Splints of classical root systems

Let \(\) be a complex semisimple Lie algebra and let \(\), resp. \(\), be its symmetric, resp. enveloping algebra. It is shown in [1] that finite \(\)-algebras can be realized, up to a central extension, as the algebra of invariants \(\), resp. \(\), for the adjoint action of a Lie subalgebra \(\) of \(\). For \(\) nilpotent, we use the Taylor lemma to give a description (generators and relations) for these algebras.

It is well known that the Weyl groups play a key role in the theory of the structure of Lie algebras and their linear representations, as well as in the theory of Chevalley groups. Suppose g is a complex semisimple Lie algebra, h its Cartan subalgebra, Σ and Π denote its root system and the basis of the root system, respectively, and w α i indicates the reflection determined by αi ∈ Π. The so-called Weyl group, denoted by W,refers to the finite group generated by all w α i ( αi ∈ Π). There have been many works published on the structure and properties of W [6, 7]. The following Molien identity related to the global structure of W is well known [3]: $$\underset{w\in W}{\mathop{\Sigma}}\,\frac{1}{\det\left( 1-tw \right)}=\frac{1}{\prod{{{\left( 1-{{t}^{{{d}_{i}}}} \right)}^{,}}}}$$ (8.1) where d i signify the Poincare indices of g. In 1956, by topological methods based on Morse’s theory, Bott proved another identity [2]: $$\underset{w\in W}{\mathop \sum }\,{{t}^{l\left(w \right)}}=\prod{\frac{1-{{t}^{{{d}_{i}}}}}{1-{{t}^{,}}}}$$ (8.2) where l(w) refers to the Π-length of w;i.e.,wmay be expressed by the number of minimal products of w αi ( αi ∈ Π). This important formula may shed new light on our understanding the structure of W. In 1966, Solomon offered a purely algebraic proof of (8.2), thus solving the order problem of the Chevalley simple groups [5, 3].

Déformation par quantification et rigidité des algèbres enveloppantes

Generalized Serre relations for Lie algebras associated with positive unit forms

We associate to an arbitrary positive root [Formula: see text] of a complex semisimple finite-dimensional Lie algebra [Formula: see text] a twisting endofunctor [Formula: see text] of the category of [Formula: see text]-modules. We apply this functor to generalized Verma modules in the category [Formula: see text] and construct a family of [Formula: see text]-Gelfand–Tsetlin modules with finite [Formula: see text]-multiplicities, where [Formula: see text] is a commutative [Formula: see text]-subalgebra of the universal enveloping algebra of [Formula: see text] generated by a Cartan subalgebra of [Formula: see text] and by the Casimir element of the [Formula: see text]-subalgebra corresponding to the root [Formula: see text]. This covers classical results of Andersen and Stroppel when [Formula: see text] is a simple root and previous results of the authors in the case when [Formula: see text] is a complex simple Lie algebra and [Formula: see text] is the maximal root of [Formula: see text]. The significance of constructed modules is that they are Gelfand–Tsetlin modules with respect to any commutative [Formula: see text]-subalgebra of the universal enveloping algebra of [Formula: see text] containing [Formula: see text]. Using the Beilinson–Bernstein correspondence we give a geometric realization of these modules together with their explicit description. We also identify a tensor subcategory of the category of [Formula: see text]-Gelfand–Tsetlin modules which contains constructed modules as well as the category [Formula: see text].

Splint is a decomposition of root system into union of root systems. Splint of root system for simple Lie algebra appears naturally in studies of (regular) embeddings of reductive subalgebras. Splint can be used to construct branching rules. We consider special embedding of Lie subalgebra to Lie algebra. We classify projections of algebra root systems using extended Dynkin diagrams and single out the conditions of splint appearance and coincidence of branching coefficients with weight multiplicities. While such a coincidence is not very common it is connected with Gelfand-Tsetlin basis.

Given a semisimple (preferably simple) complex Lie algebra $L$, we consider the monoid $\Gamma=\Gamma(L)$ of equivalence classes of the finite dimensional reducible complex representations of $L$. Here $\Gamma$ is identified with the lattice of the corresponding highest weights. (This equips $\Gamma$ with the monoid structure.) For $\pi\in\Gamma$ one considers the symmetric algebra $\displaystyle S(\pi)=\bigoplus_{n=0}^{\infty}S^n(\pi)$ (here regarded as a representation). The elements of $\Gamma$ ``occurring'' in $S(\pi)$ -- i.e., which are the highest weights of some irreducible component of the representation $S(\pi)$ -- form a subsemigroup $M(\pi)$ of $\Gamma$. Such a $M(\pi)$ has a naturally defined rank $r(\pi)$ with $1\leq r(\pi)\leq r = \text{rank of }L$. In this paper we give a classification, for all the simple $L=A_r$ and $L=B_r$ of all the $\pi$ with $r(\pi)< r$.

