text stringlengths 128 2.05k |
|---|
[EQUATION] for every [MATH] , where [MATH] is the [MATH] -th cohomology object of [MATH] in the [MATH] -structure [MATH] and [MATH] are the slices of the heart [MATH] of [MATH] for the abelian [MATH] -slicing [MATH] |
Proof. Denote by [MATH] the [MATH] -structure on [MATH] associated with the perversity function [MATH] . Then the lower subcategory [MATH] and the upper subcategory [MATH] of [MATH] are defined, by Proposition 2.42 , as |
[EQUATION] These can be equivalently described as [EQUATION] see, e.g., FLM15 , Remark 4.27] . Therefore [EQUATION] Equivalently, this means that |
[EQUATION] Example 2.55 Let [MATH] and let [MATH] the characteristic function of the interval [MATH] . Seen as a function from [MATH] to [MATH] , the function [MATH] is a perversity function of a very special kind: it is a perversity function taking exactly two values. Moreover, it is easy to see that –up to an additiv... |
[EQUATION] Therefore the perverse heart [MATH] of [MATH] is the full subcategory of [MATH] on those objects [MATH] such that [EQUATION] |
for every [MATH] . In other words, [MATH] is (up to a shift by 1) the heart of the tilted [MATH] -structure obrained by tilting [MATH] with the torsion theory on [MATH] given by [MATH] and [MATH] |
A zoo of examples The upshot of Theorem 2.49 is that out of a gluing abelian [MATH] -slicing on the heart of a bounded [MATH] -structure on a stable [MATH] -category [MATH] |
we explicitly get a natural morphism of [MATH] -posets [EQUATION] which, on the subset of perversities, acts as [EQUATION] see Proposition 2.42 . We can build this way whole new classes of ‘perverse’ [MATH] -structures on [MATH] . In this section we present a few examples, reinterpreting and rediscovering a few classic... |
3.1 An example from algebra and one from geometry 3.1.1 The nonstandard [MATH] -structure from Koszul duality An instance of a nonstandard construction of a [MATH] -structure lies within the theory of Koszul duality. Namely, under suitable finiteness and semisimplicity assumptions, if [MATH] and [MATH] are Koszul dual ... |
[EQUATION] for [MATH] , for any modules [MATH] and [MATH] . In particular, if [MATH] and [MATH] we have [MATH] , and so [MATH] is a gluable slicing. Therefore, for every perversity [MATH] we have a bounded perverse [MATH] -structure [MATH] on [MATH] . By choosing [MATH] to be the identity perversity [MATH] we get a dis... |
3.1.2 Strictly perverse coherent sheaves Let [MATH] be a smooth projective variety over [MATH] and let [MATH] , the (bounded) derived category of coherent sheaves on [MATH] , endowed with its canonical heart [MATH] . Then there is an abelian [MATH] -slicing on [MATH] given by defining [MATH] as the full subcategory of ... |
[EQUATION] Equivalently, this can be rewritten as [MATH] for [MATH] , for any [MATH] in [MATH] and [MATH] in [MATH] , i.e., [MATH] for [MATH] . In other words, [MATH] is a perverse abelian [MATH] -slicing on [MATH] . Therefore, by Theorem 2.49 and Remark 2.51 , with any strict perversity function [MATH] is associated a... |
3.2 Gluable slicings from baric structures The notion of a bounded Bridgeland [MATH] -slicing is not new: it already appears in literature under other names. Namely, it is no more than an infinite version of a semiorthogonal decomposition in the sense of Kuz14 , or a ‘ baric structure ’ as defined in AT11 . These are a... |
[EQUATION] and so it is a gluable abelian slicing. Moreover, and remarkably, the gluability of [MATH] can be easily explicited. Namely, spelling out Definition 2.12 , we see that [MATH] is gluable if and only if [MATH] for any [MATH] . As [MATH] is [MATH] for [MATH] , this is equivalent to [MATH] whenever [MATH] and [M... |
3.2.1 Gluability and the Beilinson-Soulé conjecture The existence of motives, which is still an open question in general, was conjectured by Grothendieck in order to build a universal Weil cohomology theory for schemes: the ‘motivic cohomology’. Following this input, Deligne observed that it could be easier to construc... |
[MATH] the triangulated subcategory of [MATH] [MATH] . One has an isomorphisms of groups [EQUATION] where [MATH] is the [MATH] -th higher [MATH] -theory group of the point [MATH] and [MATH] is the weight [MATH] summand of [MATH] with respect to the Adams action. |
