Nilpotent Varieties Research Articles

Preprojective algebras and MV polytopes

The purpose of this paper is to apply the theory of MV polytopes to the study of components of Lusztig’s nilpotent varieties. Along the way, we introduce reflection functors for modules over the non-deformed preprojective algebra of a quiver.

Abelian and nilpotent varieties of power groups

Abstract The present paper continues the series of the papers [Amaglobeli and Bokelavadze, Exponential groups. III. Groups which are exact for tensor complement, Herald of Omsk University, 2009, Myasnikov and Remeslennikov, Sibirsk. Mat. Zh. 35: 1106–1118, 1994, Myasnikov and Remeslennikov, Internat. J. Algebra Comput. 6: 687–711, 1996] and is dedicated to the construction of basic principles of the theory of power group varieties and tensor completions of groups in a variety. We study the relationship between free groups of a given variety for various rings of scalars. Varieties of abelian power groups are described.

Codimension Growth of Strong Lie Nilpotent Associative Algebras

We study Lie nilpotent varieties of associative algebras. We explicitly compute the codimension growth for the variety of strong Lie nilpotent associative algebras. The codimension growth is polynomial and found in terms of Stirling numbers of the first kind. To achieve the result we take the free Lie algebra of countable rank L(X), consider its filtration by the lower central series and shift it. Next we apply generating functions of special type to the induced filtration of the universal enveloping algebra U(L(X)) = A(X).

A Hessenberg Generalization of the Garsia-Procesi Basis for the Cohomology Ring of Springer Varieties

The Springer variety is the set of flags stabilized by a nilpotent operator. In 1976, T.A. Springer observed that this variety's cohomology ring carries a symmetric group action, and he offered a deep geometric construction of this action. Sixteen years later, Garsia and Procesi made Springer's work more transparent and accessible by presenting the cohomology ring as a graded quotient of a polynomial ring. They combinatorially describe an explicit basis for this quotient. The goal of this paper is to generalize their work. Our main result deepens their analysis of Springer varieties and extends it to a family of varieties called Hessenberg varieties, a two-parameter generalization of Springer varieties. Little is known about their cohomology. For the class of regular nilpotent Hessenberg varieties, we conjecture a quotient presentation for the cohomology ring and exhibit an explicit basis. Tantalizing new evidence supports our conjecture for a subclass of regular nilpotent varieties called Peterson varieties.

Nilpotent centralizers and Springer isomorphisms

Let G be a semisimple algebraic group over a field K whose characteristic is very good for G , and let σ be any G -equivariant isomorphism from the nilpotent variety to the unipotent variety; the map σ is known as a Springer isomorphism. Let y ∈ G ( K ) , let Y ∈ Lie ( G ) ( K ) , and write C y = C G ( y ) and C Y = C G ( Y ) for the centralizers. We show that the center of C y and the center of C Y are smooth group schemes over K . The existence of a Springer isomorphism is used to treat the crucial cases where y is unipotent and where Y is nilpotent. Now suppose G to be quasisplit, and write C for the centralizer of a rational regular nilpotent element. We obtain a description of the normalizer N G ( C ) of C , and we show that the automorphism of Lie ( C ) determined by the differential of σ at zero is a scalar multiple of the identity; these results verify observations of J.-P. Serre.

Nilpotent orbits in classical Lie algebras over F 2 n and the Springer correspondence

We give the number of nilpotent orbits in the Lie algebras of orthogonal groups under the adjoint action of the groups over F(2(n)). Let G be an adjoint algebraic group of type B, C, or D defined over an algebraically closed field of characteristic 2. We construct the Springer correspondence for the nilpotent variety in the Lie algebra of G.

Nilpotent variety of a reductive monoid

In this paper we study the variety M nil of nilpotent elements of a reductive monoid M. In general this variety has a completely different structure than the variety G uni of unipotent elements of the unit group G of M. When M has a unique non-trivial minimal or maximal G×G-orbit, we find a precise description of the irreducible components of M nil via the combinatorics of the Renner monoid of M and the Weyl group of G. In particular for a semisimple monoid M, we find necessary and sufficient conditions for the variety M nil to be irreducible.

