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Matrices named after Élie Cartan
mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the
Cartan_matrix
248-dimensional exceptional simple Lie group
to the eight nodes in the Dynkin diagram in the order chosen for the Cartan matrix below, i.e., the nodes are read in the seven-node chain first, with
E8_(mathematics)
Simple Lie group; the automorphism group of the octonions
that is simply connected. The Dynkin diagram for G2 is given by . Its Cartan matrix is: [ 2 − 3 − 1 2 ] {\displaystyle \left[{\begin{array}{rr}2&-3\\-1
G2_(mathematics)
133-dimensional exceptional simple Lie group
corresponding root lattice, which has rank 7. The designation E7 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into
E7_(mathematics)
Studies linear representations of finite groups over fields of positive characteristic
transpose of D with D itself results in the Cartan matrix, usually denoted C; this is a symmetric matrix such that the entries in its j-th row are the
Modular_representation_theory
78-dimensional exceptional simple Lie group
generator, and is the sixth Cartan generator. One choice of simple roots for E6 is given by the rows of the following matrix, indexed in the order : [ 1
E6_(mathematics)
52-dimensional exceptional simple Lie group
1 {\displaystyle C_{2}=1} with three distinct principal curvatures, E. Cartan, 1939). The characters of finite dimensional representations of the real
F4_(mathematics)
Rotation group in 8-dimensional Euclidean space
In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space. It could be either a real or complex simple Lie group
SO(8)
Lie algebra, usually infinite-dimensional
that can be defined by generators and relations through a generalized Cartan matrix. These algebras form a generalization of finite-dimensional semisimple
Kac–Moody_algebra
Matrix with the same number of rows and columns
the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix. Cartan matrix Cayley–Hamilton theorem Brown 1991
Square_matrix
Lie algebra with imaginary simple roots
character formula, Cartan subalgebra, roots, weights, and so on. A symmetrized Cartan matrix is a (possibly infinite) square matrix with entries c i j
Generalized_Kac–Moody_algebra
Array of numbers
In mathematics, a matrix (pl.: matrices) is a rectangular array of numbers or other mathematical objects with elements or entries arranged in rows and
Matrix_(mathematics)
Pictorial representation of symmetry
generalized Cartan matrices, as shown in this table of rank 2 Dynkin diagrams with their corresponding 2 × 2 Cartan matrices. For rank 2, the Cartan matrix form
Dynkin_diagram
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
simply laced, but no group of type B, C, F, or G is simply laced. Cartan matrix Coxeter matrix Weyl group Coxeter group Kac–Moody algebra Catastrophe theory
Simple_Lie_group
The determinant of the Cartan matrix for En is 9 − n. E3 is another name for the Lie algebra A1A2 of dimension 11, with Cartan determinant 6. [ 2 − 1
En_(Lie_algebra)
Special quantum field theory
The sine-Gordon model is the model with the same Cartan matrix but an imaginary β. This Cartan matrix corresponds to the Lie algebra s u ( 2 ) {\displaystyle
Toda_field_theory
of a semisimple Lie algebra from a Cartan matrix can be generalized by weakening the definition of a Cartan matrix. The (generally infinite-dimensional)
Serre's theorem on a semisimple Lie algebra
Serre's_theorem_on_a_semisimple_Lie_algebra
Mathematical concept
In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information
Maurer–Cartan_form
Classical theory of gravitation
In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory, is a classical theory of gravitation, one of
Einstein–Cartan_theory
Generalized matrix decomposition for Lie groups and Lie algebras
transpose matrix of X {\displaystyle X} . The identity map on g {\displaystyle {\mathfrak {g}}} is an involution. It is the unique Cartan involution
Cartan_decomposition
such entries in the matrix are non-negative integers. The decomposition matrix, multiplied by its transpose, forms the Cartan matrix, listing the composition
Decomposition_matrix
Cartan relations Cartan map Cartan matrix Cartan pair Cartan subalgebra Cartan subgroup Cartan's method of moving frames Cartan's theorem, a name for the
List of things named after Élie Cartan
List_of_things_named_after_Élie_Cartan
Nilpotent subalgebra of a Lie algebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra h {\displaystyle {\mathfrak {h}}} of a Lie algebra g {\displaystyle
Cartan_subalgebra
Matrix equal to its conjugate-transpose
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose
Hermitian_matrix
Mathematical discipline
special case of their general construction of a quantum group from a Cartan matrix. One of their principal applications has been to the theory of solvable
Quantum_affine_algebra
Mathematical concept
except for type I0) The intersection matrix of the components. This is either a 1×1 zero matrix, or an affine Cartan matrix, whose Dynkin diagram is given.
