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In mathematics, a modular Lie algebra is a Lie algebra over a field of positive characteristic. The theory of modular Lie algebras is significantly different
Modular_Lie_algebra
infinite-dimensional Lie algebras Free Lie algebra Graded Lie algebra Differential graded Lie algebra Homotopy Lie algebra Malcev Lie algebra Modular Lie algebra Monster
List of things named after Sophus Lie
List_of_things_named_after_Sophus_Lie
enveloping algebra Baker–Campbell–Hausdorff formula Casimir invariant Killing form Kac–Moody algebra Affine Lie algebra Loop algebra Graded Lie algebra One-parameter
List_of_Lie_groups_topics
Mathematical group
group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with
Group_of_Lie_type
Type of monoidal category
topological quantum field theory, conformal field theory, and quantum algebra. Modular tensor categories were introduced in 1989 by the physicists Greg Moore
Modular_tensor_category
Type of Kac–Moody algebras
affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given
Affine_Lie_algebra
Algebra used in certain conformal field theories
fusion of primary fields. In the context of modular tensor categories, there is also a Verlinde algebra. It is defined to have a basis of elements [
Verlinde_algebra
Algebraic variety
In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of
Modular_curve
Infinitesimal version of Lie groupoid
Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra. Lie algebroids play a similar same role in the theory of Lie
Lie_algebroid
Mathematical group
\mathbb {F} )} is considered a simple Lie group. The real rank of the corresponding Lie algebra, and hence of the Lie group Sp ( 2 n , F ) {\displaystyle
Symplectic_group
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
Group that is also a differentiable manifold with group operations that are smooth
matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra. On the other hand, Lie groups with isomorphic Lie algebras
Lie_group
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
American mathematician (1947–2022)
memoirs in four broad categories: modular Lie algebras; combinatorics of Lie algebra representations; graded algebras and superalgebras; and quantum groups
Georgia_Benkart
248-dimensional exceptional simple Lie group
any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the
E8_(mathematics)
133-dimensional exceptional simple Lie group
mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same
E7_(mathematics)
Integral polynomial
out that the representation theory of quantum groups, modular Lie algebras and affine Hecke algebras are all tightly controlled by appropriate analogues
Kazhdan–Lusztig_polynomial
Algebraic variety with a group structure
Similarly to the Lie group–Lie algebra correspondence, to an algebraic group over a field k {\displaystyle k} is associated a Lie algebra over k {\displaystyle
Algebraic_group
Study of Lie groups, Lie algebras and differential equations
The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The
Lie_theory
Algebra used in 2D conformal field theories and string theory
notion of vertex algebra was introduced by Richard Borcherds in 1986, motivated by a construction of an infinite-dimensional Lie algebra due to Igor Frenkel
Vertex_operator_algebra
Monster and modular connection
"Generalized Kac–Moody Lie algebras, free Lie algebras, and the structure of the Monster Lie algebra", Journal of Pure and Applied Algebra, 126 (1–3): 233–266
Monstrous_moonshine
Branch of mathematics that studies abstract algebraic structures
matrix multiplication). The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these
Representation_theory
gives conditions for a Lie algebra in characteristic 0 to be solvable, which implies a related criterion for the Lie algebra to be semisimple. It is
Cartan's_criterion
Group of unitary complex matrices with determinant of 1
domain for the Picard modular group in two complex dimensions". arXiv:math/0509708. Gilmore, Robert (1974). Lie Groups, Lie Algebras and some of their Applications
Special_unitary_group
Group of 𝑛 × 𝑛 invertible matrices
positive determinant. This is also a Lie group of dimension n 2 {\displaystyle n^{2}} ; it has the same Lie algebra as GL ( n , R ) {\displaystyle \operatorname
