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E7½ (Lie Algebra)
In mathematics, the Lie algebra E7½ is a subalgebra of E8 (mathematics), E8 containing E7 (mathematics), E7 defined by Landsberg and Manivel in order to fill the "hole" in a dimension formula for the En (Lie algebra), exceptional series E''n'' of simple Lie algebras. This hole was observed by Predrag Cvitanović, Cvitanovic, Deligne, Cohen and de Man. E7½ has dimension 190, and is not simple: as a representation of its subalgebra E7, it splits as , where (56) is the 56-dimensional irreducible representation of E7. This representation has an invariant symplectic form, and this symplectic form equips with the structure of a Heisenberg algebra; this Heisenberg algebra is the Nilradical of a Lie algebra, nilradical in E7½. See also *Vogel plane References * A.M. Cohen, R. de Man, "Computational evidence for Deligne's conjecture regarding exceptional Lie groups", ''Comptes rendus de l'Académie des Sciences'', Série I 322 (1996) 427–432. * P. Deligne, "La série exceptionn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted [x,y]. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative property, associative. Lie algebras are closely related to Lie groups, which are group (mathematics), groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected space, connected Lie group unique up to finite coverings (Lie's third theorem). This Lie group–Lie algebra correspondence, correspondence allows one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E8 (mathematics)
In mathematics, E8 is 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 corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled A''n'', B''n'', C''n'', D''n'', and five exceptional cases labeled G2, F4, E6, E7, and E8. The E8 algebra is the largest and most complicated of these exceptional cases. Basic description The Lie group E8 has dimension 248. Its rank, which is the dimension of its maximal torus, is eight. Therefore, the vectors of the root system are in eight-dimensional Euclidean space: they are described explicitly later in this article. The Weyl group of E8, which is the group of symmetries of the maximal torus which are induced by conjugations in the whole group, has order 2357 = . The compact group E8 is unique ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E7 (mathematics)
In 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 notation E7 is used for the 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 four infinite series labeled A''n'', B''n'', C''n'', D''n'', and five exceptional cases labeled E6, E7, E8, F4, and G2. The E7 algebra is thus one of the five exceptional cases. The fundamental group of the (adjoint) complex form, compact real form, or any algebraic version of E7 is the cyclic group Z/2Z, and its outer automorphism group is the trivial group. The dimension of its fundamental representation is 56. Real and complex forms There is a unique complex Lie algebra of type E7, corresponding to a complex group of complex dimension 133. The complex adjoint Lie group E7 of complex dimension 133 can be considered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
En (Lie Algebra)
In mathematics, especially in Lie theory, E''n'' is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1, 2 and ''k'', with ''k'' = ''n'' − 4. In some older books and papers, ''E''2 and ''E''4 are used as names for ''G''2 and ''F''4. Finite-dimensional Lie algebras The E''n'' group is similar to the A''n'' group, except the nth node is connected to the 3rd node. So the Cartan matrix appears similar, -1 above and below the diagonal, except for the last row and column, have −1 in the third row and column. The determinant of the Cartan matrix for E''n'' is 9 − ''n''. *E3 is another name for the Lie algebra ''A''1''A''2 of dimension 11, with Cartan determinant 6. *:\left \begin 2 & -1 & 0 \\ -1 & 2 & 0 \\ 0 & 0 & 2 \end\right /math> *E4 is another name for the Lie algebra ''A''4 of dimension 24, with Cartan determinant 5. *:\left \begin 2 & -1 & 0 & 0 \\ -1 & 2 & -1& 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predrag Cvitanović
Predrag Cvitanović (; born April 1, 1946) is a theoretical physicist regarded for his work in nonlinear dynamics, particularly his contributions to periodic orbit theory. Life Cvitanović earned his B.S. from MIT in 1969 and his Ph.D. at Cornell University in 1973. Before joining the physics department at the Georgia Institute of Technology he was the director of the Center for Chaos and Turbulence Studies of the Niels Bohr Institute in Copenhagen. Cvitanović is a member of the Royal Danish Academy of Sciences and Letters, a corresponding member of Croatian Academy of Sciences and Arts, a recipient of the Research Prize of the Danish Physical Society, and a fellow of the American Physical Society. In 2009 Cvitanović was the recipient of the prestigious Alexander von Humboldt Prize for his work in turbulence