Elliptic_surface
Matrix operation which flips a matrix over its diagonal
that flips a matrix over its diagonal; that is, transposition switches the row and column indices of the matrix A to produce another matrix, called the
Transpose
Mapping from p forms to p-1 forms
derivative of differential forms by the Cartan formula (also known as the Cartan identity, Cartan homotopy formula or Cartan magic formula): L X ω = d ( ι X ω
Interior_product
Concept in mathematics
are real. An extended Cartan matrix defines the unitary group. Shephard groups of rank n group have n generators. Ordinary Cartan matrices have diagonal
Complex_reflection_group
Group of unitary complex matrices with determinant of 1
its Dynkin diagram is given by An−1, a chain of n − 1 nodes: .... Its Cartan matrix is ( 2 − 1 0 … 0 − 1 2 − 1 … 0 0 − 1 2 … 0 ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 0 … 2 ) . {\displaystyle
Special_unitary_group
Type of Kac–Moody algebras
notion of height of a root. This allows definition of the extended Cartan matrix and extended Dynkin diagrams. The representation theory for affine Lie
Affine_Lie_algebra
Algebraic construct of interest in theoretical physics
Cartan matrix of the Kac–Moody algebra, and let q ≠ 0, 1 be a complex number, then the quantum group, Uq(G), where G is the Lie algebra whose Cartan matrix
Quantum_group
Pictorial representation of symmetry
because their root vectors are normalized. A is closely related to the Cartan matrix, used in the similar but directed graph: the Dynkin diagram, in the
Coxeter–Dynkin_diagram
Geometric arrangements of points, foundational to Lie theory
sublattice generated by long roots, D denotes the determinant of the Cartan matrix, and |W| denotes the order of the Weyl group. Let E be the subspace
Root_system
matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries
List_of_named_matrices
Direct sum of simple Lie algebras
i j ] 1 ≤ i , j ≤ l {\displaystyle [a_{ij}]_{1\leq i,j\leq l}} is a Cartan matrix). This is a theorem of Serre. In particular, two semisimple Lie algebras
Semisimple_Lie_algebra
Term in abstract algebra
\forall \lambda \in \Lambda :\phi _{\lambda }} is nondegenerate. The Cartan matrix C A {\displaystyle C_{A}} of A {\displaystyle A} is symmetric and positive
Cellular_algebra
Symmetric bilinear form in mathematics
that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show
Killing_form
Construction in graph theory
2-dimensional representation of H. If G has n irreducible characters, then the Cartan matrix cV of the representation V of dimension d is defined by c V = ( d δ
McKay_graph
Type of monoidal category
{\displaystyle D} is the biggest absolute value of an off-diagonal entry of the Cartan matrix of g {\displaystyle {\mathfrak {g}}} . From this quantum group it is
Modular_tensor_category
Matrices important in quantum mechanics and the study of spin
Pauli matrix is Hermitian, and together with the identity matrix I {\displaystyle \mathbb {I} } (sometimes considered as the zeroth Pauli matrix σ 0 {\displaystyle
Pauli_matrices
Russian mathematician
conditions were relaxed, it was still possible to associate to the Cartan matrix a Lie algebra which, necessarily, would be infinite dimensional." –
Victor_Kac
Canadian mathematician
conditions were relaxed, it was still possible to associate to the Cartan matrix a Lie algebra which, necessarily, would be infinite dimensional." -
Robert_Moody
Vector operation
In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element
Outer_product
German mathematician (1847–1923)
of classifying Lie groups, inventing the notions of a Cartan subalgebra and the Cartan matrix. He thus arrived at the conclusion that, basically, the
Wilhelm_Killing
generalized Cartan graphs as in: A generalized Cartan matrix c i j , i , j ∈ I {\displaystyle c_{ij},\;\;i,j\in I} is an integral matrix such that c i
Nichols_algebra
(\cdot ,\cdot )} is nondegenerate. 5. The Cartan matrix of the root system Φ {\displaystyle \Phi } is the matrix ( ⟨ α i , α j ⟩ ) i , j = 1 n {\displaystyle
Glossary of Lie groups and Lie algebras
Glossary_of_Lie_groups_and_Lie_algebras
Two-dimensional conformal field theory
of a Toda field theory, associated to the A 1 {\displaystyle A_{1}} Cartan matrix. More general conformal Toda theories can be viewed as generalizations
Liouville_field_theory
Shorthand notation for tensor operations
{\displaystyle u^{i}={A^{i}}_{j}v^{j}} This is a special case of matrix multiplication. The matrix product of two matrices A i j {\displaystyle A_{ij}} and B
