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Algebraic structure with "nice" duality properties
Dieudonné used this to characterize Frobenius algebras (Dieudonné 1958). Frobenius algebras were generalized to quasi-Frobenius rings, those Noetherian rings
Frobenius_algebra
Map raising elements to the pth power, in characteristic p
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with
Frobenius_endomorphism
Theorem in abstract algebra
abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over
Frobenius theorem (real division algebras)
Frobenius_theorem_(real_division_algebras)
tangent space. The concept generalizes the notion of Frobenius algebra to tangent bundles. Frobenius manifolds occur naturally in the subject of symplectic
Frobenius_manifold
Ferdinand Georg Frobenius, a German mathematician. Arithmetic and geometric Frobenius Cauchy–Frobenius lemma Frobenioid Frobenius algebra Frobenius category
List of things named after Ferdinand Georg Frobenius
List_of_things_named_after_Ferdinand_Georg_Frobenius
map and its dual bases make explicit L as a Frobenius algebra over K. More generally, separable algebras over a field K can be classified as follows:
Separable_algebra
Mathematical problem
problem (also referred to as the Frobenius coin problem or Frobenius problem, after the mathematician Ferdinand Frobenius) is a mathematical problem that
Coin_problem
Theorem in linear algebra
matrix theory, the Perron–Frobenius theorem, proved in its first part by Oskar Perron (1907) and extended by Georg Frobenius (1912), asserts that a real
Perron–Frobenius_theorem
Branch of mathematics
example, Sylow's theorem was reproven by Frobenius in 1887 directly from the laws of a finite group, although Frobenius remarked that the theorem followed from
Abstract_algebra
In ring theory and Frobenius algebra extensions, areas of mathematics, there is a notion of depth two subring or depth of a Frobenius extension. The notion
Depth of noncommutative subrings
Depth_of_noncommutative_subrings
Sum of elements on the main diagonal
B is a square matrix. The Frobenius inner product and norm arise frequently in matrix calculus and statistics. The Frobenius inner product may be extended
Trace_(linear_algebra)
Branch of mathematics
Linear algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b
Linear_algebra
Algebraic structure used in analysis
symmetric Lie algebra Poisson algebra Pre-Lie algebra Quantum groups Moyal algebra Quasi-Frobenius Lie algebra Quasi-Lie algebra Restricted Lie algebra Serre
Lie_algebra
Norm on a vector space of matrices
circular shifts. The Frobenius norm is an extension of the Euclidean norm to K n × n {\displaystyle K^{n\times n}} and comes from the Frobenius inner product
Matrix_norm
projectives and injectives coincide. It is an analog of a Frobenius algebra. The stable category of a Frobenius category is canonically a triangulated category
Frobenius_category
Topics referred to by the same term
Georg Frobenius. They include: Frobenius theorem (differential topology) in differential geometry and topology for integrable subbundles Frobenius theorem
Frobenius_theorem
quasi-Frobenius Lie algebra ( g , [ , ] , β ) {\displaystyle ({\mathfrak {g}},[\,\,\,,\,\,\,],\beta )} over a field k {\displaystyle k} is a Lie algebra (
Quasi-Frobenius_Lie_algebra
Quantum mechanics posed in terms of category theory
no-deleting theorems of quantum mechanics. Special commutative dagger Frobenius algebras model the fact that certain processes yield classical information
Categorical_quantum_mechanics
Algebra based on a vector space with a quadratic form
mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure
Clifford_algebra
Concept in abstract algebra
under the Frobenius endomorphism F*. Brion & Kumar (2005) give a detailed discussion of Frobenius splittings. A fundamental property of Frobenius-split projective
Frobenius_splitting
Algebra over a field with only invertible elements and zero
finite-dimensional as a vector space over R). The Frobenius theorem states that up to isomorphism there are three such algebras: the reals themselves (dimension 1)
Division_algebra
Canonical form of matrices over a field
In linear algebra, the Frobenius normal form or rational canonical form of a square matrix A with entries in a field F is a canonical form for matrices
Frobenius_normal_form
Mathematical ring whose elements are matrices
the three Pauli matrices. A matrix ring over a field is a Frobenius algebra, with Frobenius form given by the trace of the product: σ(A, B) = tr(AB).
