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We introduce and study a fermionisation procedure for the cohomological Hall algebra
$\mathcal{H}_{\Pi_Q}$
of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson–Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel–Moore homology of the stack of representations of the
$\unicode{x03BC}$
-deformed preprojective algebra introduced by Crawley–Boevey and Holland, for all dimension vectors. This provides a common generalisation of the results of Crawley-Boevey and Van den Bergh on the cohomology of smooth moduli schemes of representations of deformed preprojective algebras and my earlier results on the Borel–Moore homology of the stack of representations of the undeformed preprojective algebra.
We define slope subalgebras in the shuffle algebra associated to a (doubled) quiver, thus yielding a factorization of the universal R-matrix of the double of the shuffle algebra in question. We conjecture that this factorization matches the one defined by [1, 18, 32, 33, 34] using Nakajima quiver varieties.
Let K be any field of characteristic two and let
$U_1$
and
$W_1$
be the Lie algebras of the derivations of the algebra of Laurent polynomials
$K[t,t^{-1}]$
and of the polynomial ring K[t], respectively. The algebras
$U_1$
and
$W_1$
are equipped with natural
$\mathbb{Z}$
-gradings. In this paper, we provide bases for the graded identities of
$U_1$
and
$W_1$
, and we prove that they do not admit any finite basis.
For complex simple Lie algebras of types B, C, and D, we provide new explicit formulas for the generators of the commutative subalgebra $\mathfrak z(\hat {\mathfrak g})\subset {{\mathcal {U}}}(t^{-1}\mathfrak g[t^{-1}])$ known as the Feigin–Frenkel centre. These formulas make use of the symmetrisation map as well as of some well-chosen symmetric invariants of $\mathfrak g$. There are some general results on the rôle of the symmetrisation map in the explicit description of the Feigin–Frenkel centre. Our method reduces questions about elements of $\mathfrak z(\hat {\mathfrak g})$ to questions on the structure of the symmetric invariants in a type-free way. As an illustration, we deal with type G$_2$ by hand. One of our technical tools is the map ${\sf m}\!\!: {{\mathcal {S}}}^{k}(\mathfrak g)\to \Lambda ^{2}\mathfrak g \otimes {{\mathcal {S}}}^{k-3}(\mathfrak g)$ introduced here. As the results show, a better understanding of this map will lead to a better understanding of $\mathfrak z(\hat {\mathfrak g})$.
We construct and study some vertex theoretic invariants associated with Poisson varieties, specializing in the conformal weight $0$ case to the familiar package of Poisson homology and cohomology. In order to do this conceptually, we sketch a version of the calculus, in the sense of [12], adapted to the context of vertex algebras. We obtain the standard theorems of Poisson (co)homology in this chiral context. This is part of a larger project related to promoting noncommutative geometric structures to chiral versions of such.
Using crossed homomorphisms, we show that the category of weak representations (respectively admissible representations) of Lie–Rinehart algebras (respectively Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor of monoidal categories is established to give new weak representations (respectively admissible representations) of Lie–Rinehart algebras (respectively Leibniz pairs). This generalises and unifies various existing constructions of representations of many Lie algebras by using this new bifunctor. We construct some crossed homomorphisms in different situations and use our actions of monoidal categories to recover some known constructions of representations of various Lie algebras and to obtain new representations for generalised Witt algebras and their Lie subalgebras. The cohomology theory of crossed homomorphisms between Lie algebras is introduced and used to study linear deformations of crossed homomorphisms.
Let X be a finite connected poset and K a field. We study the question, when all Lie automorphisms of the incidence algebra I(X, K) are proper. Without any restriction on the length of X, we find only a sufficient condition involving certain equivalence relation on the set of maximal chains of X. For some classes of posets of length one, such as finite connected crownless posets (i.e., without weak crown subposets), crowns, and ordinal sums of two anti-chains, we give a complete answer.
Let $G= SL_{n+1}$ be defined over an algebraically closed field of characteristic $p > 2$. For each $n \geq 1$, there exists a singular block in the category of $G_1$-modules, which contains precisely $n+1$ irreducible modules. We are interested in the ‘lift’ of this block to the category of $G_1T$-modules. Imposing only mild assumptions on $p$, we will perform a number of calculations in this setting, including a complete determination of the Loewy series for the baby Verma modules and all possible extensions between the irreducible modules. In the case where $p$ is extremely large, we will also explicitly compute the Loewy series for the indecomposable projective modules.
