Semisimple Subalgebra Research Articles

Rigidity results for Lie algebras admitting a post-Lie algebra structure

We study rigidity questions for pairs of Lie algebras [Formula: see text] admitting a post-Lie algebra structure. We show that if [Formula: see text] is semisimple and [Formula: see text] is arbitrary, then we have rigidity in the sense that [Formula: see text] and [Formula: see text] must be isomorphic. The proof uses a result on the decomposition of a Lie algebra [Formula: see text] as the direct vector space sum of two semisimple subalgebras. We show that [Formula: see text] must be semisimple and hence isomorphic to the direct Lie algebra sum [Formula: see text]. This solves some open existence questions for post-Lie algebra structures on pairs of Lie algebras [Formula: see text]. We prove additional existence results for pairs [Formula: see text], where [Formula: see text] is complete, and for pairs, where [Formula: see text] is reductive with [Formula: see text]-dimensional center and [Formula: see text] is solvable or nilpotent.

Remarks on pseudo-vertex-transitive graphs with small diameter

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with vertex set $X$ and diameter $D$. Let $A$ denote the adjacency matrix of $\Gamma$. For a vertex $x\in X$ and for $0 \leq i \leq D$, let $E^*_i(x)$ denote the projection matrix to the $i$th subconstituent space of $\Gamma$ with respect to $x$. The Terwilliger algebra $T(x)$ of $\Gamma$ with respect to $x$ is the semisimple subalgebra of $\mathrm{Mat}_X(\mathbb{C})$ generated by $A, E^*_0(x), E^*_1(x), \ldots, E^*_D(x)$. Let $V$ denote a $\mathbb{C}$-vector space consisting of complex column vectors with rows indexed by $X$. We say $\Gamma$ is pseudo-vertex-transitive whenever for any vertices $x,y \in X$, there exists a $\mathbb{C}$-vector space isomorphism $\rho:V\to V$ such that $(\rho A - A \rho)V=0$ and $(\rho E^*_i(x) - E^*_i(y)\rho)V=0$ for all $0\leq i \leq D$. In this paper, we discuss pseudo-vertex transitivity for distance-regular graphs with diameter $D\in \{2,3,4\}$. For $D=2$, we show that a strongly regular graph is pseudo-vertex-transitive if and only if all its local graphs have the same spectrum. For $D = 3$, we consider the Taylor graphs and show that they are pseudo-vertex transitive. For $D=4$, we consider the antipodal tight graphs and show that they are pseudo-vertex transitive.

Lie Algebra of Killing Vector Fields and its Stationary Subalgebra

Let 𝔤 be the Lie algebra of all Killing vector fields on a locally homogeneous, analytic Riemannian manifold M, 𝔥 be a stationary subalgebra of 𝔤, G be the simply connected group generated by the algebra 𝔤, H be the subgroup of G generated by the subalgebra 𝔥, 𝔷 be the center of the algebra g, r be its radical, and [𝔤; 𝔤] be its commutator subgroup. If dim (𝔥 ∩ (𝔷 + [𝔤, 𝔤])) = dim (𝔥 ∩ [𝔤, 𝔤]), then H is closed in G. If for any semisimple subalgebra 𝔭 ⊂ 𝔤 satisfying the condition 𝔭 + 𝔯 = 𝔤, the relation (𝔭 + 𝔷) ∩ 𝔥 = 𝔭 ∩ 𝔥 holds, then H is closed in G. We also examine the analytic continuation of the given local, analytic Riemannian manifold.

Semisimple decompositions of Lie algebras and prehomogeneous modules

We study disemisimple Lie algebras, i.e., Lie algebras which can be written as a vector space sum of two semisimple subalgebras. We show that a Lie algebra g is disemisimple if and only if its solvable radical coincides with its nilradical and is a prehomogeneous s -module for a Levi subalgebra s of g . We use the classification of prehomogeneous s -modules for simple Lie algebras s given by Vinberg to show that the solvable radical of a disemisimple Lie algebra with simple Levi subalgebra is abelian. We extend this result to disemisimple Lie algebras having no simple quotients of type A .

Dispersing representations of semi-simple subalgebras of complex matrices

In this paper we consider the problem of determining the maximum dimension of P ⊥ ( A ⊕ B ) P , where A and B are unital, semi-simple subalgebras of the set M n of n × n complex matrices, and P ∈ M 2 n is a projection of rank n . We exhibit a number of equivalent formulations of this problem, including the one which occupies the majority of the paper, namely: determine the minimum dimension of the space A ∩ S − 1 B S , where S is allowed to range over the invertible group of M n . This problem in turn is seen to be equivalent to the problem of finding two automorphisms α and β of M n for which the dimension of α ( A ) + β ( B ) is maximised. It is this phenomenon which gives rise to the title of the paper.

