Twisting Theory of Lie Algebras

Overview

This project will, firstly, summarize my summer research done under the supervision of Dr. Curtis Wendlandt. This will include the basic theories of Lie algebras and their representations, Lie coalgebras and bialgebras. I will then introduce some basic (co)homological ideas, and use them to re-state the definitions for Lie algebras and coalgebras in ways that will be useful for twisting theory. I intend on building the basic twisting conditions for Lie (co)algebras mostly from scratch using the restated definitions. I should note here that I define a twist to be a function $\rho$, so that given an algebra or coalgebra $(V,\psi)$, so that $\psi+\rho$ satisfies the same axioms as $\psi$, and $(V,\psi+\rho)$ is an algebra. The point at which these restated definitions will become useful is when we begin talking about twisting by one coboundaries, at which point we will also need the notion of a casimir element, the Killing form, and the classical $r$-matrix. I will then give proofs for theorem 8.1.7 and exercise 8.1.8 of Majid [6].

References

1. Christian Kassel, Quantum Groups (Springer, 1995). https://doi.org/10.1007/978-1-4612-0783-2
2. Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction (Springer, 2015). https://doi.org/10.1007/978-3-319-13467-3
3. David S. Dummit and Richard M. Foote, Abstract Algebra, 3rd ed. (John Wiley & Sons, 2004).
4. Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups (Cambridge University Press, 1994).
5. Edward Frenkel and David Ben-Zvi, Vertex Algebras and Algebraic Curves, 2nd ed., Mathematical Surveys and Monographs, vol. 88 (American Mathematical Society, 2004). https://doi.org/10.1090/surv/088
6. Shahn Majid, Foundations of Quantum Group Theory (Cambridge University Press, 1995).
7. Karin Erdmann and Mark J. Wildon, Introduction to Lie Algebras, Springer Undergraduate Mathematics Series (Springer, 2006). https://doi.org/10.1007/1-84628-490-2.