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Let 𝔟 be the Borel subalgebra of the Lie algebra 𝔰𝔩2 and V2 be the simple two-dimensional 𝔰𝔩2-module. For the universal enveloping algebra
$\[{\cal A}: = U(\gb \ltimes {V_2})\]$
of the semi-direct product 𝔟⋉V2 of Lie algebras, the prime, primitive and maximal spectra are classified. Please approve edit to the sentence “The sets of completely prime…”.The sets of completely prime ideals of
$\[{\cal A}\]$
are described. The simple unfaithful
$\[{\cal A}\]$
-modules are classified and an explicit description of all prime factor algebras of
$\[{\cal A}\]$
is given. The following classes of simple U(𝔟⋉V2)-modules are classified: the Whittaker modules, the 𝕂[X]-torsion modules and the 𝕂[E]-torsion modules.
Let
$\mathfrak{g}=\mathfrak{g}\mathfrak{l}_{N}(\Bbbk )$
, where
$\Bbbk$
is an algebraically closed field of characteristic
$p>0$
, and
$N\in \mathbb{Z}_{{\geqslant}1}$
. Let
$\unicode[STIX]{x1D712}\in \mathfrak{g}^{\ast }$
and denote by
$U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$
the corresponding reduced enveloping algebra. The Kac–Weisfeiler conjecture, which was proved by Premet, asserts that any finite-dimensional
$U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$
-module has dimension divisible by
$p^{d_{\unicode[STIX]{x1D712}}}$
, where
$d_{\unicode[STIX]{x1D712}}$
is half the dimension of the coadjoint orbit of
$\unicode[STIX]{x1D712}$
. Our main theorem gives a classification of
$U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$
-modules of dimension
$p^{d_{\unicode[STIX]{x1D712}}}$
. As a consequence, we deduce that they are all parabolically induced from a one-dimensional module for
$U_{0}(\mathfrak{h})$
for a certain Levi subalgebra
$\mathfrak{h}$
of
$\mathfrak{g}$
; we view this as a modular analogue of Mœglin’s theorem on completely primitive ideals in
$U(\mathfrak{g}\mathfrak{l}_{N}(\mathbb{C}))$
. To obtain these results, we reduce to the case where
$\unicode[STIX]{x1D712}$
is nilpotent, and then classify the one-dimensional modules for the corresponding restricted
$W$
-algebra.
We give a new and useful approach to study the representations of symmetric Leibniz algebras. Using this approach, we obtain some results on the representations of these algebras.
We introduce the oriented Brauer–Clifford and degenerate affine oriented Brauer–Clifford supercategories. These are diagrammatically defined monoidal supercategories that provide combinatorial models for certain natural monoidal supercategories of supermodules and endosuperfunctors, respectively, for the Lie superalgebras of type Q. Our main results are basis theorems for these diagram supercategories. We also discuss connections and applications to the representation theory of the Lie superalgebra of type Q.
In this paper, we complete the ADE-like classification of simple transitive 2-representations of Soergel bimodules in finite dihedral type, under the assumption of gradeability. In particular, we use bipartite graphs and zigzag algebras of ADE type to give an explicit construction of a graded (non-strict) version of all these 2-representations.
Moreover, we give simple combinatorial criteria for when two such 2-representations are equivalent and for when their Grothendieck groups give rise to isomorphic representations.
Finally, our construction also gives a large class of simple transitive 2-representations in infinite dihedral type for general bipartite graphs.
The prime, completely prime, maximal, and primitive spectra are classified for the universal enveloping algebra of the Schrödinger algebra. The explicit generators are given for all of these ideals. A counterexample is constructed to the conjecture of Cheng and Zhang about nonexistence of simple singular Whittaker modules for the Schrödinger algebra (and all such modules are classified). It is proved that the conjecture holds ‘generically’.
We solve two problems in representation theory for the periplectic Lie superalgebra
$\mathfrak{p}\mathfrak{e}(n)$
, namely, the description of the primitive spectrum in terms of functorial realisations of the braid group and the decomposition of category
${\mathcal{O}}$
into indecomposable blocks.
To solve the first problem, we establish a new type of equivalence between category
${\mathcal{O}}$
for all (not just simple or basic) classical Lie superalgebras and a category of Harish-Chandra bimodules. The latter bimodules have a left action of the Lie superalgebra but a right action of the underlying Lie algebra. To solve the second problem, we establish a BGG reciprocity result for the periplectic Lie superalgebra.