General_linear_group
American mathematician (1927–2024)
1959 Modular Lie Algebras, Springer Verlag 1967 Rational methods in Lie algebras, Marcel Dekker 1976 Rational constructions of modules for simple Lie algebras
George_Seligman
Branch of mathematics
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations
Abstract_algebra
Overview of and topical guide to algebraic structures
types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures
Outline of algebraic structures
Outline_of_algebraic_structures
conjectures Albanese variety Picard group Modular form Moduli space Modular equation J-invariant Algebraic function Algebraic form Addition theorem Invariant theory
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Infinite-dimensional generalized Kac-Moody algebra
In mathematics, the monster Lie algebra is an infinite-dimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove
Monster_Lie_algebra
Matrices named after Élie Cartan
mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form
Cartan_matrix
Topological algebra associated to continuous groups
{f(s^{-1})}}\,\Delta (s^{-1})} where Δ is the modular function on G. With this involution, it is a *-algebra. Theorem. With the norm: ‖ f ‖ 1 := ∫ G | f
Group algebra of a locally compact group
Group_algebra_of_a_locally_compact_group
Simple Lie group; the automorphism group of the octonions
mathematics, G2 is three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras g 2 , {\displaystyle {\mathfrak
G2_(mathematics)
Deformation of the group algebra of a Coxeter group
algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a deformation of the group algebra of a Coxeter group. The Hecke algebra can
Iwahori–Hecke_algebra
In mathematics, the Hecke algebra of a pair (G, K) of locally compact or reductive Lie groups is an algebra of measures under convolution. It can also
Hecke_algebra_of_a_pair
Algebraic construct of interest in theoretical physics
class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct"
Quantum_group
extension Representation of a Lie group Lie algebra representation, Representation of a Lie superalgebra Universal enveloping algebra Casimir element Infinitesimal
List of representation theory topics
List_of_representation_theory_topics
French mathematician (born 1926)
French mathematician who has made contributions to algebraic topology, algebraic geometry and algebraic number theory. He was awarded the Fields Medal in
Jean-Pierre_Serre
Topic in group theory and harmonic analysis (Niemeier lattice-mock theta connection)
sigma-model conformal field theory has an action of the N=(4,4) superconformal algebra, arising from a hyperkähler structure. When Tohru Eguchi, Hirosi Ooguri
Umbral_moonshine
Mathematical object studied in the field of algebraic geometry
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as
Algebraic_variety
American mathematician (1939–2020)
2020) was an American mathematician who worked in algebraic groups, Lie groups, and Lie algebras and applications of these mathematical structures. He
James_E._Humphreys
Analytic function on the upper half-plane with a certain behavior under the modular group
also appear in other areas, such as algebraic topology, sphere packing, and string theory. More precisely, a modular form is a holomorphic function on the
Modular_form
Alebraic concept
in general a Lie algebra over L, but is a Lie algebra over K of dimension n[L:K] = npn. A purely inseparable extension is called a modular extension if
Purely_inseparable_extension
Generalization of the BRST formalism
Hamiltonian formulation has constraints not related to a Lie algebra (i.e., the role of Lie algebra structure constants are played by more general structure
Batalin–Vilkovisky_formalism
Branch of number theory
of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. One of the founding works of algebraic number
Algebraic_number_theory
52-dimensional exceptional simple Lie group
In mathematics, F4 is a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups. F4 has rank 4 and dimension 52. The
F4_(mathematics)
Complex-differentiable part of a Maass wave function
lie in certain explicit finite-dimensional spaces, which reduces the long and hard proofs of many identities between them to routine linear algebra.