Einstein_notation
Number line and triangular tiling's symmetry mathematical structure
(untwisted) type A n − 1 ( 1 ) {\displaystyle A_{n-1}^{(1)}} , with Cartan matrix [ 2 − 2 − 2 2 ] {\displaystyle \left[{\begin{array}{rr}2&-2\\-2&2\end{array}}\right]}
Affine_symmetric_group
Geometric transformation that preserves lines but not angles nor the origin
Springer-Verlag, ISBN 0-387-96752-4 Sharpe, R. W. (1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. New York: Springer. ISBN 0-387-94732-9
Affine_transformation
Manifold with inversion symmetry
inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric
Hermitian_symmetric_space
Type of matrix representation
complex matrix A {\displaystyle A} is a factorization of the form A = U P {\displaystyle A=UP} , where U {\displaystyle U} is a unitary matrix, and P {\displaystyle
Polar_decomposition
One of several theorems in different areas of mathematics
curves in Cartan-Hadamard manifolds. In linear algebra, Schur’s theorem is referred to as either the triangularization of a square matrix with complex
Schur's_theorem
Construct allowing differentiation of tangent vector fields of manifolds
tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used
Affine_connection
Mathematical function, in linear algebra
standard example of a linear map is an m × n {\displaystyle m\times n} matrix, which takes vectors in n {\displaystyle n} -dimensions into vectors in
Linear_map
Function in harmonic analysis on groups
that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G. The key examples are the matrix coefficients of the
Zonal_spherical_function
Associative algebra generalizing the Virasoro algebra
j = ( e i , e j ) {\displaystyle K_{ij}=(e_{i},e_{j})} given by the Cartan matrix of s l N {\displaystyle {\mathfrak {sl}}_{N}} , whose nonzero elements
W-algebra
group for more details. The Cartan decomposition writes a semisimple real Lie algebra as the sum of eigenspaces of a Cartan involution. The Iwasawa decomposition
Lie_group_decomposition
American mathematician
Philippe; Kedem, Rinat (2009). "Q-systems as Cluster Algebras II: Cartan Matrix of Finite Type and the Polynomial Property". Letters in Mathematical
Rinat_Kedem
Approach to general relativity
{\displaystyle B_{mn}} on the Lie group is the Cartan metric, aka the Killing form. Note that, as a matrix, the second W is the transpose. For N {\displaystyle
Tetrad_formalism
Group that is also a differentiable manifold with group operations that are smooth
Killing, later refined and generalized by Élie Cartan, led to classification of semisimple Lie algebras, Cartan's theory of symmetric spaces, and Hermann Weyl's
Lie_group
Universal construction of a complex Lie group from a real Lie group
complex characters of the Hopf algebra of representative functions, i.e. the matrix coefficients of finite-dimensional representations of the group. In any
Complexification_(Lie_group)
German mathematician
known today as matrix groups. A survey of his important contributions is contained in chapter V, §4 of (Borel 2001). Maurer–Cartan form Borel, Armand
Ludwig_Maurer
product – for entry-wise matrix multiplication Hadamard space – a geodesically complete metric space of non-positive curvature Cartan–Hadamard theorem – a
List of things named after Jacques Hadamard
List_of_things_named_after_Jacques_Hadamard
Non-tensorial representation of the spin group
quadratically from a spinor. Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to
Spinor
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
inertia matrix is a constant real symmetric matrix. A real symmetric matrix has the eigendecomposition into the product of a rotation matrix Q {\displaystyle
Moment_of_inertia
Algebraic structure used in analysis
of g l ( n , F ) {\displaystyle {\mathfrak {gl}}(n,F)} . Cartan's criterion (by Élie Cartan) gives conditions for a finite-dimensional Lie algebra of
Lie_algebra
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
models. In linear algebra, the n × n {\displaystyle n\times n} identity matrix I {\displaystyle \mathbf {I} } has entries equal to the Kronecker delta:
Kronecker_delta
Group theory theorem
In mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if H is
Closed-subgroup_theorem
Cauchy–Riemann equations Chapman–Kolmogorov equation Maurer–Cartan equation Pell's equation Poisson's equation Riccati equation sine-Gordon equation Verhulst