By dimensional reasons, the right hand side vanishes for [MATH] and for [MATH] with [MATH] . In other words, the Tate objects form an infinite exceptional collection on [MATH] which is clearly full by definition. By the general theory of semiorthogonal decomposition, this implies that the triangulated subcategories [MA... |
[EQUATION] whenever [MATH] and [MATH] . As [MATH] is [MATH] , the gluability condition is equivalent to [EQUATION] whenever [MATH] and [MATH] , and therefore to |
[EQUATION] whenever [MATH] and [MATH] . This is exactly the Beilinson-Soulé standard vanishing conjecture, which is known to hold, for instance, when [MATH] is a number field due to Borel’s computation of the ranks of K-theory groups in this case Bor74 . When the conjecture holds, by applying [MATH] we get a Bridgeland... |
Moreover, following the reasoning recalled at the beginning of this Section, we also get a [MATH] -structure on [MATH] for each perversity function on [MATH] . These are the ‘ perverse motives ’ appearing in SW18 |
3.2.2 Three more examples There are a number of other constructions in literature which are a particular case of the one we presented here. Just to mention a few, in Bei Beilinson defines a notion of ’filtered structure’ on a triangualted category. This is no more than a baric structure [MATH] with some additional data... |
In Mac07 , Macrí starts with a finite ‘Ext exceptional’ collection on a certain triangualted cateogory [MATH] and get a distinguished [MATH] -structure on [MATH] . This construction actually goes along the exact lines sketched in Subsection 3.2.1 . Namely, when translated into the language of this note, a finite except... |
Finally, a possibly more exotic instance is in Hen17 . Here, starting with a suitable [MATH] -slicing on the Fukaya category [MATH] of a symplectic manifold [MATH] , Hensel builds a [MATH] -structure on the Fukaya category [MATH] of [MATH] . This is done by embedding [MATH] into a triangulated category [MATH] as the ze... |
# Source: arxiv 1806.00893 # Title: On Dynkin gradings in simple Lie algebras # Sections: all # Downloaded: 2026-03-03T02:33:38.906906+00:00 |
On Dynkin gradings in simple Lie algebras To Anthony Joseph for his 75th birthday Introduction In this paper we study Dynkin gradings on simple Lie algebras arising from nilpotent elements. Specifically, we investigate abelian subalgebras which are degree 1 homogeneous with respect to these gradings. |
The study of gradings associated to nilpotent elements of simple Lie algebras is important since the finite and affine classical and quantum W-algebras are defined using these gradings. In order to study integrable systems associated to these W-algebras, it is useful to have their free field realizations. One of the wa... |
. This construction can be further improved by choosing an abelian subalgebra in the term [MATH] of the grading. That is why the description of such subalgebras, especially the ones of dimension equal half of the dimension of [MATH] (which is maximal possible), is important. |
We show that for each odd nilpotent orbit there always exists a canonically associated “strictly odd” nilpotent orbit, which allows us to reduce our investigations to the latter. (Strictly odd means that all Dynkin labels are either 0 or 1.) The rest of the paper is devoted to the investigation of maximal abelian subal... |
Recollections Let us recall the nomenclature for nilpotents in a semisimple Lie algebra [MATH] Given such a nilpotent [MATH] , one chooses an [MATH] -triple [MATH] for it, that is, another nilpotent [MATH] such that [MATH] is semisimple and the identities [MATH] [MATH] hold (Jacobson-Morozov theorem; see e. g. |
). The Dynkin grading is the eigenspace decomposition for [MATH] [EQUATION] Then, to [MATH] one assigns a combinatorial object which determines it up to isomorphism. It is the weighted Dynkin diagram corresponding to [MATH] , which is the Dynkin diagram of [MATH] with numbers assigned to each node. These numbers are th... |
The nilpotent is called even if there are no [MATH] ’s in its weighted Dynkin diagram, odd if it is not even, and strictly odd if there are no [MATH] ’s. |
