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Lapidus
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How come that a Lie algebra is defined as being non-associative and at same time it describes a Lie group locally? I wonder because groups are, again by definition, asscioative.
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The way it "describes a Lie group locally" is somewhat involved. In any case, the product operation of the Lie group and the bracket operation of the corresponding Lie algebra are by no means the same. Specifically, for a Lie group G, the corresponding Lie algebra is the tangent space of G at the identity, which 'is' the space of left invariant vector fields. The Lie bracket is the usual commutator of two vector fields. This is an entirely different operation than the group multiplication of G.Lapidus said:a Lie algebra is defined as being non-associative and at same time it describes a Lie group locally?
A non-associative Lie algebra is a mathematical structure that combines the concepts of a non-associative algebra and a Lie algebra. It is a vector space equipped with a bilinear operation and a Lie bracket operation, satisfying specific properties.
The main difference between a non-associative Lie algebra and a regular Lie algebra is the property of associativity. In a regular Lie algebra, the Lie bracket operation satisfies the associative property, while in a non-associative Lie algebra, this property does not hold.
Some examples of non-associative Lie algebras include the octonion algebra, the Jordan algebra, and the Malcev algebra. These algebras have important applications in fields such as physics, geometry, and mathematics.
Non-associative Lie algebras are used in physics to describe the symmetries and transformations of physical systems. They are particularly useful in the study of non-associative physical models, such as quantum mechanics and string theory.
Non-associative Lie algebras have important applications in mathematics, particularly in the study of algebraic structures. They provide a generalization of the classical Lie algebras and have connections to other areas such as representation theory, differential geometry, and topology.