Matrix_ring
especially ring theory, the class of Frobenius rings and their generalizations are the extension of work done on Frobenius algebras. Perhaps the most important
Quasi-Frobenius_ring
Szlachányi (J. Algebra) in 2004 for tensor categorical reasons and to accommodate examples associated to depth two Frobenius algebra extensions. The
Hopf_algebroid
On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system
Frobenius theorem (differential topology)
Frobenius_theorem_(differential_topology)
Dimension of the column space of a matrix
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal
Rank_(linear_algebra)
Generalization of quaternions to other fields
local class field theory. It is a theorem of Frobenius that there are only two real quaternion algebras: 2 × 2 matrices over the reals and Hamilton's
Quaternion_algebra
Algebraic structure with addition and multiplication
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication and denoted
Ring_(mathematics)
positive-semidefinite matrix Pfaffian Projection Spectral theorem Perron–Frobenius theorem List of matrices Diagonal matrix, main diagonal Diagonalizable
Outline_of_linear_algebra
Duality between the process of restricting and inducting in representation theory
equivalent to the theory of modules over the group algebra K[G]. Therefore, there is a corresponding Frobenius reciprocity theorem for K[G]-modules. Let G be
Frobenius_reciprocity
Non-associative algebras with positive-definite quadratic form
possibilities. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras. The theory of composition algebras has subsequently
Hurwitz's theorem (composition algebras)
Hurwitz's_theorem_(composition_algebras)
commuting with the group action is a real associative division algebra and by the Frobenius theorem can only be isomorphic to either the real numbers, or
Frobenius–Schur_indicator
Four-dimensional number system
\operatorname {Cl} _{3,0}^{+}(\mathbb {R} ).} According to the Frobenius theorem, the algebra H {\displaystyle \mathbb {H} } is one of only two finite-dimensional
Quaternion
Smallest integer n for which n equals 0 in a ring
map x ↦ xp then defines a ring homomorphism R → R, which is called the Frobenius homomorphism. If R is also an integral domain, the homomorphism is injective
Characteristic_(algebra)
Method for producing composition algebras
algebras over the real numbers, while the Frobenius theorem states that the first three are the only finite-dimensional associative division algebras
Cayley–Dickson_construction
Algebraic structure with addition, multiplication, and division
the Frobenius theorem. The octonions O, for which multiplication is neither commutative nor associative, is a normed alternative division algebra, but
Field_(mathematics)
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)
Field theory involving topological effects in physics
topological quantum field theories and the category of commutative Frobenius algebras. To consider all spacetimes at once, it is necessary to replace hBordM
Topological quantum field theory
Topological_quantum_field_theory
Type of monoidal category
categories and topology I: Frobenius algebras in and Morita equivalence of tensor categories". Journal of Pure and Applied Algebra. 180 (1): 81–157. arXiv:math/0111204
Modular_tensor_category
Number in {..., –2, –1, 0, 1, 2, ...}
numbers. In algebraic number theory, integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In
Integer
German mathematician
"Orbifolding Frobenius Algebras". Internat. J. of Math. 14 (2003), 573-619 Kaufmann, Ralph M. "Singularities with Symmetries, Orbifold Frobenius algebras and Mirror
Ralph_Kaufmann
Representation theory of groups
general, such a structure is called a Frobenius algebra. As the name implies, these were introduced by Frobenius in the nineteenth century. They have been
Regular_representation
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
Graphical representation of a morphism
of identity of Peirce's existential graphs can be axiomatised as a Frobenius algebra, the cuts are unary operators on homsets that axiomatise logical negation
String_diagram
Binary operation, takes two matrices and returns a scalar
In mathematics, the Frobenius inner product (also known as the Double-dot product) is a binary operation that takes two matrices and returns a scalar
Frobenius_inner_product
Mathematical object in abstract algebra
quasi-Frobenius ring, and is two-sided Artinian and two-sided injective, (Lam 1999, Th. 15.1). An important module theoretic property of quasi-Frobenius rings
Injective_module
Ring that is also a vector space or a module
In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center
Associative_algebra
Applications of category theory