It is clear that for even nilpotents the question about abelian subspaces in [MATH] is trivial since [MATH] is zero. We will also need the following fact from |
Proposition 1.1 The degree [MATH] part [MATH] of [MATH] with respect to the grading induced by a nilpotent [MATH] is generated as a [MATH] -module by those simple root vectors of [MATH] which have weight [MATH] in the weighted Dynkin diagram corresponding to [MATH] |
If [MATH] is a simple Lie algebra of classical type, one can assign to [MATH] another combinatorial object — a partition [MATH] which records dimensions of irreducible representations of [MATH] into which the standard representation of [MATH] decomposes as a module over its subalgebra [MATH] . Alternatively, the partit... |
Let us recall how one switches from a partition representing a nilpotent to its weighted Dynkin diagram (cf. ). Each [MATH] in the partition represents a copy of the [MATH] -dimensional irreducible representation of [MATH] , with eigenvalues of [MATH] equal to |
[EQUATION] To obtain the weighted Dynkin diagram one collects from each [MATH] those eigenvalues, arranges them in decreasing order, and takes consecutive differences. |
For example, take the partition [MATH] . This gives the following eigenvalues of [MATH] [EQUATION] Arranging all numbers from this table in the decreasing order gives |
[EQUATION] Taking the consecutive differences then gives [EQUATION] which is already the weighted Dynkin diagram of the nilpotent in case of type A. |
For types B, C, D one has to leave only left half of the obtained sequence (which obviously is centrally symmetric); more precisely, for an algebra of rank [MATH] , the first [MATH] nodes of the weighted Dynkin diagram are as stated, while the rightmost node is defined in a specific way, depending on the type. We skip ... |
For example, the same partition [MATH] also encodes a nilpotent orbit in a simple Lie algebra of type C, since all of its odd parts come with even multiplicities. Then, the weighted Dynkin diagram of this nilpotent is |
[EQUATION] It is easy to see from the above procedure that the resulting weighted Dynkin diagram begins with certain sequence of [MATH] ’s and [MATH] ’s; if the largest part of the partition is [MATH] with multiplicity [MATH] , and the parts of the same parity following it are [MATH] with multiplicity [MATH] [MATH] wit... |
According to the above procedure for assigning to a partition a weighted Dynkin diagram, it is easy to see the following Proposition 1.2 |
A nilpotent in a simple Lie algebra of classical type is even iff all the parts of the corresponding partition are of the same parity, is odd iff there are some parts with different parities, and strictly odd iff the largest part and the next largest part differ by [MATH] |
Important reduction Let [MATH] and [MATH] be finite-dimensional modules over a reductive Lie algebra [MATH] and let [MATH] be a [MATH] -module homomorphism. It is thus a [MATH] -equivariant algebra structure on [MATH] with values in [MATH] |
Proposition 2.1 Suppose that there exists an abelian subalgebra of dimension [MATH] of the algebra [MATH] . Then there exists an abelian subalgebra of the algebra [MATH] of dimension [MATH] , spanned by weight vectors of [MATH] |
Proof (proposed by the referee) It follows from Borel’s fixed point theorem. Indeed, the Cartan subgroup acts on the complete variety of [MATH] -dimensional abelian subalgebras of [MATH] , hence has a fixed point. |
Using this, in what follows we will assume throughout that for a simple Lie algebra of classical type we are given a basis in the standard representation consisting of weight vectors corresponding to the weights [MATH] [MATH] and moreover, for the type B, to the zero weight. In the adjoint representation, accordingly, ... |
Proposition 2.2 For any weighted Dynkin diagram corresponding to a nilpotent [MATH] in a simple Lie algebra [MATH] , consider a subdiagram obtained as a result of erasing all nodes with weight [MATH] . Consider the resulting subdiagram together with the remaining weights. Then all connected components of this subdiagra... |