In [10] Patera and Sharp conceived a new relation, subjoining, between semisimple Lie algebras. Our objective in this paper is twofold. Firstly, to lay down a mathematical formalization of this concept for arbitrary Lie algebras. Secondly, to give a complete classification of all maximal subjoinings between Lie algebras of the same rank, of which many examples were already known to the above authors.The notion of subjoining is a generalization of the subalgebra relation between Lie algebras. To give an intuitive idea of what is involved we take a simple example. Suppose is a complex simple Lie algebra of type B2. Let be a Cartan subalgebra of and Δ the corresponding root system. We have the standard root diagramInside B2 there lies the subalgebra A1 × A1 which can be identified with the sum of and the root spaces corresponding to the long roots of B2.

Goldie rank in the enveloping algebra of a semisimple Lie algebra, I

Invariant eigendistributions on a semisimple Lie algebra and homology classes on the conormal variety. I. An integral formula

The universal Lie algebra associatedto a root system I

The connection between Bessel series and SU(2) is reviewed from the standpoint of the outer multiplicity problem for this group. Its extension to any complex semisimple Lie algebras allows one to introduce new objects called ’’generalized Bessel series.’’ Some applications concerning the special function theory (addition theorems) are given.

We prove that the quantum moduli algebra associated to a possibly punctured compact oriented surface and a complex semisimple Lie algebra \mathfrak{g} is a Noetherian and finitely generated ring. If the surface has punctures, we also prove that it has no non-trivial zero divisors (i.e., it is a domain). Moreover, we show that the quantum moduli algebra is isomorphic to the skein algebra of the surface, defined by means of the Reshetikhin–Turaev functor for the quantum group U_{q}(\mathfrak{g}) , and which coincides with the Kauffman bracket skein algebra when \mathfrak{g}=\mathfrak{sl}_{2} . We obtain these results by a similar study of quantum graph algebras, which we show to be isomorphic to stated skein algebras.

Sur la méthode des orbites pour les orbites régulières d'une algèbre de Lie semi-simple

Let \(\mathfrak {g}\) be a complex semisimple Lie algebra and \(Y_{\hbar }(\mathfrak {g})\) its Yangian. Drinfeld proved that the universal R-matrix \(\mathcal {R}(s)\) of \(Y_{\hbar }(\mathfrak {g})\) gives rise to rational solutions of the \(\operatorname {QYBE}\) on irreducible, finite-dimensional representations of \(Y_{\hbar }(\mathfrak {g})\). This result was recently extended by Maulik–Okounkov to symmetric Kac–Moody algebras and representations arising from geometry. We show that rationality ceases to hold on arbitrary finite-dimensional representations, if one requires such solutions to be natural and compatible with tensor products. Equivalently, the tensor category of finite-dimensional representations of \(Y_{\hbar }(\mathfrak {g})\) does not admit rational commutativity constraints. We construct instead two meromorphic commutativity constraints, which are related by a unitarity condition. Each possesses an asymptotic expansion in s which has the same formal properties as \(\mathcal {R}(s)\), and therefore coincides with it by uniqueness. In particular, we give a constructive proof of the existence of \(\mathcal {R}(s)\). Our construction relies on the Gauss decomposition \(\mathcal {R}^+(s)\cdot \mathcal {R}^0(s)\cdot \mathcal {R}^-(s)\) of \(\mathcal {R}(s)\). The divergent abelian term \(\mathcal {R}^0\) was resummed on finite-dimensional representations by the first two authors in Gautam and Toledano Laredo (Publ Math Inst Hautes Études Sci 125:267–337, 2017). In the present paper, we construct \(\mathcal {R}^{\pm }(s)\), prove that they are rational on finite-dimensional representations, and that they intertwine the standard coproduct of \(Y_{\hbar }(\mathfrak {g})\) and the deformed Drinfeld coproduct introduced in loc. cit.