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1. Only an introduction not rigorously involved

2. By the end of the presentation, listener should

be able to have some broad notion what LieAlgebra/Groups are, and why are they so famous

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E8

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- Left: Evariste Galois - one of the first to provide

foundation for group theory, who used it as a

connection to solving algebraic equations- Right: Our guy! Sophus Lie - Norwegian

mathematician - tried to apply Galois methods to

the problem of integrating differential equations in

1891

- Relation between integration/differentiation andgroups will be clear soon!

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http://en.wikipedia.org/wiki/Norwayhttp://en.wikipedia.org/wiki/Mathematicianhttp://en.wikipedia.org/wiki/Mathematicianhttp://en.wikipedia.org/wiki/Norway

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Why do study Lie groups? for you guys to keeplistening, I feel I have to give you an incentive. On a

side note, I really believe you cannot really bepassionate about something ONLY because it is new,there must be a reason grabbing your attention.1. one of the most beautiful and elegantmathematical theories - analyzes symmetry2. Lie groups have found extensive use in applicationssuch as formulation of

-symmetries of Hamiltonian systems (used byQuantum Mechanics)-the description of atomic, molecular and nuclearspectra,-theories of gravity (General Relativity)- superstring theory.

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Why are continuous groups special?- consider a set of elements that depend on a number ofreal continuous parameters, ()(1,2,...,).Theseelements are said to form a continuous group if they fulfillthe requirements of a group (rigorously mentioned later)and if there is some notion of `proximity' or `continuity'imposed on the elements of the group in the sense that asmall change in one of the factors of a product produces acorrespondingly small change in their product.-Like the introduction of dx in functions. Using dx, any

element y = x + ndx-- Unlike discrete groups, using these tools (later on, we willknow that a specific type of tool is very important thegroup infinitesimal generators) any element (ortransformation) can then be constructed by the repeatedapplication, or integration," of this tool

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We can treat the group using Algebra + Analysis!

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a Cartesian product is a mathematical operation

which returns a set (or product set) from multiple

sets. The Cartesian product is the result of crossingmembers of each set with one another.

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Derivation from document

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If is an analytic function, i.e., a function with a

convergent Taylor series expansion

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If is an analytic function, i.e., a function with a convergent Taylor seriesexpansion

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Where are Lie Algebras? Where are the tools?

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PDF!

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-instead of having to consider the group as a whole,for many purposes it is sufficient to consider an

infinitesimal transformation around the identity. Infact, with Lie groups, being continuous, it is possible ifwe start by an element, say the identity, then weapply such infinitesimal transformations again andagain, and eventually we could possible produce anyfinite transformation

--This process of applying the infinitesimaltransformations has been of tremendous importanceand they actually have occupied most of Lies results.In other words, they are the reason why Lie groupsare important.

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-instead of having to consider the group as a whole,for many purposes it is sufficient to consider an

infinitesimal transformation around the identity. Infact, with Lie groups, being continuous, it is possible ifwe start by an element, say the identity, then weapply such infinitesimal transformations again andagain, and eventually we could possible produce anyfinite transformation

--This process of applying the infinitesimaltransformations has been of tremendous importanceand they actually have occupied most of Lies results.In other words, they are the reason why Lie groupsare important.

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-instead of having to consider the group as a whole,for many purposes it is sufficient to consider an

infinitesimal transformation around the identity. Infact, with Lie groups, being continuous, it is possible ifwe start by an element, say the identity, then weapply such infinitesimal transformations again andagain, and eventually we could possible produce anyfinite transformation

--This process of applying the infinitesimaltransformations has been of tremendous importanceand they actually have occupied most of Lies results.In other words, they are the reason why Lie groupsare important.

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-instead of having to consider the group as a whole,for many purposes it is sufficient to consider an

infinitesimal transformation around the identity. Infact, with Lie groups, being continuous, it is possible ifwe start by an element, say the identity, then weapply such infinitesimal transformations again andagain, and eventually we could possible produce anyfinite transformation

--This process of applying the infinitesimaltransformations has been of tremendous importanceand they actually have occupied most of Lies results.In other words, they are the reason why Lie groupsare important.

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-instead of having to consider the group as a whole,for many purposes it is sufficient to consider an

infinitesimal transformation around the identity. Infact, with Lie groups, being continuous, it is possible ifwe start by an element, say the identity, then weapply such infinitesimal transformations again andagain, and eventually we could possible produce anyfinite transformation

--This process of applying the infinitesimaltransformations has been of tremendous importanceand they actually have occupied most of Lies results.In other words, they are the reason why Lie groupsare important.