Proof. For algebras of classical type, this is proved in 3.6 below. For algebras of type G this is clear as all nilpotents in them are either even or strictly odd. As for exceptional Lie algebras of types E or F, the assertion can be seen to be true directly from looking at the tables F4o, E6o, E7o, E8o given in the la... |
Corollary 2.3 For any odd nilpotent [MATH] in a simple Lie algebra [MATH] there exists a simple diagram subalgebra [MATH] and a strictly odd nilpotent [MATH] such that |
[EQUATION] i. e. the degree 1 homogeneous parts for the grading on [MATH] induced by [MATH] and for the grading on [MATH] induced by [MATH] coincide. In particular, these degree 1 homogeneous parts have the same abelian subspaces. |
Proof. Take for [MATH] the subalgebra corresponding to the connected component of the weighted Dynkin diagram of [MATH] as described in 2.2 above. Moreover let [MATH] be any representative from the orbit corresponding to the weights on this connected component — it exists by 2.2 |
By construction this subalgebra contains all simple root vectors of degree 1, and moreover they will be precisely the root vectors of those simple roots of [MATH] which contribute to degree [MATH] part in the grading induced by [MATH] . From 1.1 we know that [MATH] is the [MATH] -module [MATH] is the [MATH] -module |
Now observe that the only removed nodes which connect with an edge to some node in the remaining connected component have weight [MATH] , so that all simple root vectors corresponding to removed nodes with weight [MATH] commute with every simple root vector in this component. |
It follows that the [MATH] -module [MATH] -module [MATH] coincides with [MATH] Definition 2.4 For the orbit of an odd nilpotent in a simple Lie algebra [MATH] , call its strictly odd reduction the nilpotent orbit in the simple Lie algebra [MATH] obtained as in 2.3 |
Given a nilpotent [MATH] as in 2.2 , one can explicitly produce a nilpotent [MATH] from the orbit corresponding to its strictly odd reduction in the sense of 2.4 as follows. The nilpotent [MATH] clearly lies in the degree 2 subspace [MATH] for the corresponding grading. This subspace is a [MATH] -module and decomposes ... |
[MATH] corresponding to simple roots with weight [MATH] Proposition 2.5 Given a nilpotent [MATH] , represent it (in a unique way) as a sum [MATH] with [MATH] and [MATH] . Then the weighted Dynkin diagram of [MATH] in the subalgebra corresponding to the subdiagram described in 2.2 is given by weights on that subdiagram. |
Proof. We have a reductive group [MATH] corresponding to [MATH] acting on [MATH] , with the element [MATH] having an open orbit in [MATH] This means that [MATH] . But this implies that [MATH] (and similarly for [MATH] ). Hence [MATH] is an open orbit in [MATH] |
Let us consider an intermediate diagram subalgebra [MATH] corresponding to the (in general disconnected) diagram, obtained by erasing the nodes with weight [MATH] but leaving all other nodes together with their weights intact. It is clear from 2.2 that [MATH] is a direct sum of [MATH] and some simple algebras of type A... |
On the other hand from 2.2 we know that there exists a (strictly odd) nilpotent element [MATH] in [MATH] , which has the needed Dynkin diagram. Then just as [MATH] , we can view [MATH] as a nilpotent in [MATH] , having zero summands in all remaining type A components of [MATH] . It is then clear that this nilpotent wil... |
Remark 2.6 It would be convenient to supplement 2.3 with the explicit construction, from an [MATH] -triple [MATH] corresponding to a given nilpotent orbit in [MATH] , of an [MATH] -triple [MATH] for its strictly odd reduction as in 2.4 . Since [MATH] comes with a grading (determined by the weights on the corresponding ... |
[MATH] Example 2.7 For [MATH] of type D , consider the nilpotent orbit corresponding to the weighted Dynkin diagram [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] (and to the partition 5,3,2,2). One of the nilpotents in this orbit is the following sum of positive root vectors |
[EQUATION] where the subscripts denote the linear combinations of simple roots that give the corresponding positive roots. The corresponding [MATH] in the [MATH] -triple for [MATH] is the following combination of negative root vectors: |