Tensor Structure on the Kazhdan–Lusztig Category for Affine 𝔤𝔩(1|1)

AbstractWe show that the Kazhdan–Lusztig category $KL_k$ of level-$k$ finite-length modules with highest-weight composition factors for the affine Lie superalgebra $\widehat{\mathfrak{gl}(1|1)}$ has vertex algebraic braided tensor supercategory structure and that its full subcategory $\mathcal{O}_k^{fin}$ of objects with semisimple Cartan subalgebra actions is a tensor subcategory. We show that every simple $\widehat{\mathfrak{gl}(1|1)}$-module in $KL_k$ has a projective cover in ${\mathcal{O}}_k^{fin}$, and we determine all fusion rules involving simple and projective objects in ${\mathcal{O}}_k^{fin}$. Then using Knizhnik–Zamolodchikov equations, we prove that $KL_k$ and $\mathcal{O}_k^{fin}$ are rigid. As an application of the tensor supercategory structure on $\mathcal{O}_k^{fin}$, we study certain module categories for the affine Lie superalgebra $\widehat{\mathfrak{sl}(2|1)}$ at levels $1$ and $-\frac{1}{2}$. In particular, we obtain a tensor category of $\widehat{\mathfrak{sl}(2|1)}$-modules at level $-\frac{1}{2}$ that includes relaxed highest-weight modules and their images under spectral flow.

Tensor categories of affine Lie algebras beyond admissible levels

We show that if V is a vertex operator algebra such that all the irreducible ordinary V-modules are $$C_1$$ -cofinite and all the grading-restricted generalized Verma modules for V are of finite length, then the category of finite length generalized V-modules has a braided tensor category structure. By applying the general theorem to the simple affine vertex operator algebra (resp. superalgebra) associated to a finite simple Lie algebra (resp. Lie superalgebra) $$\mathfrak {g}$$ at level k and the category $$KL_k(\mathfrak {g})$$ of its finite length generalized modules, we discover several families of $$KL_k(\mathfrak {g})$$ at non-admissible levels k, having braided tensor category structures. In particular, $$KL_k(\mathfrak {g})$$ has a braided tensor category structure if the category of ordinary modules is semisimple or more generally if the category of ordinary modules is of finite length. We also prove the rigidity and determine the fusion rules of some categories $$KL_k(\mathfrak {g})$$ , including the category $$KL_{-1}(\mathfrak {sl}_n)$$ . Using these results, we construct a rigid tensor category structure on a full subcategory of $$KL_1(\mathfrak {sl}(n|m))$$ consisting of objects with semisimple Cartan subalgebra actions.

$${\varvec{\pi }}$$-systems of symmetrizable Kac–Moody algebras

As part of his classification of regular semisimple subalgebras of semisimple Lie algebras, Dynkin introduced the notion of a $$\pi $$ -system. This is a subset of the set of roots such that pairwise differences of its elements are not roots. Such systems arise as simple systems of regular semisimple subalgebras. Morita and Naito generalized this notion to all symmetrizable Kac–Moody algebras. In this work, we systematically develop the theory of $$\pi $$ -systems of symmetrizable Kac–Moody algebras and establish their fundamental properties. For several Kac–Moody algebras with physical significance, we study the orbits of the Weyl group on $$\pi $$ -systems and completely determine the number of orbits. In particular, we show that there is a unique $$\pi $$ -system of type $${A}^{++}_1$$ (the Feingold–Frenkel rank 3 hyperbolic algebra) in $$E_{10}$$ (the rank 10 hyperbolic algebra) up to Weyl group action and negation.

Computing the real Weyl group

Let g be a semisimple Lie algebra over the field of real numbers. Let G be a real Lie group with Lie algebra g. The real Weyl group of G with respect to a Cartan subalgebra h of g is defined as W(G,h)=NG(h)/ZG(h). We describe an explicit construction of W(G,h) for Lie groups G that arise as the set of real points of connected algebraic groups. We show that this also gives a construction of W(G,h) when G is the adjoint group of g. This algorithm is important for the classification of regular semisimple subalgebras, real carrier algebras, and real nilpotent orbits associated with g; the latter have various applications in theoretical physics.

Branches on division algebras

We describe the set of maximal orders in a 2-by-2 matrix algebra over a non-commutative local division algebra B containing a given suborder, for certain important families of such suborders, including rings of integers of division subalgebras of B or most maximal semisimple commutative subalgebras.

Integrals in Left Coideal Subalgebras and Group-Like Projections

We develop a theory of right group-like projections in Hopf algebras linking them with the theory of left coideal subalgebras with two sided counital integrals. Every right group-like projection is associated with a left coideal subalgebra, maximal among the ones containing the given group-like projection as an integral, and we show that that subalgebra is finite dimensional. We observe that in a semisimple Hopf algebra H every left coideal subalgebra has an integral and we prove a 1-1 correspondence between right group-like projections and left coideal subalgebras of H. We provide a number of equivalent conditions for a right group-like projections to be left group-like projection and prove a 1-1 correspondence between semisimple left coideal subalgebras preserved by the squared antipode and two sided group-like projections. We also classify left coideal subalgebras in Taft Hopf algebras H_{n^{2}} over a field {mathbbm {k}}, showing that the automorphism group splits them intoa class of cardinality |{mathbbm {k}}|-1 of semisimple ones which correspond to right group-like projections which are not two sided;finitely many semisimple singletons, each corresponding to two sided group-like projection; the number of those singletons for H_{n^{2}} is equal to the number of divisors of n;finitely many singletons, each non-semisimple and admitting no right group-like projection; the number of those singletons for H_{n^{2}} is equal to the number of divisors of n;In particular we answer the question of Landstad and Van Daele showing that there do exist right group-like projections which are not left group-like projections.

