SPECTRAL DECOMPOSITION OF R-MATRICES FOR EXCEPTIONAL LIE ALGEBRAS

AbstractThe exceptional simple Lie algebras of types E7 and E8 are endowed with optimal $\mathsf{SL}_2^n$ -structures, and are thus described in terms of the corresponding coordinate algebras. These are nonassociative algebras which much resemble the so-called code algebras.

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Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700023636 ◽

1985 ◽

Vol 38 (3) ◽

pp. 330-350 ◽

Cited By ~ 15

Author(s):

D. F. Holt ◽

N. Spaltenstein

Keyword(s):

Finite Field ◽

Lie Algebras ◽

Closed Field ◽

Nilpotent Orbits ◽

Reductive Algebraic Group ◽

Exceptional Lie Algebras ◽

Closed Fields ◽

Characteristic 3 ◽

Algebraically Closed Fields

AbstractThe classification of the nilpotent orbits in the Lie algebra of a reductive algebraic group (over an algebraically closed field) is given in all the cases where it was not previously known (E7 and E8 in bad characteristic, F4 in characteristic 3). The paper exploits the tight relation with the corresponding situation over a finite field. A computer is used to study this case for suitable choices of the finite field.

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Real Hamiltonian Forms of Affine Toda Models Related to Exceptional Lie Algebras

Symmetry Integrability and Geometry Methods and Applications ◽

10.3842/sigma.2006.022 ◽

2006 ◽

Author(s):

Vladimir S. Gerdjikov

Keyword(s):

Lie Algebras ◽

Exceptional Lie Algebras ◽

Hamiltonian Forms

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Spectral Decomposition of the Elasticity Tensor

Journal of Applied Mechanics ◽

10.1115/1.2894040 ◽

1992 ◽

Vol 59 (4) ◽

pp. 762-773 ◽

Cited By ~ 44

Author(s):

S. Sutcliffe

Keyword(s):

Spectral Decomposition ◽

Dimensional Space ◽

Anisotropic Elasticity ◽

Elasticity Tensor ◽

Symmetry Groups ◽

Stored Energy ◽

Equilibrium Equations ◽

Classical Symmetry ◽

Spectral Decompositions ◽

Acoustic Tensor

The elasticity tensor in anisotropic elasticity can be regarded as a symmetric linear transformation on the nine-dimensional space of second-order tensors. This allows the elasticity tensor to be expressed in terms of its spectral decomposition. The structures of the spectral decompositions are determined by the sets of invariant subspaces that are consistent with material symmetry. Eigenvalues always depend on the values of the elastic constants, but the eigenvectors are, in part, independent of these values. The structures of the spectral decompositions are presented for the classical symmetry groups of crystallography, and numerical results are presented for representative materials in each group. Spectral forms for the equilibrium equations, the acoustic tensor, and the stored energy function are also derived.

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On the minimal modules for exceptional Lie algebras: Jordan blocks and stabilizers

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157016000103 ◽

2016 ◽

Vol 19 (1) ◽

pp. 235-258 ◽

Cited By ~ 3

Author(s):

David I. Stewart

Keyword(s):

Algebraic Group ◽

Lie Algebras ◽

Nilpotent Orbit ◽

Closed Field ◽

Algebraically Closed Field ◽

Exceptional Lie Algebras ◽

Induced Module ◽

Simply Connected

Let $G$ be a simple simply connected exceptional algebraic group of type $G_{2}$, $F_{4}$, $E_{6}$ or $E_{7}$ over an algebraically closed field $k$ of characteristic $p>0$ with $\mathfrak{g}=\text{Lie}(G)$. For each nilpotent orbit $G\cdot e$ of $\mathfrak{g}$, we list the Jordan blocks of the action of $e$ on the minimal induced module $V_{\text{min}}$ of $\mathfrak{g}$. We also establish when the centralizers $G_{v}$ of vectors $v\in V_{\text{min}}$ and stabilizers $\text{Stab}_{G}\langle v\rangle$ of $1$-spaces $\langle v\rangle \subset V_{\text{min}}$ are smooth; that is, when $\dim G_{v}=\dim \mathfrak{g}_{v}$ or $\dim \text{Stab}_{G}\langle v\rangle =\dim \text{Stab}_{\mathfrak{g}}\langle v\rangle$.

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