Reductive Subalgebra Research Articles

Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic

Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic

Category {mathcal {O}} for Takiff Lie algebras

We study category {mathcal {O}} for Takiff Lie algebras {mathfrak {g}}otimes {mathbb {C}}[epsilon ]/(epsilon ^2) where {mathfrak {g}} is the Lie algebra of a reductive algebraic group over {mathbb {C}}. We decompose this category as a direct sum of certain subcategories and use an analogue of parabolic induction functors and twisting functors for BGG category {mathcal {O}} to prove equivalences between these subcategories. We then use these equivalences to compute the composition multiplicities of the simple modules in the Verma modules in terms of composition multiplicities in the BGG category {mathcal {O}} for reductive subalgebras of {mathfrak {g}}. We conclude that the composition multiplicities are given in terms of the Kazhdan–Lusztig polynomials.

Branching Rules for Splint Root Systems

A root system is splint if it is a decomposition into a union of two disjoint root systems. Examples of such root systems arise naturally in studying embeddings of reductive Lie subalgebras into simple Lie algebras. Given a splint root system, one can try to understand its branching rule. In this paper we discuss methods to understand such branching rules, and give precise formulas for specific cases, including the restriction functor from the exceptional Lie algebra \(\mathfrak {g}_{2}\) to \(\mathfrak {sl}_{3}\).

MONOMIAL BASES AND BRANCHING RULES

Following a question of Vinberg, a general method to construct monomial bases for finite-dimensional irreducible representations of a reductive Lie algebra was developed in a series of papers by Feigin, Fourier, and Littelmann. Relying on this method, we construct monomial bases of multiplicity spaces associated with the restriction of the representation to a reductive subalgebra. As an application, we produce monomial bases for representations of the general linear and symplectic Lie algebras associated with natural chains of subalgebras. We also show that our basis in type A is related to both the Gelfand-Tsetlin basis and the Littelmann basis via triangular transition matrices which implies that the triangularity property extends to the matrix connecting the Gelfand-Tsetlin and canonical bases. A similar relationship holds between our basis in type C and a suitably modified version of the basis constructed earlier by the first author.

On some classes of $${\mathbb {Z}}$$-graded Lie algebras

We study finite dimensional almost- and quasi-effective prolongations of nilpotent $${\mathbb {Z}}$$ -graded Lie algebras, especially focusing on those having a decomposable reductive structural subalgebra. Our assumptions generalize effectiveness and algebraicity and are appropriate to obtain Levi–Malcev and Levi–Chevalley decompositions and precisions on the heigth and other properties of the prolongations in a very natural way. In a last section we consider the semisimple case and discuss some examples in which the structural algebras are central extensions of orthogonal Lie algebras and their degree $$(-\,1)$$ components arise from spin representations.

Real forms of embeddings of maximal reductive subalgebras of the complex simple Lie algebras of rank up to 8

We consider the problem of determining the noncompact real forms of maximal reductive subalgebras of complex simple Lie algebras. We briefly describe two algorithms for this purpose that are taken from the literature. We discuss applications in theoretical physics of these embeddings. The supplementary material to this paper contains the tables of embeddings that we have obtained for all real forms of the semisimple Lie algebras of rank up to 8.

On an Infinite Limit of BGG Categories O

We study a version of the BGG category O for Dynkin Borel subalgebras of root-reductive Lie algebras, such as g=gl(infinity). We prove results about extension fullness and compute the higher extensions of simple modules by Verma modules. We also show that the category is Ringel self-dual and initiate the study of Koszul duality. An important tool in obtaining these results is an equivalence we establish between appropriate Serre subquotients of category O for g and category O for finite dimensional reductive subalgebras of g.

Splints of root systems of basic Lie superalgebras

Splints [9] of root systems of simple Lie algebras appear naturally on the studies of embedding of reductive subalgebras. A splint can be used to construct branching rules, as implementation of this idea simplifies calculation of branching coefficients. We extend the concept of splints to basic Lie superalgebras case as these algebras have wide applications in physics. In this context we have determined the splints of root system of all basic Lie superalgebras and hope to contribute towards a small step in the direction of representation of these algebras.

Expanding solitons to the Hermitian curvature flow on complex Lie groups

We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. Furthermore, we give a structural result concerning expanding semi-algebraic solitons on complex Lie groups. It turns out that the restriction of the soliton metric to the nilradical is also an expanding algebraic soliton and we explain how to construct expanding solitons on complex Lie groups starting from expanding solitons on their nilradicals.

CLASSIFICATION OF REDUCTIVE REAL SPHERICAL PAIRS II. THE SEMISIMPLE CASE

If ${\mathfrak g}$ is a real reductive Lie algebra and ${\mathfrak h} < {\mathfrak g}$ is a subalgebra, then $({\mathfrak g}, {\mathfrak h})$ is called real spherical provided that ${\mathfrak g} = {\mathfrak h} + {\mathfrak p}$ for some choice of a minimal parabolic subalgebra ${\mathfrak p} \subset {\mathfrak g}$. In this paper we classify all real spherical pairs $({\mathfrak g}, {\mathfrak h})$ where ${\mathfrak g}$ is semi-simple but not simple and ${\mathfrak h}$ is a reductive real algebraic subalgebra. The paper is based on the classification of the case where ${\mathfrak g}$ is simple (see arXiv:1609.00963) and generalizes the results of Brion and Mikityuk in the (complex) spherical case.

