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Kantor pairs arise naturally in the study of 5-graded Lie algebras. In this article, we introduce and study Kantor pairs with short Peirce gradings and relate themto Lie algebras graded by the root system of type
$\text{B}{{\text{C}}_{2}}$
. This relationship allows us to define so-called Weyl images of short Peirce graded Kantor pairs. We use Weyl images to construct new examples of Kantor pairs, including a class of infinite dimensional central simple Kantor pairs over a field of characteristic
$\ne$
2 or 3, as well as a family of forms of a split Kantor pair of type
${{\text{E}}_{6}}$
.
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
$\mathfrak{k}$
.
The trace (or zeroth Hochschild homology) of Khovanov’s Heisenberg category is identified with a quotient of the algebra
$W_{1+\infty }$
. This induces an action of
$W_{1+\infty }$
on the center of the categorified Fock space representation, which can be identified with the action of
$W_{1+\infty }$
on symmetric functions.
We prove that the group of automorphisms of the Lie algebra DerK(Qn) of derivations of the field of rational functions Qn = K(x1, . . ., xn) over a field of characteristic zero is canonically isomorphic to the group of automorphisms of the K-algebra Qn.
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.
We study the monomial crystal defined by the second author. We show that each component of the monomial crystal can be embedded into a crystal of an extremal weight module introduced by Kashiwara. And we determine all monomials appearing in the components corresponding to all level 0 fundamental representations of quantum affine algebras except for some nodes of . Thus we obtain explicit descriptions of the crystals in these examples. We also give those for the corresponding finite dimensional representations. For classical types, we give them in terms of tableaux. For exceptional types, we list up all monomials.
We study the monomial crystal defined by the second author. We show that each component of the monomial crystal can be embedded into a crystal of an extremal weight module introduced by Kashiwara. And we determine all monomials appearing in the components corresponding to all level 0 fundamental representations of quantum affine algebras except for some nodes of . Thus we obtain explicit descriptions of the crystals in these examples. We also give those for the corresponding finite dimensional representations. For classical types, we give them in terms of tableaux. For exceptional types, we list up all monomials.
Modular forms for the Weil representation of SL2 (ℤ) play an important role in the theory of automorphic forms on orthogonal groups. In this paper we give some explicit constructions of these functions. As an application, we construct new examples of generalized Kac-Moody algebras whose denominator identities are holomorphic automorphic products of singular weight. They correspond naturally to the Niemeier lattices with root systems and to the Leech lattice.
For Dewitt super groups
$G$
modeled via an underlying finitely generated Grassmann algebra it is well known that when there exists a body group
$BG$
compatible with the group operation on
$G$
, then, generically, the kernel
$K$
of the body homomorphism is nilpotent. This is not true when the underlying Grassmann algebra is infinitely generated. We show that it is quasi-nilpotent in the sense that as a Banach Lie group its Lie algebra
$\kappa$
has the property that for each
$a\,\in \,\kappa ,\,\text{a}{{\text{d}}_{a}}$
has a zero spectrum. We also show that the exponential mapping from
$\kappa$
to
$K$
is surjective and that
$K$
is a quotient manifold of the Banach space
$\kappa$
via a lattice in
$\kappa$
.
We construct categorical braid group actions from 2-representations of a Heisenberg algebra. These actions are induced by certain complexes which generalize spherical (Seidel–Thomas) twists and are reminiscent of the Rickard complexes defined by Chuang–Rouquier. Conjecturally, one can relate our complexes to Rickard complexes using categorical vertex operators.
We prove that for simple complex finite dimensional Lie algebras, affine Kac–Moody Lie algebras, the Virasoro algebra, and the Heisenberg–Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro algebras and for Heisenberg algebras.
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).
Let G be an arbitrary non-zero additive subgroup of the complex number field ℂ, and let Vir[G] be the corresponding generalized Virasoro algebra over ℂ. In this paper we determine all irreducible weight modules with finite-dimensional weight spaces over Vir[G]. The classification strongly depends on the index group G. If G does not have a direct summand isomorphic to ℤ (the integers), then such irreducible modules over Vir[G] are only modules of intermediate series whose weight spaces are all one dimensional. Otherwise, there is one further class of modules that are constructed by using intermediate series modules over a generalized Virasoro subalgebra Vir[G0] of Vir[G] for a direct summand G0 of G with G = G0 ⊕ ℤb, where b ∈ G \ G0. This class of irreducible weight modules do not have corresponding weight modules for the classical Virasoro algebra.
Representations of various one-dimensional central extensions of quantum tori (called quantum torus Lie algebras) were studied by several authors. Now we define a central extension of quantum tori so that all known representations can be regarded as representations of the new quantum torus Lie algebras
${{\mathfrak{L}}_{q}}$
. The center of
${{\mathfrak{L}}_{q}}$
now is generally infinite dimensional.
In this paper,
$\mathbb{Z}$
-graded Verma modules
$\tilde{V}\left( \varphi \right)$
over
${{\mathfrak{L}}_{q}}$
and their corresponding irreducible highest weight modules
$V\left( \varphi \right)$
are defined for some linear functions
$\varphi $
. Necessary and sufficient conditions for
$V\left( \varphi \right)$
to have all finite dimensional weight spaces are given. Also necessary and sufficient conditions for Verma modules
$\tilde{V}\left( \varphi \right)$
to be irreducible are obtained.
Let
$A$
be a unital commutative associative algebra over a field of characteristic zero,
$\mathfrak{k}$
a Lie algebra, and
$\mathfrak{z}$
a vector space, considered as a trivial module of the Lie algebra
$\mathfrak{g}:=A\otimes \mathfrak{k}$
. In this paper, we give a description of the cohomology space
${{H}^{2}}(\mathfrak{g},\mathfrak{z})$
in terms of easily accessible data associated with
$A$
and
$\mathfrak{k}$
. We also discuss the topological situation, where
$A$
and
$\mathfrak{k}$
are locally convex algebras.
In this paper we prove that the Kostrikin radical of an extended affine Lie algebra of reduced type coincides with the center of its core, and use this characterization to get a type-free description of the core of such algebras. As a consequence we get that the core of an extended affine Lie algebra of reduced type is invariant under the automorphisms of the algebra.
Let K be a field of characteristic 0, G the direct sum of two copies of the additive group of integers. For a total order ≺ on G, which is compatible with the addition, and for any ċ1, ċ2 ∈ K, we define G-graded highest weight modules M(ċ1, ċ2, ≺) over the Virasoro-like algebra , indexed by G. It is natural to call them Verma modules. In the present paper, the irreducibility of M (ċ1, ċ2, ≺) is completely determined and the structure of reducible module M (ċ1, ċ2, ≺)is also described.
We investigate a class of Lie algebras which we call generalized reductive Lie algebras. These are generalizations of semi-simple, reductive, and affine Kac–Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible root system is said to be irreducible and we note that this class of algebras has been under intensive investigation in recent years. They have also been called extended affine Lie algebras. The larger class of generalized reductive Lie algebras has not been so intensively investigated. We study them in this paper and note that one way they arise is as fixed point subalgebras of finite order automorphisms. We show that the core modulo the center of a generalized reductive Lie algebra is a direct sum of centerless Lie tori. Therefore one can use the results known about the classification of centerless Lie tori to classify the cores modulo centers of generalized reductive Lie algebras.