On symmetric invariants of centralisers in reductive Lie algebras

Let g be a finite-dimensional simple Lie algebra of rank l over an algebraically closed field of characteristic 0. Let e be a nilpotent element of g and let g e be the centraliser of e in g . In this paper we study the algebra S ( g e ) g e of symmetric invariants of g e . We prove that if g is of type A or C, then S ( g e ) g e is always a graded polynomial algebra in l variables, and we show that this continues to hold for some nilpotent elements in the Lie algebras of other types. In type A we prove that the invariant algebra S ( g e ) g e is freely generated by a regular sequence in S ( g e ) and describe the tangent cone at e to the nilpotent variety of g .

Kazhdan-Lusztig Basis and a Geometric Filtration of an Affine Hecke Algebra

AbstractAccording to Kazhdan-Lusztig and Ginzburg, the Hecke algebra of an affine Weyl group is identified with the equivariant K-group of Steinberg’s triple variety. The K-group is equipped with a filtration indexed by closed G-stable subvarieties of the nilpotent variety, where G is the corresponding reductive algebraic group over ℂ. In this paper we will show in the case of type A that the filtration is compatible with the Kazhdan-Lusztig basis of the Hecke algebra.

Nilpotent Orbits of $ℤ_4$-Graded Lie Algebra and Geometry of Moment Maps Associated to the Dual Pair $(U(p,q), U(r,s))$

Let \mathfrak s^1 ← L_+ → \mathfrak s^2 be the K_ℂ -versions of the moment maps associated to the dual pair (U(p,q), U(r,s)) and \mathcal N(\mathfrak s^1) ← \mathcal N(L_+) → \mathcal N(\mathfrak s^2) their restrictions to the nilpotent varieties. In this paper, we first describe the nilpotent orbit correspondence via the moment maps explicitly. Second, under the condition \min\{p, q\} ≥ \max\{r, s\} , we show that there are open subvariety L'_+ (resp. ( (\mathfrak s^2)' ) of L_+ (resp. \mathfrak s^2 ) and locally closed subvariety (\mathfrak s^1)' of \mathfrak s^1 such that the restrictions of the moment maps \mathcal N((\mathfrak s^1)') ← \mathcal N(L'_+) → \mathcal N((\mathfrak s^2)') give bijections of nilpotent orbits. Furthermore, we show that the bijections preserve the closure relation and the equivalence class of singularities.

Words and Almost Nilpotent Varieties of Groups#

For a word of a free group of rank n , the author obtains an invariant called its standard exponent, and shows that if any residually finite group satisfying the law defined by such a word is almost nilpotent, then the standard exponent of the word equals 1 . Conversely, if the standard exponent of a word ω is 1 , then any residually finite or soluble group and any locally finite or soluble group satisfying the group law ω≡ 1 is nilpotent-of-bounded-class-by-bounded-exponent.

Relative Springer isomorphisms

Let G be a simple algebraic group over the algebraically closed field k. A slightly strengthened version of a theorem of T.A. Springer says that (under some mild restrictions on G and k) there exists a G-equivariant isomorphism of varieties ϕ : U → N , where U denotes the unipotent variety of G and N denotes the nilpotent variety of g = Lie G . Such ϕ is called a Springer isomorphism. Let B be a Borel subgroup of G, U the unipotent radical of B and u the Lie algebra of U. In this note we show that a Springer isomorphism ϕ induces a B-equivariant isomorphism ϕ ˜ : U / M → u / m , where M is any unipotent normal subgroup of B and m = Lie M . We call such a map ϕ ˜ a relative Springer isomorphism. We also use relative Springer isomorphisms to describe the geometry of U-orbits in u .

Quiver varieties of type A

We prove a conjecture of Nakajima describing the relation between quiver varieties of type A and the geometry of partial flag varieties and of the nilpotent variety.

Semicanonical bases and preprojective algebras

We study the multiplicative properties of the dual of Lusztig's semicanonical basis. The elements of this basis are naturally indexed by the irreducible components of Lusztig's nilpotent varieties, which can be interpreted as varieties of modules over preprojective algebras. We prove that the product of two dual semicanonical basis vectors ρ Z ′ and ρ Z ″ is again a dual semicanonical basis vector provided the closure of the direct sum of the corresponding two irreducible components Z ′ and Z ″ is again an irreducible component. It follows that the semicanonical basis and the canonical basis coincide if and only if we are in Dynkin type A n with n ⩽ 4 . Finally, we provide a detailed study of the varieties of modules over the preprojective algebra of type A 5 . We show that in this case the multiplicative properties of the dual semicanonical basis are controlled by the Ringel form of a certain tubular algebra of type ( 6 , 3 , 2 ) and by the corresponding elliptic root system of type E 8 ( 1 , 1 ) .