in natural language with compact closed categories and Frobenius algebras", Logic and Algebraic Structures in Quantum Computing, Cambridge University Press
Applied_category_theory
Number of solutions of linear systems in terms of matrix ranks
Rouché–Fontené theorem in France; Rouché–Frobenius theorem in Spain and many countries in Latin America; Frobenius theorem in Czechia and Slovakia. A system
Rouché–Capelli_theorem
stabilizer of any regular element in g. If ind g = 0, then g is called Frobenius Lie algebra. This is equivalent to the fact that the Kirillov form K ξ : g ⊗
Index_of_a_Lie_algebra
Branch of mathematics that studies algebraic structures
(& outline) Algebra representation Group representation Lie algebra representation Maschke's theorem Schur's lemma Equivariant map Frobenius reciprocity
List of abstract algebra topics
List_of_abstract_algebra_topics
Algebraic structure
the Frobenius endomorphism x ↦ x p {\displaystyle x\mapsto x^{p}} is an automorphism. The separable closure of K {\displaystyle K} is algebraically closed
Perfect_field
Algebraic variety with a group structure
mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus
Algebraic_group
developed modular representation theory. Lam 1998. Cayley 1854. Frobenius 1896, Frobenius 1897. Burnside 1904. In the first edition of his famous treatise
History of representation theory
History_of_representation_theory
Algebraic structure
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more
Polynomial_ring
Concepts from linear algebra
In linear algebra, an eigenvector (/ˈaɪɡən-/ EYE-gən-) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by
Eigenvalues_and_eigenvectors
Vector space in mathematics
of multiplication and comultiplication, include Lie bialgebras and Frobenius algebras. Additional examples are given in the article on coalgebras. Quasi-bialgebra
Bialgebra
Mathematical structure in abstract algebra
mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra; read as "star-algebra") is a mathematical structure consisting of
*-algebra
Foundational object in quantum communication theory
quantum mechanics, the classical information is carried in a Frobenius algebra or Frobenius category. For a purely quantum system, the time evolution, at
Quantum_channel
Group of unitary complex matrices with determinant of 1
structure of this Lie algebra can be found below in § Lie algebra structure. In the physics literature, it is common to identify the Lie algebra with the space
Special_unitary_group
Prime number with a certain relationship to an elliptic curve
supersingular for E {\displaystyle E} if and only if the trace of the Frobenius endomorphism a p = p + 1 − # E ( F p ) {\displaystyle a_{p}=p+1-\#E(\mathbb
Supersingular prime (algebraic number theory)
Supersingular_prime_(algebraic_number_theory)
248-dimensional exceptional simple Lie group
several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding
E8_(mathematics)
American mathematician
are states of the bracket polynomial decorated with elements of a Frobenius algebra. The Kauffman polynomial is a 2-variable knot polynomial due to Louis
Louis_Kauffman
Element of a unital algebra over the field of real numbers
Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: Hurwitz's theorem says finite-dimensional real composition algebras are the reals
Hypercomplex_number
Category in mathematics
kG-modules. More generally, the stable module category is defined for any Frobenius algebra in place of kG. Some experts suspectpg 190 (see, for example, (Gelfand
Triangulated_category
of embedding. Deformation theory Differential graded Lie algebra Kodaira–Spencer map Frobenius splitting Relative effective Cartier divisor M. Artin, Lectures
Degeneration (algebraic geometry)
Degeneration_(algebraic_geometry)
named after Issai Schur. Frobenius–Schur indicator Herz–Schur multiplier Jordan–Schur theorem Lehmer–Schur algorithm Schur algebra Schur class Schur's conjecture
List of things named after Issai Schur
List_of_things_named_after_Issai_Schur
Mathematical group
known algebraic groups. Ree (1960, 1961) knew that the algebraic group B2 had an "extra" automorphism in characteristic 2 whose square was the Frobenius automorphism
Group_of_Lie_type
Branch of algebra that studies commutative rings
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both
Commutative_algebra
Submodule of a mathematical ring
non-associative rings. For algebras, we additionally assume that an ideal is a linear subspace. If a k {\displaystyle k} -algebra A {\displaystyle A} is unital
Ideal_(ring_theory)
Finite extension of the rationals
tools such as the Frobenius map, it is always possible to explicitly compute such a basis, and it is now standard for computer algebra systems to have built-in
Algebraic_number_field