theory. He currently holds the Glen P. Robinson Chair in Non-Linear Science in from Georgia Institute of Technology. Scientific work Perhaps his best-known work is h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal. Early life and education Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles (ULB), writing a dissertation titled ''Théorème de Lefschetz et critères de dégénérescence de suites spectrales'' (Theorem of Lefschetz and criteria of degeneration of spectral sequences). He completed his doctorate at the University of Paris-Sud in Orsay 1972 under the supervision of Alexander Grothendieck, with a thesis titled ''Théorie de Hodge''. Career Starting in 1972, Deligne worked with Grothendieck at the Institut des Hautes Études Scientifiques (IHÉS) near Paris, initially on the generalization within scheme theory of Zariski's main theorem. In 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreducible Representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W), with W \subset V closed under the action of \. Every finite-dimensional unitary representation on a Hilbert space V is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices is indecomposable but reducible. History Group representation theory was generalized by Richard Brauer from the 1940s to give modular representation theory, in which the matrix operators act on a vector space over a field K of arbitrary characteristic, rather than a vector space over the field of real numbers or o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Form
In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument separately; ; Alternating: holds for all ; and ; Non-degenerate: for all implies that . If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa. Working in a fixed basis, ''ω'' can be represented by a matrix. The conditions above are equivalent to this matrix being skew-symmetric, nonsingular, and hollow (all diagonal entries are zero). This should not be confused with a symplectic matrix, which represents a symplectic transformation of the space. If ''V'' is finite-dimensional, then its dimension must necessarily be even sinc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heisenberg Algebra
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ''a, b'' and ''c'' can be taken from any commutative ring with identity, often taken to be the ring of real numbers (resulting in the "continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group"). The continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems, especially in the context of the Stone–von Neumann theorem. More generally, one can consider Heisenberg groups associated to ''n''-dimensional systems, and most generally, to any symplectic vector space. The three-dimensional case In the three-dimensional case, the product of two Heisenberg matrices is given by: :\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end \begin 1 & a' & c'\\ 0 & 1 & b'\\ 0 & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilradical Of A Lie Algebra
In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible. The nilradical \mathfrak(\mathfrak g) of a finite-dimensional Lie algebra \mathfrak is its maximal nilpotent ideal, which exists because the sum of any two nilpotent ideals is nilpotent. It is an ideal in the radical \mathfrak(\mathfrak) of the Lie algebra \mathfrak. The quotient of a Lie algebra by its nilradical is a reductive Lie algebra \mathfrak^. However, the corresponding short exact sequence : 0 \to \mathfrak(\mathfrak g)\to \mathfrak g\to \mathfrak^\to 0 does not split in general (i.e., there isn't always a ''subalgebra'' complementary to \mathfrak(\mathfrak g) in \mathfrak). This is in contrast to the Levi decomposition: the short exact sequence : 0 \to \mathfrak(\mathfrak g)\to \mathfrak g\to \mathfrak^\to 0 does split (essentially because the quotient \mathfrak^ is semisimple). See also * Levi decomposition * Nilradical of a ring In algebra, the nilradical of a commutativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vogel Plane
In mathematics, the Vogel plane is a method of parameterizing simple Lie algebras by eigenvalues α, β, γ of the Casimir operator on the symmetric square of the Lie algebra, which gives a point (α: β: γ) of ''P''2/''S''3, the projective plane ''P''2 divided out by the symmetric group 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 operation is the composition of functions. In particular, the finite symmetric group ... ''S''3 of permutations of coordinates. It was introduced by , and is related by some observations made by . generalized Vogel's work to higher symmetric powers. The point of the projective plane (modulo permutations) corresponding to a simple complex Lie algebra is given by three eigenvalues α, β, γ of the Casimir operator acting on spaces ''A'', ''B'', ''C'', where the symmetric square of the Lie algebra (usually) decomposes as a sum of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
