algebras. The isomorphism maps the center Z ( U ( g ) ) {\displaystyle {\mathcal {Z}}(U({\mathfrak {g}}))} of the universal enveloping algebra U (
Harish-Chandra_isomorphism
Added a basic definition in group theory and algebra
Definition 2.1) (Kisilevsky, Neftin & Sonn 2010) (Kisilevsky & Sonn 2010) (De Witt 2014) (Thompson 1984) (Neftin 2009, Definition 1.1.) (Blum-Smith 2014) (Legrand
Semiabelian_group
Quadratic form for which there is a non-zero vector on which the form evaluates to zero
10016. Emil Artin (1957) Geometric Algebra, page 119 via Internet Archive Pete L. Clark, Quadratic forms chapter I: Witts theory from University of Miami
Isotropic_quadratic_form
Sporadic simple group
In the area of modern algebra known as group theory, the Mathieu group M11 is a sporadic simple group of order 7,920 = 11 · 10 · 9 · 8 = 24 · 32 ·
Mathieu_group_M11
Algebraic structure
structures Witt vectors Hecke algebra (used in Wiles's proof of Fermat's Last Theorem) Fontaine's period rings Cluster algebra Convolution algebra (of a commutative
Commutative_ring
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