Athanase Peltier Perlin noise – Ken Perlin Perron–Frobenius theorem – Oskar Perron, and Ferdinand Georg Frobenius Petkau effect – Abram Petkau Petri dish – Julius
Scientific phenomena named after people
Scientific_phenomena_named_after_people
Japanese mathematician
Tadasi Nakayama. Orthogonality relation for Frobenius- and quasi-Frobenius-algebras . Proc. Amer. Math. Soc. 3 (1952) 183–195. MR 0049876 doi:10
Tadashi Nakayama (mathematician)
Tadashi_Nakayama_(mathematician)
Branch of number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations
Algebraic_number_theory
Generalization of vector spaces from fields to rings
central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. In a vector space, the
Module_(mathematics)
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
Algebra over a field where binary multiplication is not necessarily associative
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative
Non-associative_algebra
Differential mapping
In mathematics, more specifically differential algebra, a p-derivation (for p a prime number) on a ring R, is a mapping from R to R that satisfies certain
P-derivation
from the theory of real and complex Lie algebras. This difference can be traced to the properties of Frobenius automorphism and to the failure of the exponential
Modular_Lie_algebra
Three-holed sphere
correspond to Frobenius algebras, where the circle (the only connected closed 1-manifold) maps to the underlying vector space of the algebra, while the pair
Pair_of_pants_(mathematics)
Concept in mathematics
G. Frobenius. Suppose G is a Frobenius group consisting of permutations of a set X. A subgroup H of G fixing a point of X is called a Frobenius complement
Frobenius_group
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
Mathematical concept
value of the Frobenius characteristic map ch {\displaystyle \operatorname {ch} } at f {\displaystyle f} , which is also called the Frobenius image of f
Frobenius_characteristic_map
Commutative group (mathematics)
abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally
Abelian_group
Commutative ring with no zero divisors other than zero
domain of prime characteristic p, then the Frobenius endomorphism x ↦ xp is injective. The Wikibook Abstract algebra has a page on the topic of: Integral domains
Integral_domain
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
Algebraic structure
{\displaystyle \mathrm {GF} (p)} . It is called the Frobenius automorphism, after Ferdinand Georg Frobenius. Denoting by φk the composition of φ with itself
Finite_field
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
Concept in ring theory
In mathematics, an Azumaya algebra is a generalization of central simple algebras to R {\displaystyle R} -algebras where R {\displaystyle R} need not
Azumaya_algebra
Describes the objects of a given type, up to some equivalence
Statement in abstract algebra Jordan normal form – Form of a matrix indicating its eigenvalues and their algebraic multiplicities Frobenius normal form – Canonical
Classification_theorem
Equivalence under a change of basis (linear algebra)
In linear algebra, two n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix P such that B = P − 1 A P . {\displaystyle
Matrix_similarity
Group of 𝑛 × 𝑛 invertible matrices
Semigroup Algebras. Springer Science & Business Media. 2.3: Full linear semigroup. ISBN 978-1-4020-5810-3. Meinolf Geck (2013). An Introduction to Algebraic Geometry
General_linear_group
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Albert–Brauer–Hasse–Noether theorem we may take to be a cyclic algebra (L,φ,πk) for some k mod n, where φ is the Frobenius map and π is a uniformiser. The invariant map
Hasse_invariant_of_an_algebra
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
Process of extending a representation of a subgroup to the parent group
program. Restricted representation Nonlinear realization Frobenius character formula Frobenius reciprocity, an important result that relates induced representations
Induced_representation
Type of ring in commutative algebra
related to Frobenius splitting: A Noetherian local ring A {\displaystyle A} of positive characteristic p is regular if and only if the Frobenius morphism
Regular_local_ring
History of maths
Categories of abstract algebraic structures including representation theory and universal algebra; Homological algebra; Homotopical algebra; Topology using categories
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
In mathematics, the Frobenius determinant theorem states that if one takes the multiplication table of a finite group G and replaces each entry g with
Frobenius_determinant_theorem
FROBENIUS ALGEBRA
FROBENIUS ALGEBRA
FROBENIUS ALGEBRA
FROBENIUS ALGEBRA
Girl/Female
Hebrew English
Lily.
