Questions tagged [lie-algebras]
Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
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Finite groups of Lie type, modular Lie algebras and root systems of positive characteristic
In the sources I know, the basic theory of root systems is always developed over $\mathbb{Z}$ or $\mathbb{R}$. This is the case even when algebraic groups $G$ over fields $K$ of positive ...
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Reductive fixed points in simple algebraic groups
Let $G$ be a connected simple algebraic group over $\mathbb{C}$.
Then $\mathrm{Aut}(G)$ is a linear algebraic group, so every
automorphism $\sigma \in \mathrm{Aut}(G)$ admits a Jordan
decomposition
$$
...
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Any known decomposition of the 240 4_21 vertices into four sets of 24, 84, 96, and 36 vertices?
I seem to remember that there is a known decomposition of 4_21 (or equivalently, the roots of E8) in which a set of 84 vertices plays a key role.
In any such decomposition, is there a "natural&...
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Does every element of the center $Z(\mathfrak g)$ acts by a scalar on each irreducible $\mathfrak g$-submodule of $V$?
Let $\mathfrak g$ be a Lie algebra over $\mathbb Q_p$ acting semisimply on a
finite-dimensional $\mathbb Q_p$-vector space $V$, that is, $V$ is a semi-simple $\mathfrak g$-module.
Does every element ...
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product on cohomology of a Lie algebra
Assume $g$ is a Lie algebra (f.dimensional for simplicity)
Then its cohomology with trivial coefficients can be defined as cohomology of the Chevalley-Eilenberg complex $\wedge^* (g^*)$ and by ...
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Does this parabolic subgroup act transitively on this subset of positive roots?
While our team was studying a geometric problem, we encountered a seemingly elementary Lie theoretic statement which turned out to be equivalent to our conjecture. The statement goes as follows.
Let $...
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Integrating ternary Lie algebras
Let $(\mathfrak g,[\,,\,,\,])$ be a ternary Lie algebra with a skew-symmetric multilinear bracket $[\,,\,,\,]:\mathfrak g\wedge\mathfrak g\wedge\mathfrak g\to\mathfrak g$ satisfying the Jacobi ...
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Rational all-pass interpolation of unitary-valued functions on the circle modulo torus–Weyl action
Let $\mathcal U(m)$ denote the unitary group, $T = \mathcal U(1)^m$ be the maximal torus of diagonal unitary matrices, and
$W = S_m$ be the Weyl group acting by column permutation.
Consider the right ...
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Has the hydrogen representation as minimal K-type the trivial representation?
The irreducible representations of the compact group $SO(4,R)$ are classified by pairs of so called “spin-quantum numbers” $(j_1, j_2 )$ with $j_1, j_2$ non-negative integers or half-integers.
The ...
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How many copies of the Standard Model gauge group are there in Spin(10)?
Define the Standard Model gauge group to be $\text{S}(\text{U}(2) \times \text{U}(3))$, the subgroup of $\text{SU}(5)$ consisting of block diagonal matrices with a $2 \times 2$ block and then a $3 \...
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Injectivity of derivations from the middle transvectant in the free Lie algebra on $\operatorname{Sym}^m$ for $\mathrm{SL}_2$
Let $G=\mathrm{SL}_2(\mathbb C)$, $V$ its standard representation, and $V_m=\operatorname{Sym}^m(V)$ with $m\equiv 2 \pmod 4$. It is classical that
$$
\Lambda^2 V_m \;\cong\; \bigoplus_{\substack{1\le ...
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Algebra isomorphic to its complex conjugate
Let $A$ be a finite-dimensional algebra (not necessarily unital nor associative, e.g. a Lie algebra) over $\mathbb{C}$. Define the complex conjugate $\overline{A}$ by choosing a basis for $A$ and ...
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How to realize this polytope as the weight polytope of a Demazure module
In $R^{2n-2}$ with basis $\Delta^+_1,\cdots,\Delta^+_{n-1},\Delta^-_1,\cdots,\Delta^-_{n-1}$, define a convex polytope with vertices $\Delta_i^++\Delta_{j}^-$ for all $0\leq i,j\leq n-1$ such that $i+...
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Simplicity for exterior powers of $\mathfrak{g}$-modules
Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Denote by $V_{\pi_m}$ the $m$-th fundamental representation of $\mathfrak{g}$. When is it true that the $k$-th exterior power of $\Lambda^k(V_{\...
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Even permutations on geodesic between even permutations $p,q\in \mathrm{SO}(n)$
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\supp{supp}$Let $n\in\mathbb{N}$ and $P(n)$ be the group of even permutations on $n$ symbols and $\SO(n)$ the special orthogonal group. It is well know ...
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