Integrable sl(∞)‐modules and category O for gl(m|n)

We introduce and study new categories T(g,k)of integrable sl(\infty)-modules which depend on the choice of a certain reductive subalgebra k in g=sl(\infty). The simple objects of these categories are tensor modules as in the previously studied category, however, the choice of k provides more flexibility of nonsimple modules. We then choose k to have two infinite-dimensional diagonal blocks, and show that a certain injective object K(m|n) in T(g,k) realizes a categorical sl(\infty)-action on the integral category O(m|n) of the Lie superalgebra gl(m|n). We show that the socle of K(m|n) is generated by the projective modules in O(m|n), and compute the socle filtration of K(m|n) explicitly. We conjecture that the socle filtration of K(m|n) reflects a "degree of atypicality filtration" on the category O(m|n). We also conjecture that a natural tensor filtration on K(m|n) arises via the Duflo--Serganova functor sending the category O(m|n) to O(m-1|n-1). We prove this latter conjecture for a direct summand of K(m|n) corresponding to the finite-dimensional gl(m|n)-modules.

On the classification of non-equal rank affine conformal embeddings and applications

We complete the classification of conformal embeddings of a maximally reductive subalgebra $$\mathfrak {k}$$ into a simple Lie algebra $$\mathfrak {g}$$ at non-integrable non-critical levels k by dealing with the case when $$\mathfrak {k}$$ has rank less than that of $$\mathfrak {g}$$ . We describe some remarkable instances of decomposition of the vertex algebra $$V_{k}(\mathfrak {g})$$ as a module for the vertex subalgebra generated by $$\mathfrak {k}$$ . We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. In particular, we study an example of conformal embeddings $$A_1 \times A_1 \hookrightarrow C_3$$ at level $$k=-1/2$$ , and obtain explicit branching rules by applying certain q-series identity. In the analysis of conformal embedding $$A_1 \times D_4 \hookrightarrow C_8$$ at level $$k=-1/2$$ we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs.

CLASSIFICATION OF REDUCTIVE REAL SPHERICAL PAIRS I. THE SIMPLE CASE

This paper gives a classification of all pairs $(\mathfrak g, \mathfrak h)$ with $\mathfrak g$ a simple real Lie algebra and $\mathfrak h < \mathfrak g$ a reductive subalgebra for which there exists a minimal parabolic subalgebra $\mathfrak p < \mathfrak g$ such that $\mathfrak g = \mathfrak h + \mathfrak p$ as vector sum.

Finite vs. Infinite Decompositions in Conformal Embeddings

Building on work of the first and last author, we prove that an embedding of simple affine vertex algebras $V_{\mathbf{k}}(\mathfrak g^0)\subset V_{k}(\mathfrak g)$, corresponding to an embedding of a maximal equal rank reductive subalgebra $\mathfrak g^0$ into a simple Lie algebra $\mathfrak g$, is conformal if and only if the corresponding central charges are equal. We classify the equal rank conformal embeddings. Furthermore we describe, in almost all cases, when $V_{k}(\mathfrak g)$ decomposes finitely as a $V_{\mathbf{k}}(\mathfrak g^0)$-module.

Nilradicals of parabolic subalgebras admitting symplectic structures

In this paper we describe all the nilradicals of parabolic subalgebras of split real simple Lie algebras admitting symplectic structures.The main tools used to obtain this list are Kostant's description of the highest weight vectors (hwv) of the cohomology of these nilradicals and some necessary conditions obtained for the g-hwv's of H2(n) for a finite dimensional real symplectic nilpotent Lie algebra n with a reductive Lie subalgebra of derivations g acting on it.

Splints of root systems for special Lie subalgebras

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.

Distinguished nilpotent orbits, Kostant pairs and normalizers of Lie algebras

A pair of Lie algebras (g,g1) will be called a Kostant pair if g is semisimple, g1 is reductive in g and the restriction of the Killing form Bg to g1 is nondegenerate. We study the class of such (nonsymmetric) pairs and obtain some useful and new structural results. We study the structure of the normalizers Ng(g1), and as a consequence we obtain some corresponding worthy results about algebraic groups. In particular we consider an interesting case when g1 is a distinguished sl2-subalgebra of g. Combined with the research due to V.L. Popov we observe that the notions of self-normalizing (reductive) subalgebras of a semisimple Lie algebra and projective self-dual algebraic subvarieties of the usual nilpotent cones are closely related.

Fan, splint, and branching rules

A splint of a root system for a simple Lie algebra appears naturally in studies of (regular) embeddings of reductive subalgebras. A splint can be used to construct branching rules. We demonstrate that implementation of splint properties drastically simplifies calculations of branching coefficients. Bibliography: 5 titles.

Rings of Fractions of Reduction Algebras

We establish the absence of zero divisors in the reduction algebra of a Lie algebra ${\mathfrak{g}}$ with respect to its reductive Lie subalgebra ${\mathfrak{k}}$ . We identify the field of fractions of the diagonal reduction algebra of ${\mathfrak{sl}}_2$ with the standard skew field; as a by-product we obtain a two-parametric family of realizations of this diagonal reduction algebra by differential operators. We also present a new proof of the Poincare–Birkhoff–Witt theorem for reduction algebras.

On affine extension of splint root systems

Splint of root system of simple Lie algebra appears naturally in the study of (regular) embeddings of reductive subalgebras. It can be used to derive branching rules. Application of splint properties drastically simplifies calculations of branching coefficients. We study affine extension of splint root system of simple Lie algebra and obtain relations on theta and branching functions.