Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0704.0165

Frobenius-Schur indicators for semisimple Lie algebras

Mohammad Abu-Hamed·Shlomo Gelaki
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let g be a finite dimensional complex semisimple Lie algebra, and let V be a finite dimensional represenation of g. We give a closed formula for the mth Frobenius-Schur indicator, m>1, of V in representation-theoretic terms. We deduce that the indicators take integer values, and that for a large enough m, the mth indicator of V equals the dimension of the zero weight space of V. For the classical Lie algebras sl(n), so(2n), so(2n+1) and sp(2n), this is the case for m greater or equal to 2n-1, 4n-5, 4n-3 and 2n+1, respectively.

Explore related subjects

math.QAmath.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Mohammad Abu-Hamed, Shlomo Gelaki. 2007-04-02. Frobenius-Schur indicators for semisimple Lie algebras. https://arxiv.org/abs/0704.0165

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Classification of finite-dimensional Nichols algebras of rank two in twisted Yetter--Drinfeld categories

Let \(G\) be a finite non-abelian group, let \(Φ\in Z^3(G,\mathbb C^\times)\) be normalized, and let \(V,W\) be finite-dimensional simple objects of \({}_G^G\mathcal{YD}^Φ\). We classify, up to interchange, the braided-indecomposable pairs \((V,W)\) whose supports generate \(G\) and for which \(\mathcal B(V\oplus W)\) is finite-dimensional. The classification consists of eight cases with five possible support quandles. In every case the Cartan graph is standard of type \(A_2\), \(B_2\), or \(G_2\), and the dimension is determined explicitly. A new phenomenon occurs {in the \(Γ_2\) case}: the twisted setting admits a family of type \(G_2\) absent from the ordinary \(Γ_2\) classification and we construct an explicit example over a non-abelian group of order \(16\).

math.QA↗
arXiv2026

Facets of nondegeneracy: central type factor groups, Heisenberg representations and Hadamard matrices

Given a pair of complementary subgroups $P$ and $N$ of a finite group $G$, we first discuss how the same type of functions $\mathbb E\colon P\times N\to \mathbb T$, which we call bi-$1$-cocycles, appear naturally in the study of projective representations of $G$ of Heisenberg type and second cohomology of $G$ and of its dual quantum group $\widehat G$. We show then that such representations are irreducible and the corresponding ordinary and dual $2$-cocycles are nondegenerate if and only if the matrix $(\mathbb E(p,n))_{p,n}$ is invertible, if and only if it is Hadamard. We illustrate the result with examples arising from symplectic nilpotent Lie algebras over finite fields with two complementary Lagrangian subalgebras.

math.QA↗