Questions tagged [exceptional-groups]
Exceptional Lie groups G2, F4, E6, E7, E8 of dimensions 14, 52, 78, 133, 248 were obtained as result of classification of simple Lie groups performed by Killing and Elie Cartan. The tool used in classification is Dynkin diagram and root system of vectors in Lie algebra of the group. The remaining Lie groups form four infinite families of transformations of n-dimensional space over real (odd and even), complex and quaternionic field.
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The Grassmannian Gr(2,8) and an E7 surprise
Are there any mathematical explanations for the following surprising facts?
$$\int_{Gr(2,8)} c_{\text{top}}(TX(-2)) = 6556 = \frac{1}{2} \deg(E_7/P(\alpha_7)) + 1,$$
and
$$\int_{Gr(2,6)} c_{\text{top}}...
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Does Deligne's exceptional series lead to an "exceptional K-theory"?
To a certain extent, Deligne's exceptional series $A_1 \subset A_2 \subset G_2 \subset D_4 \subset F_4 \subset E_6 \subset E_7 \subset E_8$ plays a role analogous to the classical series $A_n \subset ...
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Freudenthal geometries for exceptional simple Lie groups
When reading answer to this question I recall Freudenthal, Lie groups and foundations of geometry, 1964. In chapter 4 he describes 2-dim elliptic geometry, 2-dim projective geometry, 5-dim symplectic ...
user21230
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A few questions about $E_6$ and its symmetric spaces
Preface
The purpose of my question - on high level - is to understand exceptional symmetric spaces. My latest idea is to embed them into Lie group. There is quite nice embedding of 32-dimensional $E_{...
user21230
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Branching rules for E6 into SU(3)^3
I am very confused about what are the branching rules for representations of $E6$ into a $SU(3)\times SU(3)\times SU(3)$ subgroup. At least in the physics literature, there seems to be a serious ...
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Adjoint orbits of a finite group of type $G_2$
Let $q=p^\alpha$ be a prime power and $k=\mathbb{F}_q$. Let $G\subseteq \mathrm{GL}_N(k)$ be a simple finite group of Lie type, with root system of type $G_2$, and let $\mathfrak{g}\subseteq \mathfrak{...
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Exceptional symmetric spaces embedded in exceptional Lie group
In Yokota (1959) and Atsuyama (1977) papers one can find embedding of projective space $\mathbb OP^2$ into Lie group $F_4$. Lately I come to following idea to have embedding of all four projective ...
user21230
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Better names for Lie groups
After reading this question I was wondering whether mathematicians tried to invent better names for exceptional simple Lie groups $F_4, E_6, E_7, E_8$ ? These names seems a bit obscure and does not ...
user21230
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What is known about the cohomology of the U-duality group?
$\newcommand{\Es}{E_{7(7)}}\newcommand{\Z}{\mathbb Z}$Let $\Es$ denote the split form of $E_7$, which is a real Lie
group. It can be characterized as the subgroup of $\mathrm{Sp}_{56}(\mathbb R)$ ...
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Is there a smooth Weyl equivariant map from this quotient space into $G_2/T^2$?
It is known that $G_2$ acts transitively on $S^6$ with fibers $SU(3)$. Let us consider the following set $P$ of complex unitary $7 \times7$ matrices $A$, where
$$ A = (v_0, \, v_1, \, v_2, \, v_3, \, ...
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Parabolic Bruhat graphs for exceptional types
I am looking for some computer software or a reference for some parabolic Bruhat graphs. In particular, what I really need $E_8 \setminus E_7$. Does anyone know where or how I'd find this?
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Exceptional symmetric spaces with quaternionic structure
Following this and this question I found following chain of exceptional symmetric spaces being quaternionic manifolds. I listed dimensions as superscripts for reader convenience.
$F_{I}^{28}\subset ...
user21230
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A few questions about $E_7$ and its symmetric spaces
My question about $E_6$ survived, so I post next episode. From the Yokota book I found out that there is $-1$ in $E_7$ Lie group. This book defines Lie group $E_7$ using 56-dimensional Freudenthal ...
user21230
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In how many ways does a Lie algebra decompose as an orthogonal direct sum of Cartans?
For a prime $p$, the Lie algebra $\mathfrak{su}(p)$ can be decomposed into an orthogonal direct sum of $p+1$ Cartan subalgebras as follows. Consider the clock and shift matrices — these are a pair of ...
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The special embedding $\mathfrak{so}(7)\subset\mathfrak{so}(8)$
It is commonly known that we have a chain of embeddings
$$SU(4)\subset Spin(7)\subset SO(8)$$
(there is more than one possible $Spin(7)$, just take one).
Which is the explicit analog for the Lie ...
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Do Weyl groups generate the exceptional Lie groups as sequences of reflexions in the Weyl chambers?
Platonic groups of symmetry are Weyl groups for the exceptional Lie algebra E6->E8, as root systems. These can be viewed as mirrors in a kaleidoscope (Goodman).
I would like to know if one can ...
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Lie Groups and Lie algebras related to Jordan algebras
Let $J$ be a Jordan algebra. I knew three relative Lie groups/Lie algebras to $J$.
In the paper "The Capelli Identity, Tube Domains, and the Generalized Laplace Transform"
Jacobson [J] has ...
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Reference request: Commutator relations for the exceptional group F4
Is there any standard reference for the commutator relations for the exceptional group of type $F_4$?
If this question is not appropriate here, please let me know and I will delete it.
Thanks in ...
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How the exceptional simple Lie groups/ algebras were first discovered and by whom?
I am wondering whether exceptional simple Lie groups/ algebras were first discovered in order to obtain a complete list of such objects, or they appeared as answers to completely different questions.
...
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