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One of my favorite parts about Category Theory is its generalization of commonly seen structures or interactions, and then being able to witness similar behavior in surprisingly different context. This blog post pertains to one particular categorification I like: monoid objects and module objects with respect to a monoid object.
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This blog post should be accompanied with this post on Hopf algebras, but the order is up to you. At least, I recommend you read them both before proceeding to a couple of my future blog posts where I discuss their connection. Of course, there is plenty of theory one can develop for one object independent from the other. The theory of Hopf algebras and of Tensor Categories is very rich and can be studied independently from each other if the reader wants.
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My first exposure to category theory was back in my first undergraduate course on abstract algebra. It was certainly ambitious of my professor to introduce students to group theory through a categorical lens, but I was absolutely hooked. Even now my research interests are very categorical in nature. However, there was one definition above all else that seemed a little more complicated to grasp: the notion of a limit.
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I first came across the notion of root systems from Humphreys. Leading up to that section, Humphreys presents the root space decomposition of a finite-dimensional semisimple Lie algebra, and at the end he notes some interesting properties that these roots have. These proeprties seem rather forced observatio a priori, but upon learning about the general theory of root systems, it becomes rather amazing that root systems do indeed arise in the theory of semisimple Lie algebras.
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The goal of this blog post is to present the definition and basic consequences therein of a Hopf algebra. For those that read more of my blog posts, this post and this one are some preliminary notions and results for some awesome blog posts to come. Without further ado, what is a Hopf algebra?
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I first came across the notion of root systems from Humphreys. Leading up to that section, Humphreys presents the root space decomposition of a finite-dimensional semisimple Lie algebra, and at the end he notes some interesting properties that these roots have. These proeprties seem rather forced observatio a priori, but upon learning about the general theory of root systems, it becomes rather amazing that root systems do indeed arise in the theory of semisimple Lie algebras.
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The study of complex semisimple Lie algebras is a very well understand and marvelous thing. From the study of the structure of a semisimple Lie algbera, to their classification, and to their representation theory via highest weights; I recommend Humphreys for the interested reader. To those with familiarity whether through a formal class or self-studying, there are certain structures along the way that should have made an impression on you. In particular, certain ideals of a Lie algebra are very prolific and determine much of the structure of the overall Lie algebra. These include
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I discussed that any finite-dimensional Lie algebra admits a decomposition into a semi-direct product of a semisimple Lie subalgebra and a solvable ideal (Levi decomposition). Thus, it is reasonable to work with semisimple Lie algebras, and indeed they happen to have some nice properties. This blog post explores one of the crucial properties of a complex finite-dimensional semisimple Lie algebra.
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I wrote in an earlier blog post that the adjoint representation of a Lie algebra was a very informative representation. This blog post will explore one of the quintessential notions of a Lie algebra which is formulated in terms of the Lie bracket: solvability.
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I wrote in an earlier blog post that the adjoint representation of a Lie algebra was a very informative representation. This blog post will explore one of the quintessential notions of a Lie algebra which is formulated in terms of the Lie bracket (deja vu): nilpotency.
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The goal of this blog post is to present some of the basic definitions and corresponding consequences of the structure of a Lie algebra. Some statements will be given with a citation to a proof in Humphreys. Others might be straightforward enough for the reader to work out, but in future posts I might work out the proof if it is enlightening or interesting enough.
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The first topic I pursued at the start of my doctoral studies was learning about the representation theory of complex semisimple Lie algebras. My main resource was Humphreys, which, while relatively small given the complexity of the topic, was a very clear and concise treatment of the material.
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One of my favorite parts about Category Theory is its generalization of commonly seen structures or interactions, and then being able to witness similar behavior in surprisingly different context. This blog post pertains to one particular categorification I like: monoid objects and module objects with respect to a monoid object.
Published:
This blog post should be accompanied with this post on Hopf algebras, but the order is up to you. At least, I recommend you read them both before proceeding to a couple of my future blog posts where I discuss their connection. Of course, there is plenty of theory one can develop for one object independent from the other. The theory of Hopf algebras and of Tensor Categories is very rich and can be studied independently from each other if the reader wants.
Published:
The first topic I pursued at the start of my doctoral studies was learning about the representation theory of complex semisimple Lie algebras. My main resource was Humphreys, which, while relatively small given the complexity of the topic, was a very clear and concise treatment of the material.
Published:
The first topic I pursued at the start of my doctoral studies was learning about the representation theory of complex semisimple Lie algebras. My main resource was Humphreys, which, while relatively small given the complexity of the topic, was a very clear and concise treatment of the material.
Published:
The first topic I pursued at the start of my doctoral studies was learning about the representation theory of complex semisimple Lie algebras. My main resource was Humphreys, which, while relatively small given the complexity of the topic, was a very clear and concise treatment of the material.