List_of_equations
Type of transport in differential geometry
j ) = 0 {\displaystyle (w_{j})=0} . Relative to the matrix representation above, the Maurer-Cartan form of G is a system of 1-forms ( ξ , α j , α j i
Projective_connection
Function in mathematics
geometry. (See, for example, (Cartan 1926) and (Cartan 1983).) Furthermore, using the dynamics of Gaston Darboux, Cartan was able to generalize the notion
Connection_(mathematics)
Mathematical operation on vector spaces
depending on how the tensor v ⊗ w {\displaystyle v\otimes w} is vectorized, the matrix describing the tensor product f ⊗ g {\displaystyle f\otimes g} is the Kronecker
Tensor_product
Algebra associated to any vector space
by members of the French geometry school (notably Henri Poincaré, Élie Cartan, and Gaston Darboux) who applied Grassmann's ideas to the calculus of differential
Exterior_algebra
Mathematical process dealing with Lie groups
the complexification of g 0 {\displaystyle {\mathfrak {g}}_{0}} . θ is a Cartan involution of g 0 {\displaystyle {\mathfrak {g}}_{0}} g 0 = k 0 ⊕ p 0 {\displaystyle
Iwasawa_decomposition
Statement about integration on manifolds
apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about the integration of differential forms on manifolds
Generalized_Stokes_theorem
Concept in differential geometry
} which is ρ ( Ω ) ⋅ ϕ {\displaystyle \rho (\Omega )\cdot \phi } by E. Cartan's structure equation. Besse 1987, Section 1.12; Kolář, Michor & Slovák 1993
Exterior_covariant_derivative
Theorem in the mathematics of Lie's theory
matrices via the matrix exponential yields a Lie group integrating the original Lie algebra. A more geometric proof is due to Élie Cartan and was published
Lie's_third_theorem
Algebraic operation on coordinate vectors
example in this way, a 1 × 3 matrix (row vector) is multiplied by a 3 × 1 matrix (column vector) to get a 1 × 1 matrix that is identified with its unique
Dot_product
Japanese mathematician (born 1944)
Genealogy Project Saito, Kyoji (1987). "A new relation among Cartan matrix and Coveter matrix". Journal of Algebra. 105 (1): 149–158. doi:10.1016/0021-8693(87)90183-9
Kyoji_Saito
Tensor that describes the 4D geometry of spacetime
symmetric matrix with entries g μ ν {\displaystyle g_{\mu \nu }} . The nondegeneracy of g μ ν {\displaystyle g_{\mu \nu }} means that this matrix is non-singular
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Index of articles associated with the same name
In mathematics, Cartan's lemma refers to a number of results named after either Élie Cartan or his son Henri Cartan: In exterior algebra: Suppose that
Cartan's_lemma
Concept in mathematics
{\displaystyle [h,f]=-2f} , and [ h , e ] = 2 e {\displaystyle [h,e]=2e} . This is a Cartan-Weyl basis for s l 2 C {\displaystyle {\mathfrak {sl}}_{2}\mathbb {C} }
Special_linear_Lie_algebra
Object in differential geometry
relativity theory, such ideas have been implemented in the form of Einstein–Cartan theory. Let M be a manifold with an affine connection on the tangent bundle
Torsion_tensor
Spin representations of the SO(3) group
matrix was formulated by Élie Cartan. In detail, given a vector x = (x1, x2, x3) of real (or complex) numbers, one can associate the complex matrix x
Spinors_in_three_dimensions
Functions on special groups related to their matrix representations
considered by Élie Cartan. Israel Gelfand realized that many classical special functions and orthogonal polynomials are expressible as the matrix coefficients
Matrix_coefficient
{\displaystyle {\mathfrak {g}}} , which in turn equals the dimension of some Cartan subalgebra h {\displaystyle {\mathfrak {h}}} (note that in earlier papers
Regular element of a Lie algebra
Regular_element_of_a_Lie_algebra
Math/physics concept
differential forms. Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principal
Connection_form
Type of group in mathematics
matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal
Orthogonal_group
Families of matrices in mathematics, physics, and quantum information
the identity matrix are called the generalized Gell-Mann matrices, in dimension d {\displaystyle d} . The symbol ⊕ (utilized in the Cartan subalgebra above)
Generalizations of Pauli matrices
Generalizations_of_Pauli_matrices
Vector behavior under coordinate changes
transformation described by an n × n {\displaystyle n\times n} invertible matrix M, so that the basis vectors transform according to [ e 1 ′ e 2 ′ .