[EQUATION] with subscripts now designating linear combinations of negatives of simple roots. Thus [MATH] determines the grading corresponding to the above weighted Dynkin diagram. It is straightforward to check that in the degree 2 subspace [MATH] , root vectors corresponding to the combinations [MATH] [MATH] [MATH] [M... |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] , i. e. of the simple root with weight 2, while the remaining positive root vectors from [MATH] lie in [MATH] . Thus according to 2.5 , a strictly odd nilpotent [MATH] in the diagram subalgebra [MATH] of type D corresponding to the subdiagram obtained by omitting the node with ... |
[EQUATION] Now if we attempt to choose for the companion of [MATH] in the [MATH] -triple the element [MATH] obtained in the same way from [MATH] , i. e. by omitting in the sum for [MATH] the summands that lie in [MATH] , we obtain |
[EQUATION] However it turns out that [MATH] is not the semisimple element determining the needed grading of [MATH] . As a matter of fact this element is not semisimple, rather it has form |
[EQUATION] with [MATH] in the Cartan subalgebra of [MATH] . A correct [MATH] (the one with [MATH] an element in the Cartan subalgebra of [MATH] which gives the correct grading of [MATH] ) is |
[EQUATION] and is thus not obtained from [MATH] by projecting it to [MATH] or in any other readily apparent way. Let us add that there are also many examples (even for algebras of type A) when the bracket of the projections [MATH] of [MATH] and [MATH] is semisimple but does not induce the required grading on [MATH] |
Maximizing abelian subspaces We are interested in abelian subspaces of [MATH] . First of all, one has the following well-known fact. |
Proposition 3.1 Dimension of [MATH] is even, and the largest possible dimension of an abelian subspace in [MATH] is at most [MATH] |
Proof. Let [MATH] be an element of the orbit, and choose an [MATH] -triple [MATH] with [MATH] , and [MATH] inducing the grading. Then one may define a bilinear form on [MATH] via |
[EQUATION] where [MATH] is the Killing form. It is well known that the skew-symmetric form [MATH] is nondegenerate (since [MATH] is an isomorphism), so that dimension of [MATH] is indeed even. Moreover any commuting elements of [MATH] are orthogonal with respect to this form. Since such a form does not possess isotropi... |
Remark 3.2 More generally it is known that a nondegenerate skew-symmetric form exists on the homogeneous part [MATH] of each odd degree — see , Proposition 1.2] . Thus each [MATH] is even, too. |
We now consider the abelian subalgebras in [MATH] , separately for simple algebras of classical types (right now) and for algebras of exceptional types (in Section ). |
Let us thus turn to the simple algebras of classical types. For the type A, it has been proved in that a half-dimensional abelian subspace in [MATH] exists for any nilpotent orbit. |
The central result of this section is the following characterization in terms of the associated partitions, of those strictly odd nilpotent orbits in types B, C or D which admit an abelian subspace of half the dimension in [MATH] . We will then deduce the general (not necessarily strictly odd) case using strictly odd r... |
Theorem 3.3 Given a strictly odd nilpotent in a simple Lie algebra [MATH] of type [MATH] [MATH] or [MATH] , there is an abelian subspace of half dimension in [MATH] if and only if the partition corresponding to the nilpotent satisfies one of the following conditions: |
the largest part [MATH] of the partition is even and there are no other even parts; moreover if [MATH] is of type [MATH] then [MATH] has multiplicity [MATH] |
the largest part [MATH] of the partition is odd, and either there are no other odd parts, or [MATH] is not of type [MATH] , and the only other parts are [MATH] with multiplicity [MATH] and [MATH] (with any multiplicity). |
In other words, abelian subspaces of half dimension in [MATH] occur precisely for those strictly odd nilpotents which correspond to partitions of the following kind: |
type C: [MATH] [MATH] [MATH] [MATH] type B or D: [MATH] [MATH] [MATH] [MATH] type B: [MATH] [MATH] type D: [MATH] [MATH] Proof. It will be convenient to introduce the following notations: for a partition as above, let [MATH] be the multiplicity of the number [MATH] in it. Moreover let [MATH] be the [MATH] -eigensubspac... |