Boy/Male
Teutonic American Italian English German
Dark skinned.
Girl/Female
Arabic, French, Indian, Muslim, Sindhi
Bringer of Good Tidings; Glad Tiding; Happy News; Joy
Boy/Male
British, English, Swedish
Down-bearded; Youth
Girl/Female
Hindu
Girl/Female
Biblical Native American
Sleep, a sacrifice of myrrh, ascension.
Boy/Male
Tamil
Lord Shiva
Boy/Male
Tamil
Smile
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi
Error-less
Girl/Female
Tamil
Decorated
FROBENIUS ALGEBRA
FROBENIUS ALGEBRA
FROBENIUS ALGEBRA
FROBENIUS ALGEBRA
FROBENIUS ALGEBRA
n.
Any particular system of characters, symbols, or abbreviated expressions used in art or science, to express briefly technical facts, quantities, etc. Esp., the system of figures, letters, and signs used in arithmetic and algebra to express number, quantity, or operations.
v. t.
To perform by algebra; to reduce to algebraic form.
n.
Either of the two parts of an algebraic equation, connected by the sign of equality.
a.
Susceptible of being solved; as, a soluble algebraic problem; susceptible of being disentangled, unraveled, or explained; as, the mystery is perhaps soluble.
n.
Anything which is required to be done; as, in geometry, to bisect a line, to draw a perpendicular; or, in algebra, to find an unknown quantity.
n.
One of the terms in an algebraic expression.
n.
A homogeneous algebraic function of two or more variables, in general containing only positive integral powers of the variables, and called quadric, cubic, quartic, etc., according as it is of the second, third, fourth, fifth, or a higher degree. These are further called binary, ternary, quaternary, etc., according as they contain two, three, four, or more variables; thus, the quantic / is a binary cubic.
a.
That can be passed over in a single course; -- said of a curve when the coordinates of the point on the curve can be expressed as rational algebraic functions of a single parameter /.
n.
An expression of the condition of equality between two algebraic quantities or sets of quantities, the sign = being placed between them; as, a binomial equation; a quadratic equation; an algebraic equation; a transcendental equation; an exponential equation; a logarithmic equation; a differential equation, etc.
n.
An algebraic curve, so called from its resemblance to a heart.
n.
That branch of algebra which treats of quadratic equations.
a.
A branch of algebra which relates to the direct search for unknown quantities.
adv.
By algebraic process.
a.
Alt. of Algebraical
a.
That may be sqyared, or reduced to an equivalent square; -- said of a surface when the area limited by a curve can be exactly found, and expressed in a finite number of algebraic terms.
n.
One versed in algebra.
a.
Of or pertaining to algebra; containing an operation of algebra, or deduced from such operation; as, algebraic characters; algebraical writings.
n.
A single algebraic expression; that is, an expression unconnected with any other by the sign of addition, substraction, equality, or inequality.
n.
The quotient of two vectors, or of two directed right lines in space, considered as depending on four geometrical elements, and as expressible by an algebraic symbol of quadrinomial form.
v. t.
To change, as an algebraic expression or geometrical figure, into another from without altering its value.