A classification of simple weight modules over the Schrödinger algebra is given. The Krull and the global dimensions are found for the centralizer
${{C}_{S}}(H)$
(and some of its prime factor algebras) of the Cartan element
$H$
in the universal enveloping algebra
$S$
of the Schrödinger (Lie) algebra. The simple
${{C}_{S}}(H)$
-modules are classified. The Krull and the global dimensions are found for some (prime) factor algebras of the algebra
$S$
(over the centre). It is proved that some (prime) factor algebras of
$S$
and
${{C}_{S}}(H)$
are tensor homological
$/$
Krull minimal.
We prove a conjecture of Rouquier relating the decomposition numbers in category
${\mathcal{O}}$
for a cyclotomic rational Cherednik algebra to Uglov’s canonical basis of a higher level Fock space. Independent proofs of this conjecture have also recently been given by Rouquier, Shan, Varagnolo and Vasserot and by Losev, using different methods. Our approach is to develop two diagrammatic models for this category
${\mathcal{O}}$
; while inspired by geometry, these are purely diagrammatic algebras, which we believe are of some intrinsic interest. In particular, we can quite explicitly describe the representations of the Hecke algebra that are hit by projectives under the
$\mathsf{KZ}$
-functor from the Cherednik category
${\mathcal{O}}$
in this case, with an explicit basis. This algebra has a number of beautiful structures including categorifications of many aspects of Fock space. It can be understood quite explicitly using a homogeneous cellular basis which generalizes such a basis given by Hu and Mathas for cyclotomic KLR algebras. Thus, we can transfer results proven in this diagrammatic formalism to category
${\mathcal{O}}$
for a cyclotomic rational Cherednik algebra, including the connection of decomposition numbers to canonical bases mentioned above, and an action of the affine braid group by derived equivalences between different blocks.
The closest infinite-dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e., real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras
$\left( \text{LALAs} \right)$
correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras
$\mathfrak{k}$
, also called affinisations of
$\mathfrak{k}$
. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the 7 families
for some infinite set
$J$
. To each of these types corresponds a “minimal ” affinisation of some simple Hilbert-Lie algebra
$\mathfrak{k}$
, which we call standard.
In this paper, we give for each affinisation
$\mathfrak{g}$
of a simple Hilbert-Lie algebra
$\mathfrak{k}$
an explicit isomorphism from
$\mathfrak{g}$
to one of the standard affinisations of
$\mathfrak{k}$
. The existence of such an isomorphism could also be derived from the classiffication of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitly as a deformation between two twists that is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of
$\mathfrak{g}$
. In subsequent work, this paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of
$\mathfrak{k}$
.
An equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme)
$X$
to a queer Lie superalgebra
$\mathfrak{q}$
that are equivariant with respect to the action of a finite group
$\Gamma $
acting on
$X$
and
$\mathfrak{q}$
. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that
$\Gamma $
is abelian and acts freely on
$X$
. We show that such representations are parameterized by a certain set of
$\Gamma $
-equivariant finitely supported maps from
$X$
to the set of isomorphism classes of irreducible finite-dimensional representations of
$\mathfrak{q}$
. In the special case where
$X$
is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.
For a variety with a Whitney stratification by affine spaces, we study categories of motivic sheaves which are constant mixed Tate along the strata. We are particularly interested in those cases where the category of mixed Tate motives over a point is equivalent to the category of finite-dimensional bigraded vector spaces. Examples of such situations include rational motives on varieties over finite fields and modules over the spectrum representing the semisimplification of de Rham cohomology for varieties over the complex numbers. We show that our categories of stratified mixed Tate motives have a natural weight structure. Under an additional assumption of pointwise purity for objects of the heart, tilting gives an equivalence between stratified mixed Tate sheaves and the bounded homotopy category of the heart of the weight structure. Specializing to the case of flag varieties, we find natural geometric interpretations of graded category
${\mathcal{O}}$
and Koszul duality.
We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to
$\mathfrak{sl}_{n+1}$
. After a suitable twist, the limit is a module for
$\mathfrak{sl}_{n+1}[t]$
, i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in Hernandez and Leclerc (Duke Math. J.154 (2010), 265–341; Symmetries, Integrable Systems and Representations, Springer Proceedings in Mathematics & Statitics, Volume 40, pp. 175–193 (2013)), in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting) to
$\mathfrak{sl}_{n+1}[t]$
-stable prime Demazure modules in level-two integrable highest-weight representations of the classical affine Lie algebra. It was proved in Chari et al. (arXiv:1408.4090) that a stable Demazure module is isomorphic to the fusion product of stable prime Demazure modules. Our next result proves that such a fusion product is the limit of the tensor product of the corresponding irreducible prime representations of quantum affine
$\mathfrak{sl}_{n+1}$
.
