Semisimple Lie algebra

Summary

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra \mathfrak g, if nonzero, the following conditions are equivalent: *\mathfrak g is semisimple; *the Killing form, κ(x,y) = tr(ad(x)ad(y)), is non-degenerate; *\mathfrak g has no non-zero abelian ideals; *\mathfrak g has no non-zero solvable ideals;

the radical (maximal solvable ideal) of \mathfrak g is zero.

Significance The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable ideal (its radical) and a semisimple algebra. In particular, there is no nonzero Lie alg

Official source

https://en.wikipedia.org/wiki/Semisimple_Lie_algebra

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The Center of Small Quantum Groups I: The Principal Block in Type A

Anna Lachowska

We develop an elementary algebraic method to compute the center of the principal block of a small quantum group associated with a complex semisimple Lie algebra at a root of unity. The cases of sl(3) and sl(4) are computed explicitly. This allows us to formulate the conjecture that, as a bigraded vector space, the center of a regular block of the small quantum sl(m) at a root of unity is isomorphic to Haiman's diagonal coinvariant algebra for the symmetric group S-m.

OXFORD UNIV PRESS2018

Remarks on the derived center of small quantum groups

Anna Lachowska

Let u(q)(g) be the small quantum group associated with a complex semisimple Lie algebra g and a primitive root of unity q, satisfying certain restrictions. We establish the equivalence between three different actions of g on the center of u(q)(g) and on the higher derived center of u(q)(g). Based on the triviality of this action for g = sl(2), sl(3), sl(4), we conjecture that, in finite type A, central elements of the small quantum group u(q)(Sl(n)) arise as the restriction of central elements in the big quantum group u(q)(sl(n)). We also study the role of an ideal z(Hig) known as the Higman ideal in the center of u(q)(g). We show that it coincides with the intersection of the Harish-Chandra center and its Fourier transform, and compute the dimension of z(Hig) in type A. As an illustration we provide a detailed explicit description of the derived center of u(q)(sl(2)) and its various symmetries.

SPRINGER INTERNATIONAL PUBLISHING AG2021

We classify the spherical birational sheets in a complex simple simply-connected algebraic group. We use the classification to show that, when G is a connected reductive complex algebraic group with simply-connected derived subgroup, two conjugacy classes O-1, O-2 of G, with O-1 spherical, lie in the same birational sheet, up to a shift by a central element of G, if and only if the coordinate rings of O-1 and O-2 are isomorphic as G-modules. As a consequence, we prove a conjecture of Losev for the spherical subvariety of the Lie algebra of G. (C) 2021 Elsevier Inc. All rights reserved.

ACADEMIC PRESS INC ELSEVIER SCIENCE2021