Mock_modular_form
Invariant of vertex algebra
vertex algebras analogue of Cartan's criterion for semisimplicity in the theory of Lie algebras because it relates a structural property of the algebra to
Zhu_algebra
Type of group in mathematics
whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has
Orthogonal_group
Operation measuring the failure of two entities to commute
Lie bracket, 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
Commutator
78-dimensional exceptional simple Lie group
mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras e 6 {\displaystyle {\mathfrak {e}}_{6}} , all
E6_(mathematics)
Subgroup of the group of invertible n×n matrices
{\displaystyle 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
Linear_algebraic_group
36 mathematical problems stated in 1955
1955. The problems primarily focused on algebraic geometry, number theory, and the connections between modular forms and elliptic curves. Taniyama's twelfth
Taniyama's_problems
Set whose pairs have minima and maxima
studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements
Lattice_(order)
Branch of mathematics that studies algebraic structures
ring Baer ring, Rickart ring Lie ring, Lie algebra Ideal (Lie algebra) Jordan algebra Differential algebra Banach algebra Rational number, Real number
List of abstract algebra topics
List_of_abstract_algebra_topics
The first Rankin–Cohen bracket is the Lie bracket when considering a ring of modular forms as a Lie algebra. Maass–Shimura operator Cohen, Henri (1975)
Rankin–Cohen_bracket
Lie group of Lorentz transformations
matrix Lie group, its corresponding Lie algebra s o ( 1 , 3 ) {\displaystyle {\mathfrak {so}}(1,3)} is a matrix Lie algebra, which may be computed as s o (
Lorentz_group
Concept in mathematics
over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups
Reductive_group
Type of group in mathematics
\mathrm {SL} (n,F)=\{g\in \mathrm {GL} _{n}(F)\mid \det g=1\}.} Its Lie algebra is s l ( n , F ) = { X ∈ M n ( F ) ∣ tr ( X ) = 0 } . {\displaystyle
Classical_group
Algebraic object
ring of modular forms is a graded Lie algebra since the Lie bracket [ f , g ] = k f g ′ − ℓ f ′ g {\displaystyle [f,g]=kfg'-\ell f'g} of modular forms f
Ring_of_modular_forms
Kazakh mathematician and physicist (born 1956)
1. – P. 201–230. Dzhumadildaev A.S., A.I. Kostrikin, Modular Lie algebras: new trends // Algebra (Proc. Kurosh Conf. may, 1998), Walter de Gruyter, p
Askar_Dzhumadildayev
Theory of algebraic structures in general
algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in general, not specific types of algebraic structures
Universal_algebra
Mathematical theorem
so-called non-modular case). Then the following properties are equivalent: (A) The group G is generated by pseudoreflections. (B) The algebra of invariants
Chevalley–Shephard–Todd theorem
Chevalley–Shephard–Todd_theorem
Shirshov–Cohn theorem (Jordan algebras) Shirshov–Witt theorem (Lie algebras) Beck's monadicity theorem (category theory) Bruguières modularity theorem (category theory)
List_of_theorems
Orientation-preserving mapping class group of the torus
In mathematics, the modular group is the projective special linear group PSL ( 2 , Z ) {\displaystyle \operatorname {PSL} (2,\mathbb {Z} )} of 2 × 2
Modular_group
Branch of mathematics that studies the properties of groups
methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced
Group_theory
Group of real 2×2 matrices with unit determinant
geometry. The group SL(2, R) acts on its Lie algebra sl(2, R) by conjugation (remember that the Lie algebra elements are also 2 × 2 matrices), yielding
SL2(R)
Group for which a given group is a normal subgroup
Metaplectic groups also occur in quantum mechanics. Algebra extension Lie algebra extension Virasoro algebra HNN extension Group contraction Extension of a
Group_extension
Group of flat spacetime symmetries
{Spin} (1,3)} . The Poincaré algebra is the Lie algebra of the Poincaré group. It is a Lie algebra extension of the Lie algebra of the Lorentz group. More
Poincaré_group
Concept in modular arithmetic
In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent
Modular multiplicative inverse
Modular_multiplicative_inverse
Measure of network community structure
Modularity is a measure of the structure of networks or graphs which measures the strength of division of a network into modules (also called groups, clusters
Modularity_(networks)
Concept in mathematical group theory
Connecting Algebra, Modular Forms and Physics. Cambridge University Press. ISBN 978-0-521-83531-2. Hall, Brian C. (2015), Lie groups, Lie algebras, and representations:
Character_theory
Sporadic simple group
"The 5-modular characters of the covering group of the sporadic simple Fischer group Fi22 and its automorphism group", Communications in Algebra, 22 (9):
Fischer_group_Fi22
algebra. The JC algebra satisfies the additional condition that (T + T*)/2 lies in the algebra whenever T is a product of operators from the algebra.