In Bennett et al. [BGG reciprocity for current algebras, Adv. Math. 231 (2012), 276–305] it was conjectured that a BGG-type reciprocity holds for the category of graded representations with finite-dimensional graded components for the current algebra associated to a simple Lie algebra. We associate a current algebra to any indecomposable affine Lie algebra and show that, in this generality, the BGG reciprocity is true for the corresponding category of representations.
We describe all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. Let
$K/F$
be a finite separable field extension and let
$x,\,y\,\in \,K$
. When is
$F\left[ x,\,y \right]\,=\,F\left[ \alpha x\,+\,\beta y \right]$
for some nonzero elements
$\alpha ,\,\beta \,\in \,F?$
In this article, we study the homomorphisms between scalar generalized Verma modules. We conjecture that any homomorphism between scalar generalized Verma modules is a composition of elementary homomorphisms. The purpose of this article is to confirm the conjecture for some parabolic subalgebras under the assumption that the infinitesimal characters are regular.
This paper studies the
$K$
-theoretic invariants of the crossed product
${{C}^{*}}$
-algebras associated with an important family of homeomorphisms of the tori
${{\mathbb{T}}^{n}}$
called Furstenberg transformations. Using the Pimsner–Voiculescu theorem, we prove that given
$n$
, the
$K$
-groups of those crossed products whose corresponding
$n\,\times \,n$
integer matrices are unipotent of maximal degree always have the same rank
${{a}_{n}}$
. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these
$K$
-groups is false. Using the representation theory of the simple Lie algebra
$\mathfrak{s}\mathfrak{l}\left( 2,\,\mathbb{C} \right)$
, we show that, remarkably,
${{a}_{n}}$
has a combinatorial significance. For example, every
${{a}_{2n+1}}$
is just the number of ways that 0 can be represented as a sum of integers between –
$n$
and
$n$
(with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erdős), a simple explicit formula for the asymptotic behavior of the sequence
$\{{{a}_{n}}\}$
is given. Finally, we describe the order structure of the
${{K}_{0}}$
-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.
Let k be an algebraically closed field of characteristic zero. I. M. Musson and M. Van den Bergh (Mem. Amer. Math. Soc., vol. 136, 1998, p. 650) classify primitive ideals for rings of torus invariant differential operators. This classification applies in particular to subquotients of localized extended Weyl algebras
$\mathcal{A}_{r,n-r}=k[x_1,\ldots,x_r,x_{r+1}^{\pm1}, \ldots, x_{n}^{\pm1},\partial_1,\ldots,\partial_n],$
where it can be made explicit in terms of convex geometry. We recall these results and then turn to the corresponding primitive quotients and study their Goldie ranks. We prove that the primitive quotients fall into finitely many families whose Goldie ranks are given by a common quasi-polynomial and then realize these quasi-polynomials as Ehrhart quasi-polynomials arising from convex geometry.
Let d be a positive integer, q=(qij)d×d be a d×d matrix, ℂq be the quantum torus algebra associated with q. We have the semidirect product Lie algebra
$\mathfrak{g}$
=Der(ℂq)⋉Z(ℂq), where Z(ℂq) is the centre of the rational quantum torus algebra ℂq. In this paper, we construct a class of irreducible weight
$\mathfrak{g}$
-modules
$\mathcal{V}$
α (V,W) with three parameters: a vector α∈ℂd, an irreducible
$\mathfrak{gl}$
d-module V and a graded-irreducible
$\mathfrak{gl}$
N-module W. Then, we show that an irreducible Harish Chandra (uniformaly bounded)
$\mathfrak{g}$
-module M is isomorphic to
$\mathcal{V}$
α(V,W) for suitable α, V, W, if the action of Z(ℂq) on M is associative (respectively nonzero).
We give an easy diagrammatical description of the parabolic Kazhdan–Lusztig polynomials for the Weyl group Wn of type Dn with parabolic subgroup of type An and consequently an explicit counting formula for the dimension of morphism spaces between indecomposable projective objects in the corresponding category . As a by-product we categorify irreducible Wn-modules corresponding to the pairs of one-line partitions. Finally, we indicate the motivation for introducing the combinatorics by connections to the Springer theory, the category of perverse sheaves on isotropic Grassmannians, and to the Brauer algebras, which will be treated in two subsequent papers of the second author.