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Exterior algebraic map taking tensors from p forms to n-p forms
k}V} through linearity. The Gram matrix G of Gram determinants is a 2 n × 2 n {\displaystyle 2^{n}\times 2^{n}} matrix allowing the inner product on the
Hodge_star_operator
Expression that may be integrated over a region
manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics
Differential_form
Basic result in harmonic analysis on compact topological groups
compact group. The theorem has three parts. The first part states that the matrix coefficients of irreducible representations of G are dense in the space
Peter–Weyl_theorem
Graphical notation for multilinear algebra calculations
The notation widely appears in modern quantum theory, particularly in matrix product states and quantum circuits. In particular, categorical quantum
Penrose_graphical_notation
function Free Riemann gas also called primon gas Riemann invariant Riemann–Cartan geometry Riemann–Silberstein vector Riemann-Lebovitz formulation Riemann
List of things named after Bernhard Riemann
List_of_things_named_after_Bernhard_Riemann
Basis for the SU(3) Lie algebra
are conventionally normalized. In this three-dimensional matrix representation, the Cartan subalgebra is the set of linear combinations (with real coefficients)
Gell-Mann_matrices
Type of mathematical functions
character. After 1945 important work in France, in the seminar of Henri Cartan, and Germany with Hans Grauert and Reinhold Remmert, quickly changed the
Function of several complex variables
Function_of_several_complex_variables
Klein's encyclopedia by Élie Cartan (1908) and in more detail by Hermann Rothe (1916). He was also concerned with the matrix theory of Arthur Cayley and
Arthur_Buchheim
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CARTAN MATRIX
CARTAN MATRIX
Surname or Lastname
English
English : habitational name from any of various places called Carleton or Carlton, from Old Norse karl ‘common man’, ‘peasant’ + Old English tūn ‘settlement’ (compare Charlton 1). Places spelled Carl(e)ton (as opposed to Charlton) are in areas of Scandinavian settlement, mostly in northern England.Irish : Americanized and altered form of Carlin 1.
Boy/Male
Gujarati, Hindu, Indian, Malayalam, Marathi, Punjabi, Sikh
Dance
Female
English
Irish Gaelic unisex name CARLIN means "little champion."
Biblical
a general (official title)
Surname or Lastname
French
French : from Old Norman French cardon ‘thistle’ (a diminutive of carde, from Latin carduus), hence a topographic name for someone who lived on land overgrown with thistles, an occupational name for someone who carded wool (originally a process carried out with thistles and teasels), or perhaps a nickname for a prickly and unapproachable person.French : possibly from a reduced form of the personal name Ricardon, a pet form of Richard.English : variant spelling of Carden, cognate with 1.
Female
English
Pet form of English Cara, CARITA means "beloved" or "friend."
Female
English
 19th-century English elaborated form of Latin cara, CARINA means "beloved." From the constellation Carina, from Latin carina, which originally meant "shell of a nut," later "keel of a ship."
Male
Italian
Italian form of Roman Latin Caietanus, CAJETAN means "from Caieta (Gaeta, Italy)."
Male
English
Variant spelling of English Charlton, CARLTON means "settlement of the free peasants."
Male
Hungarian
Hungarian form of Greek Bartholomaios, BARTAL means "son of Talmai."
Boy/Male
Armenian, Australian
Giver of Roses
Surname or Lastname
English
English : from an Old Norse personal name Kar(l)ma{dh}r (accusative Kar(l)mann), composed of the elements karl ‘male’, ‘man’ + ma{dh}r ‘man’, ‘person’.English : occupational name for a carter, from Anglo-Norman French, Middle English car(re) ‘cart’ (Late Latin carrus) + Middle English man ‘man’.Dutch : variant spelling of Karman.Altered spelling of Germann or Korman.
Boy/Male
Australian, French, Irish
Lord of the Castle
Boy/Male
Australian, French, Irish, Latin
Dedicated to Mars; Warlike
Boy/Male
Biblical
A general (official title).
Male
English
Anglicized form of Irish Gaelic Ciarán, CARRAN means "little black one."Â
Male
Portuguese
Portuguese form of Latin Caietanus, CAETANO means "from Caieta (Gaeta, Italy)."
Female
English
Feminine variant spelling of Irish Gaelic unisex Carlin, CARLYN means "little champion."Â
Boy/Male
Indian
Doer
Surname or Lastname
English
English : variant spelling of Barton.
CARTAN MATRIX
CARTAN MATRIX
CARTAN MATRIX
CARTAN MATRIX
CARTAN MATRIX
CARTAN MATRIX
CARTAN MATRIX
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