Jordan_operator_algebra
Sporadic simple group
acts on a vertex operator algebra over the field with 3 elements. This vertex operator algebra contains the E8 Lie algebra over F3, giving the embedding
Thompson_sporadic_group
Representation theory
representation of a semisimple Lie algebra. In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character
Weyl_character_formula
Group of matrices with determinant 1
F)} is a Lie subgroup of GL ( n , F ) {\displaystyle \operatorname {GL} (n,F)} of dimension n 2 − 1 {\displaystyle n^{2}-1} . The Lie algebra s l ( n
Special_linear_group
Jordan algebra is the Albert algebra of 3×3 self-adjoint matrices over the octonions. The simple Lie groups form a number of series (classical Lie groups)
Exceptional_object
Concept in mathematical group theory
Pseudo-Euclidean space R p , q {\displaystyle \mathbb {R} ^{p,q}} , the Lie algebra of the conformal group is given by the basis { M μ ν , P μ , K μ , D
Conformal_group
Series of mathematics books by Nicolas Bourbaki
treated in the series include set theory, abstract algebra, topology, analysis, Lie groups and Lie algebras. The unusual singular "mathématique" (mathematic)
Éléments_de_mathématique
Belgian mathematician
work on algebraic geometry. In joint work with George Lusztig, Deligne applied étale cohomology to construct representations of finite groups of Lie type;
Pierre_Deligne
Mathematical group of loops in a Lie group
pointwise. When G is a compact Lie group, LG is a basic example of an infinite-dimensional Lie group, with Lie algebra L𝔤 = C∞(S1, 𝔤). The subgroup
Loop_group
This is a list of named algebraic surfaces, compact complex surfaces, and families thereof, sorted according to their Kodaira dimension following Enriques–Kodaira
List of complex and algebraic surfaces
List_of_complex_and_algebraic_surfaces
Elements taken to zero by a homomorphism
In algebra, the kernel of a homomorphism is the relation describing how elements in the domain of the homomorphism become related in the image. A homomorphism
Kernel_(algebra)
Concept in mathematics
13. A heyting algebra is residuated. 14. A residuated lattice is a lattice. (def) 15. A distributive lattice is modular. 16. A modular complemented lattice
Map_of_lattices
Matrix group
subgroups of 2 × 2 matrices are fundamental objects in the classical theory of modular forms; the modern theory of automorphic forms makes a similar use of congruence
Congruence_subgroup
Mathematical concept
analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined
Shimura_variety
Type of group in group theory
Picard modular group. When G {\displaystyle G} is a Lie group one can define an arithmetic lattice in G {\displaystyle G} as follows: for any algebraic group
Arithmetic_group
Type of group in abstract algebra
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group
Symmetric_group
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
Discrete subgroup in a locally compact topological group
is particularly rich for lattices in semisimple Lie groups or more generally in semisimple algebraic groups over local fields. In particular there is
Lattice_(discrete_subgroup)
Mathematical structure in differential geometry
M {\displaystyle M} , making it into a Lie algebra subject to a Leibniz rule (also known as a Poisson algebra). Poisson structures on manifolds were introduced
Poisson_manifold
Set without nontrivial polynomial equalities
In abstract algebra, a subset S {\displaystyle S} of a field L {\displaystyle L} is algebraically independent over a subfield K {\displaystyle K} if the
Algebraic_independence
Reduction of a ring by one of its ideals
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite
Quotient_ring
German mathematician (born 1948)
specializing in the areas of algebra known as representation theory (especially modular representation theory) and homological algebra (especially Hochschild
Karin_Erdmann
Modular function in mathematics
In mathematics, the j-invariant or j function is a modular function of weight zero for the special linear group SL ( 2 , Z ) {\displaystyle \operatorname
J-invariant
Mathematical group based upon a finite number of elements
chapter of linear algebra. A group of Lie type is a group closely related to the group G(k) of rational points of a reductive linear algebraic group G with
Finite_group
elements of N {\displaystyle N} . Let U be the universal enveloping algebra of a Lie algebra g {\displaystyle {\mathfrak {g}}} over a field k; it is filtered
Associated_graded_ring
Vectors mapped to 0 by a linear map
Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0-387-98259-0. Lay, David C. (2005), Linear Algebra and Its Applications (3rd ed
Kernel_(linear_algebra)
History of a branch of mathematics
theory was built up by Klein, Lie, Henri Poincaré, and Charles Émile Picard, in connection in particular with modular forms and monodromy. The third
History_of_group_theory
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MODULAR LIE-ALGEBRA
MODULAR LIE-ALGEBRA
MODULAR LIE-ALGEBRA
MODULAR LIE-ALGEBRA
MODULAR LIE-ALGEBRA
MODULAR LIE-ALGEBRA
MODULAR LIE-ALGEBRA
MODULAR LIE-ALGEBRA
MODULAR LIE-ALGEBRA
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