A linear étale representation of a complex algebraic group G is given by a complex algebraic G-module V such that G has a Zariski-open orbit in V and
$\dim G=\dim V$
. A current line of research investigates which reductive algebraic groups admit such étale representations, with a focus on understanding common features of étale representations. One source of new examples arises from the classification theory of nilpotent orbits in semisimple Lie algebras. We survey what is known about reductive algebraic groups with étale representations and then discuss two classical constructions for nilpotent orbit classifications due to Vinberg and to Bala and Carter. We determine which reductive groups and étale representations arise in these constructions and we work out in detail the relation between these two constructions.
For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule
$\Delta (\lambda )$
to be such that every nonzero homomorphism from another Verma supermodule to
$\Delta (\lambda )$
is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras
$\mathfrak {pe} (n)$
and, furthermore, to reduce the problem of description of
$\mathrm {Ext}^1_{\mathcal O}(L(\mu ),\Delta (\lambda ))$
for
$\mathfrak {pe} (n)$
to the similar problem for the Lie algebra
$\mathfrak {gl}(n)$
. Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category
$\mathcal O^{\mathfrak {p}}$
for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra
$\mathfrak {pe} (n)$
and the orthosymplectic Lie superalgebra
$\mathfrak {osp}(2|2n)$
.
A thin Lie algebra is a Lie algebra $L$, graded over the positive integers, with its first homogeneous component $L_1$ of dimension two and generating $L$, and such that each non-zero ideal of $L$ lies between consecutive terms of its lower central series. All homogeneous components of a thin Lie algebra have dimension one or two, and the two-dimensional components are called diamonds. Suppose the second diamond of $L$ (that is, the next diamond past $L_1$) occurs in degree $k$. We prove that if $k>5$, then $[Lyy]=0$ for some non-zero element $y$ of $L_1$. In characteristic different from two this means $y$ is a sandwich element of $L$. We discuss the relevance of this fact in connection with an important theorem of Premet on sandwich elements in modular Lie algebras.
Let $U_q'({\mathfrak {g}})$ be a quantum affine algebra with an indeterminate $q$, and let $\mathscr {C}_{\mathfrak {g}}$ be the category of finite-dimensional integrable $U_q'({\mathfrak {g}})$-modules. We write $\mathscr {C}_{\mathfrak {g}}^0$ for the monoidal subcategory of $\mathscr {C}_{\mathfrak {g}}$ introduced by Hernandez and Leclerc. In this paper, we associate a simply laced finite-type root system to each quantum affine algebra $U_q'({\mathfrak {g}})$ in a natural way and show that the block decompositions of $\mathscr {C}_{\mathfrak {g}}$ and $\mathscr {C}_{\mathfrak {g}}^0$ are parameterized by the lattices associated with the root system. We first define a certain abelian group $\mathcal {W}$ (respectively $\mathcal {W} _0$) arising from simple modules of $\mathscr {C}_{\mathfrak {g}}$ (respectively $\mathscr {C}_{\mathfrak {g}}^0$) by using the invariant $\Lambda ^\infty$ introduced in previous work by the authors. The groups $\mathcal {W}$ and $\mathcal {W} _0$ have subsets $\Delta$ and $\Delta _0$ determined by the fundamental representations in $\mathscr {C}_{\mathfrak {g}}$ and $\mathscr {C}_{\mathfrak {g}}^0$, respectively. We prove that the pair $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$ is an irreducible simply laced root system of finite type and that the pair $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} , \Delta )$ is isomorphic to the direct sum of infinite copies of $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$ as a root system.
We compare the $K$-theory stable bases of the Springer resolution associated to different affine Weyl alcoves. We prove that (up to relabelling) the change of alcoves operators are given by the Demazure–Lusztig operators in the affine Hecke algebra. We then show that these bases are categorified by the Verma modules of the Lie algebra, under the localization of Lie algebras in positive characteristic of Bezrukavnikov, Mirković, and Rumynin. As an application, we prove that the wall-crossing matrices of the $K$-theory stable bases coincide with the monodromy matrices of the quantum cohomology of the Springer resolution.