As recalled in Section 1 above, the adjoint representation can be identified with the symmetric square of the standard one for type C, and with its exterior square for types B and D. |
Because of this, clearly the degree 1 part of the adjoint representation is the direct sum of spaces of the form [MATH] with [MATH] [MATH] , and |
[EQUATION] Now, from the correspondence described in Section , one has [EQUATION] Dimension of the subspace [MATH] of grading 1 with respect to the corresponding [MATH] -triple is thus given by |
[EQUATION] Given an abelian subspace in [MATH] , using 2.1 we may assume it has a basis consisting of root vectors. In particular, each of our basis vectors is situated in one of the direct summands [MATH] |
Note that any elements in [MATH] and [MATH] commute for [MATH] ; whereas when [MATH] , we will obtain a non-commuting pair as soon as our basis contains any elements of the form [MATH] and [MATH] with [MATH] and [MATH] mutually dual basis elements. We are thus forced to choose non-intersecting subsets [MATH] [MATH] in ... |
Moreover any such choice of non-intersecting subsets [MATH] [MATH] of bases of [MATH] gives indeed an abelian subspace, and we may further assume that [MATH] is the whole basis, since otherwise our abelian subspace would not be maximal. |
The case [MATH] is special, and depends on the type considered. Namely, it may happen that two basis vectors, both from [MATH] , do not commute. Two basis elements of this subspace, being the tensor products of basis vectors corresponding to [MATH] and [MATH] respectively, will commute if and only if the sum [MATH] is ... |
This is the only restriction on [MATH] for type D. For type C, there is an additional restriction that an abelian subspace of [MATH] cannot contain root vectors corresponding to both [MATH] and [MATH] (since the sum of these is the root [MATH] ). For type B, an additional restriction is that an abelian subspace of [MAT... |
It follows that to obtain a maximal abelian subspace of [MATH] , in addition to splitting the weight vector basis of [MATH] into nonintersecting subsets ( [MATH] and its complement [MATH] ), for any weights [MATH] and [MATH] corresponding to a weight basis vector in [MATH] we have to pick in [MATH] the root basis eleme... |
Thus for the maximal dimension of the piece of an abelian subspace corresponding to [MATH] we have the following possibilities: [MATH] |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] This results in the following possibilities for the maximal dimension of an abelian subspace in [MATH] |
[EQUATION] where [MATH] is the largest part of the partition. We thus want to maximize each of these quantities for [MATH] [MATH] Note that each of them is linear in all of the [MATH] separately, hence any possible maxima are attained when every [MATH] is either [MATH] or [MATH] . In fact, more is true: |
Lemma 3.4 An abelian subspace of maximal possible dimension in [MATH] can be obtained either with [MATH] [MATH] or with [MATH] [MATH] for all [MATH] |
Proof. Looking at the subsum [EQUATION] determining dimension of the abelian subspace, it is easy to see that each of the following changes: |
[EQUATION] does not decrease the dimension of the abelian subspace. Indeed, these changes do not affect any other summands except those in the above subsum. The first change transforms |
[EQUATION] i. e. changes the sum by the amount equal to the change from [MATH] to [MATH] . But [MATH] , and [MATH] by ( ), so that indeed the sum does not decrease. |
Similarly, the second change transforms [EQUATION] i. e. changes the sum by the amount equal to the change from [MATH] to [MATH] , which is obviously a nondecreasing change. |
Now using the above changes we may arrive at one of the needed choices. For simplicity, let us encode a given choice of [MATH] ’s by a sequence of zeroes and ones (at the [MATH] th place of the sequence stands zero if [MATH] and one if [MATH] ). We are allowed to perform “local transformations” of the kind [MATH] and [... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.