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...

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...

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...

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...

that can be defined by generators and relations through a generalized Cartan matrix. These algebras form a generalization of finite-dimensional semisimple...

choice of simple roots for F4, , is given by the rows of the following matrix: [ 0 1 − 1 0 0 0 1 − 1 0 0 0 1 1 2 − 1 2 − 1 2 − 1 2 ] {\displaystyle...

The designation E6 comes from the Cartan–Killing classification of the complex simple Lie algebras (see Élie Cartan § Work). This classifies Lie algebras...

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...

character formula, Cartan subalgebra, roots, weights, and so on. A symmetrized Cartan matrix is a (possibly infinite) square matrix with entries c i j...

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...

the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix. Cartan matrix Cayley-Hamilton theorem Brown 1991...

In mathematics, a matrix (pl.: matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows...

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...

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...

transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing...

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...

In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory, is a classical theory of gravitation, one of...

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...

In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra h {\displaystyle {\mathfrak {h}}} of a Lie algebra g {\displaystyle...

matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries...

In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure...

notion of height of a root. This allows definition of the extended Cartan matrix and extended Dynkin diagrams. The representation theory for affine Lie...

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...

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...

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...

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...

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...

such entries in the matrix are non-negative integers. The decomposition matrix, multiplied by its transpose, forms the Cartan matrix, listing the composition...

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....

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...