Semisimple subalgebras in simple Lie algebras and a computational approach to the compact Clifford–Klein forms problem

In this article, we develop the algorithm for finding homogeneous spaces of semisimple non-compact Lie groups which do not admit compact Clifford–Klein forms. We propose a computer program which checks if the given homogeneous space has a non-vanishing cohomological obstruction (found by Tholozan) to compact Clifford–Klein forms. By a numerical experiment, we show that there is a large class of homogeneous spaces satisfying Tholozan’s condition.

Glider representations of chains of semisimple Lie algebra

ABSTRACTWe start the study of glider representations in the setting of semisimple Lie algebras. A glider representation is defined for some positively filtered ring FR and here we consider the right bounded algebra filtration FU(𝔤) on the universal enveloping algebra U(𝔤) of some semisimple Lie algebra 𝔤 given by a fixed chain of semisimple Lie subalgebras . Inspired by the classical representation theory, we introduce so called Verma glider representations. Their existence is related to the relations between the root systems of the appearing Lie algebras 𝔤i. In particular, we consider chains of simple Lie algebras of the same type A,B,C and D.

Wedderburn–Malcev decomposition of one-sided ideals of finite dimensional algebras

ABSTRACTLet A be a finite dimensional associative algebra over a perfect field and let R be the radical of A. We show that for every one-sided ideal I of A there exists a semisimple subalgebra S of A such that where IS = I∩S and IR = I∩R.

Subalgebras of the rank two semisimple Lie algebras

ABSTRACTIn this expository article, we describe the classification of the subalgebras of the rank 2 semisimple Lie algebras. Their semisimple subalgebras are well-known, and in a recent series of papers, we completed the classification of the subalgebras of the classical rank 2 semisimple Lie algebras. Finally, Mayanskiy finished the classification of the subalgebras of the remaining rank 2 semisimple Lie algebra, the exceptional Lie algebra . We identify subalgebras of the classification in terms of a uniform classification scheme of Lie algebras of low dimension. The classification is up to inner automorphism, and the ground field is the complex numbers.

Minimal superalgebras generating minimal supervarieties

It has been shown that in characteristic zero the generators of the minimal supervarieties of finite basic rank belong to the class of minimal superalgebras introduced by Giambruno and Zaicev (Trans Am Math Soc 355:5091–5117, 2003). In the present paper the complete list of minimal supervarieties generated by minimal superalgebras whose maximal semisimple homogeneous subalgebra is the sum of three graded simple algebras is provided. As a consequence, we negatively answer the question of whether any minimal superalgebra generates a minimal supervariety.

Pseudo-direct sums and wreath products of loose-coherent algebras with applications to coherent configurations

We introduce the notion of a loose-coherent algebra, which is a special semisimple subalgebra of the matrix algebra, and define two operations to obtain new loose-coherent algebras from the old ones: the pseudo-direct sum and the wreath product. For two arbitrary coherent configurations C , D and their wreath product C ≀ D , it is difficult to express the Terwilliger algebra T ( x , y ) ( C ≀ D ) in terms of the Terwilliger algebras T x ( C ) and T y ( D ) . By using the concept and operations of loose-coherent algebras, we find a very simple such expression. As a direct consequence of this expression, we obtain the central primitive idempotents of T ( x , y ) ( C ≀ D ) in terms of the central primitive idempotents of T x ( C ) and T y ( D ) . Many results in [4] , [6] , [10] , [12] are special cases of the results in this paper.

The subalgebras of the rank two symplectic Lie algebra

The semisimple subalgebras of the rank 2 symplectic Lie algebra sp ( 4 , C ) are well-known, and we recently classified its Levi decomposable subalgebras. In this article, we classify the solvable subalgebras of sp ( 4 , C ) , up to inner automorphism. This completes the classification of the subalgebras of sp ( 4 , C ) . More broadly speaking, in completing the classification of the subalgebras of sp ( 4 , C ) we have completed the classification of the subalgebras of the rank 2 semisimple Lie algebras.

Integrable representations of root-graded Lie algebras

In this paper we study the category of representations of a root-graded Lie algebra L which are integrable as representations of a finite-dimensional semisimple subalgebra g and whose weights are bounded by some dominant weight of g . We link this category to the module category of an associative algebra, whose structure we determine for map algebras and sl n ( A ) . Our approach unifies recent work of Chari and her collaborators on map algebras, of Fourier and Savage and their collaborators on equivariant map algebras, as well as the classical work of Seligman on isotropic Lie algebras.

The Subalgebras of

We classify the solvable subalgebras, semisimple subalgebras, and Levi decomposable subalgebras of , up to inner automorphism. By Levi's Theorem, this is a full classification of the subalgebras of .