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Articles 1 - 6 of 6
Full-Text Articles in Algebra
What's New In Differentialgeometry Release Dg2022, Ian M. Anderson, Charles G. Torre
What's New In Differentialgeometry Release Dg2022, Ian M. Anderson, Charles G. Torre
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This Maple worksheet demonstrates the salient new features and functionalities of the 2022 release of the DifferentialGeometry software package.
Non-Isomorphic Real Simple Lie Algebras Of The Same Complex Type And Character, Ian M. Anderson
Non-Isomorphic Real Simple Lie Algebras Of The Same Complex Type And Character, Ian M. Anderson
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Complex simple Lie algebras are classified by their root types. The type of a real simple Lie algebra is the root type of the associated complex algebra. The character of a real simple Lie algebra is the signature of its Killing form.
For many root types, the character is sufficient to uniquely classify the corresponding real Lie algebras. However, one should not take this statement to be literally true – there are a few cases where the character does not suffice to distinguish all possible real forms.
In this worksheet we will show that the 2 real non-isomorphic Lie algebras …
A Rank 7 Pfaffian System On A 15-Dimensional Manifold With F4 Symmetry Algebra, Ian M. Anderson
A Rank 7 Pfaffian System On A 15-Dimensional Manifold With F4 Symmetry Algebra, Ian M. Anderson
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Let I be a differential system on a manifold M. The infinitesimal symmetry algebra of I is the set of all vectors fields X on M such that preserve I. In this worksheet we present an example, due to E. Cartan of a rank 7 Pfaffian system on a 15-dimensional manifold whose infinitesimal symmetry algebra is the split real form of the exceptional Lie algebra f4 .
Cartan Involutions And Cartan Decompositions Of A Semi-Simple Lie Algebra, Ian M. Anderson
Cartan Involutions And Cartan Decompositions Of A Semi-Simple Lie Algebra, Ian M. Anderson
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In this worksheet we shall review the basic definitions and properties of Cartan involutions and Cartan decompositions and illustrate these using the DifferentialGeometry software package for Lie algebras.
Jordan Algebras And The Exceptional Lie Algebra F4, Ian M. Anderson
Jordan Algebras And The Exceptional Lie Algebra F4, Ian M. Anderson
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This worksheet analyzes the structure of the Jordan algebra J(3, O) and its split and exceptional versions. The algebra of derivations is related to the exceptional Lie algebra f4.
Cartan Subalgebras, Compact Roots And The Satake Diagram For Su(2, 2), Ian M. Anderson
Cartan Subalgebras, Compact Roots And The Satake Diagram For Su(2, 2), Ian M. Anderson
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In this worksheet we use the 15-dimensional real Lie algebra su(2, 2) to illustrate some important points regarding the general structure theory and classification of real semi-simple Lie algebras.
1. Recall that a real semi-simple Lie algebra g is called a compact Lie algebra if the Killing form is negative definite. The Lie algebra g is compact if and only if all the root vectors for any Cartan subalgebra are purely imaginary. However, if the root vectors are purely imaginary for some choice of Cartan subalgebra it is not necessarily true that the Lie algebra is compact.
2. A real …