Mathematics

Questions tagged [root-systems]

For questions involving abstract root systems, their associated Weyl groups and Dynkin diagrams, as well as their applications to Lie theory, graph theory, or other related fields.

Given a finite-dimensional Euclidean vector space V, a root system in V is a subset of nonzero vectors satisfying some fairly restrictive axioms. As such, in any given dimension there are only finitely many possible isomorphism classes of root systems.

Irreducible root systems are those which cannot be 'built' from smaller root systems in a definable way. Irreducible root systems are completely classified by their associated connected Dynkin diagrams, with four infinite families (called the classical root systems) and 5 exceptional cases.

Weyl groups are subgroups of the isometry group of root systems.

Cartan subalgebras (h) of semisimple Lie algebras (g) have a root system ($\Phi$) relation, as indicated by $\dim g = \dim h+ |\Phi|$.

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Proving that there exists a saturated set with given highest weight

This is an question about an exercise in Humphreys book on Lie algebras. First of all a bunch of definitions and notation, see §13 in Humphreys for details. Let $\Phi$ be a root system, $\Delta$ a base for $\Phi$. The set of weights of $\Phi$ is the…
spin
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Is there a definition of a dual Lie algebra?

Let $L$ be a Lie algebra. For vector spaces, modules, Banach spaces, etc. we have the notion of a dual. Question: Is it possible to define naturally a Lie algebra $L^*$ that is in some sense dual to $L$? If $L$ is semisimple over an algebraically…
spin
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Relation between root systems and representations of complex semisimple Lie algebras

I'm trying to understand the machinery of root systems for the purpose of classifying complex semisimple Lie algebras. During this process i lost the overview, espacially when it came to highest weight representations and all that, so i'm looking…
user83496
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How to prove that $B^\vee$ is a base for coroots?

Let $\Phi$ be a root system in a real inner product space $E$. Define $\alpha^\vee = \frac{2\alpha}{(\alpha, \alpha)}$. Then the set $\Phi^\vee = \{\alpha^\vee: \alpha \in \Phi \}$ is also a root system. Let $B$ be a base for the root system $\Phi$,…
spin
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Length of root strings is at most 4

Let $\Phi$ be a root system. In his Introduction to Lie algebras and Representation Theory , J. Humphreys proves that if $$\beta-p\alpha,\dots,\beta,\dots,\beta+q\alpha$$ is the $\alpha$-root string through $\beta$, then…
Klaus
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How to understand Weyl chambers?

Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane that is perpendicular to $\alpha$ the elements of…
user
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Finding a dominant root $\alpha$ for a semisimple, irreducible Lie-algebra $\mathfrak{g}$.

I´m working myself right now through this article and have trouble understanding a part of the proof of Proposition 6.4, case II) on page 345-346. I try to give a short outline of the situation: Suppose $V$ is an irreducible $\mathfrak{g}$ module,…
user12345
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Why is a root system called a "root" system?

Root systems plays an important role in, among other things, classifying semisimple Lie Algebras. Their name suggest that they have something to do with "roots" of a polynomial. Are they the roots of some polynomial? Where does the name " root…
Ma Joad
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How do I use the fact that $\langle \gamma, \lambda \rangle > 0$?

If I post all the details of my question up to the point where I am stuck, no one is going to want to read all that. So I hope there is someone reading this question who is familiar with Weyl chambers and systems of positive roots will immediately…
D_S
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$\Delta \subset \Phi$ is a base in a root system imples $\Delta^\vee \subset \Phi^\vee$ is a base in a root system

(the notation here is compatible with J.E. Humphrey's "Introduction to Lie Algebras and Representation Theory" ) Let $\Phi \subset E$ be a root system. Let $\Delta \subset \Phi$ be a base. I already know that $\Phi^\vee \subset E$ is a root system.…
Gils
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How to visualize intuitively root systems and Weyl group?

I am just learning about root systems for the first time and I am wondering how people visualize intuitively the notion of a root system and the Weyl group.
Patrick Fraser
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How can one assume that an isomorphism of root spaces $\Phi\to\Phi'$ comes from an isometry?

By definition, if $\Phi$ and $\Phi'$ are root systems of the Euclidean spaces $E$ and $E'$, respectively, then an isomorphism $\Phi\to\Phi'$ is one that is induced by an isomorphism $E\to E'$ which preserves the inner product on pairs of roots, but…
Kristina Thai
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There are no semisimple Lie algebras of dimension $4$, $5$, or $7$

I came across the claim here which states that there are no complex semisimple Lie algebras of dimension $4$ , $5$ , or $7$ . As the problem suggests, we can take a Cartan subalgebra $H$ and root system $\Phi$ so that…
Anonymous
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Recover Lie algebra bracket from the root system

I have coordinates of the $n$ roots in the $d$-dimensional space forming the root system of the $(n+d)$-dimensional Lie algebra. I want to implement the algorithm to recover the Lie bracket of the algebra. The Lie bracket between the elements of the…
Prof. Legolasov
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What is the root system of $\mathfrak{sl}(3,\mathbb C)$?

I want to determine the root-system of the lie algebra $sl(3,\mathbb C)$. Does someone know a good (and complete) reference for this problem? I know that the root-system is $A_2$ but I want to see a complete proof (calculation) for this. Thanks in…
Marc
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