We compare crystal combinatorics of the level $2$ Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. We show that the crystal operators that remove boxes preserve the combinatorial conditions for unitarity, and that the parabolic restriction functors categorifying the crystals send irreducible unitary representations to unitary representations. Furthermore, we find the supports of the unitary representations.
Let
$\mathcal {X}$
be a Banach space over the complex field
$\mathbb {C}$
and
$\mathcal {B(X)}$
be the algebra of all bounded linear operators on
$\mathcal {X}$
. Let
$\mathcal {N}$
be a nontrivial nest on
$\mathcal {X}$
,
$\text {Alg}\mathcal {N}$
be the nest algebra associated with
$\mathcal {N}$
, and
$L\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$
be a linear mapping. Suppose that
$p_n(x_1,x_2,\ldots ,x_n)$
is an
$(n-1)\,$
th commutator defined by n indeterminates
$x_1, x_2, \ldots , x_n$
. It is shown that L satisfies the rule
for all
$A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$
if and only if there exist a linear derivation
$D\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$
and a linear mapping
$H\colon \text {Alg}\mathcal {N}\longrightarrow \mathbb {C}I$
vanishing on each
$(n-1)\,$
th commutator
$p_n(A_1,A_2,\ldots , A_n)$
for all
$A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$
such that
$L(A)=D(A)+H(A)$
for all
$A\in \text {Alg}\mathcal {N}$
. We also propose some related topics for future research.
We show that the Tits index
$E_{8,1}^{133}$
cannot be obtained by means of the Tits construction over a field with no odd degree extensions. The proof uses a general method coming from the theory of symmetric spaces. We construct two cohomological invariants, in degrees
$6$
and
$8$
, of the Tits construction and the more symmetric Allison–Faulkner construction of Lie algebras of type
$E_8$
and show that these invariants can be used to detect the isotropy rank.
In [2], Beilinson–Lusztig–MacPherson (BLM) gave a beautiful realization for quantum
$\mathfrak {gl}_n$
via a geometric setting of quantum Schur algebras. We introduce the notion of affine Schur superalgebras and use them as a bridge to link the structure and representations of the universal enveloping superalgebra
${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$
of the loop algebra
$\widehat {\mathfrak {gl}}_{m|n}$
of
${\mathfrak {gl}}_{m|n}$
with those of affine symmetric groups
${\widehat {{\mathfrak S}}_{r}}$
. Then, we give a BLM type realization of
${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$
via affine Schur superalgebras.
The first application of the realization of
${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$
is to determine the action of
${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$
on tensor spaces of the natural representation of
$\widehat {\mathfrak {gl}}_{m|n}$
. These results in epimorphisms from
$\;{\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$
to affine Schur superalgebras so that the bridging relation between representations of
${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$
and
${\widehat {{\mathfrak S}}_{r}}$
is established. As a second application, we construct a Kostant type
$\mathbb Z$
-form for
${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$
whose images under the epimorphisms above are exactly the integral affine Schur superalgebras. In this way, we obtain essentially the super affine Schur–Weyl duality in arbitrary characteristics.
We establish some cohomological bounds in $D$-module theory that are known in the holonomic case and folklore in general. The method rests on a generalization of the $b$-function lemma for non-holonomic $D$-modules.
For a finite-dimensional Lie algebra
$\mathfrak {L}$
over
$\mathbb {C}$
with a fixed Levi decomposition
$\mathfrak {L} = \mathfrak {g} \ltimes \mathfrak {r}$
, where
$\mathfrak {g}$
is semisimple, we investigate
$\mathfrak {L}$
-modules which decompose, as
$\mathfrak {g}$
-modules, into a direct sum of simple finite-dimensional
$\mathfrak {g}$
-modules with finite multiplicities. We call such modules
$\mathfrak {g}$
-Harish-Chandra modules. We give a complete classification of simple
$\mathfrak {g}$
-Harish-Chandra modules for the Takiff Lie algebra associated to
$\mathfrak {g} = \mathfrak {sl}_2$
, and for the Schrödinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright’s and Arkhipov’s completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple
$\mathfrak {g}$
-Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff
$\mathfrak {sl}_2$
and the Schrödinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple
$\mathfrak {g}$
-Harish-Chandra modules.
We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup to the category of $\operatorname{GL}(N-1,{\mathbb {C}}[\![t]\!])$-equivariant perverse sheaves on the affine Grassmannian of $\operatorname{GL}_N$. We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.