Quantum groups, the loop Grassmannian, and the Springer resolution

We establish equivalences of the following three triangulated categories: \[ D quantum ( g ) ⟷ D coherent G ( N ~ ) ⟷ D perverse ( G r ) . D_\text {quantum}(\mathfrak {g})\enspace \longleftrightarrow \enspace D^G_\text {coherent}(\widetilde {{\mathcal N}})\enspace \longleftrightarrow \enspace D_\text {perverse}(\mathsf {Gr}). \] Here, D quantum ( g ) D_\text {quantum}(\mathfrak {g}) is the derived category of the principal block of finite-dimensional representations of the quantized enveloping algebra (at an odd root of unity) of a complex semisimple Lie algebra g \mathfrak {g} ; the category D coherent G ( N ~ ) D^G_\text {coherent}(\widetilde {{\mathcal N}}) is defined in terms of coherent sheaves on the cotangent bundle on the (finite-dimensional) flag manifold for G G ( = = semisimple group with Lie algebra g \mathfrak {g} ), and the category D perverse ( G r ) D_\text {perverse}({\mathsf {Gr}}) is the derived category of perverse sheaves on the Grassmannian G r {\mathsf {Gr}} associated with the loop group L G ∨ LG^\vee , where G ∨ G^\vee is the Langlands dual group, smooth along the Schubert stratification. The equivalence between D quantum ( g ) D_\text {quantum}(\mathfrak {g}) and D coherent G ( N ~ ) D^G_\text {coherent}(\widetilde {{\mathcal N}}) is an “enhancement” of the known expression (due to Ginzburg and Kumar) for quantum group cohomology in terms of nilpotent variety. The equivalence between D perverse ( G r ) D_\text {perverse}(\mathsf {Gr}) and D coherent G ( N ~ ) D^G_\text {coherent}(\widetilde {{\mathcal N}}) can be viewed as a “categorification” of the isomorphism between two completely different geometric realizations of the (fundamental polynomial representation of the) affine Hecke algebra that has played a key role in the proof of the Deligne-Langlands-Lusztig conjecture. One realization is in terms of locally constant functions on the flag manifold of a p p -adic reductive group, while the other is in terms of equivariant K K -theory of a complex (Steinberg) variety for the dual group. The composite of the two equivalences above yields an equivalence between abelian categories of quantum group representations and perverse sheaves. A similar equivalence at an even root of unity can be deduced, following the Lusztig program, from earlier deep results of Kazhdan-Lusztig and Kashiwara-Tanisaki. Our approach is independent of these results and is totally different (it does not rely on the representation theory of Kac-Moody algebras). It also gives way to proving Humphreys’ conjectures on tilting U q ( g ) U_q(\mathfrak {g}) -modules, as will be explained in a separate paper.

Normality of nilpotent varieties in E6

We determine which nilpotent orbits in E 6 have closures which are normal varieties and which do not. At the same time we are able to verify a conjecture in [E. Sommers, Comm. Math. Univ. Sancti Pauli 49 (1) (2000) 101–104] concerning functions on non-special nilpotent orbits for E 6.

Certain locally nilpotent varieties of groups

Let c ≥ 0, d ≥ 2 be integers and be the variety of groups in which every d-generator subgroup is nilpotent of class at most c. N.D. Gupta asked for what values of c and d is it true that is locally nilpotent? We prove that if c ≤ 2d + 2d−1 − 3 then the variety is locally nilpotent and we reduce the question of Gupta about the periodic groups in to the prime power exponent groups in this variety.

Solid varieties of arbitrary type

A solid variety is an equational class in which every identity holds as a hyperidentity as well, meaning that it is satisfied not just by the fundamental operations but also by all terms of the appropriate arity. For type (2), an infinite number of solid varieties (of semigroups) are known, but for other types very few examples of solid varieties are known. In this paper we present several constructions which produce infinite chains of solid varieties. One construction generalizes the normalization of a variety, and gives a method to produce a chain of solid varieties from any given solid variety of type (n). The second construction generalizes the rectangular nilpotent varieties of type (2) to type (n). Finally, we use identities which are consequences of idempotency to construct an infinite chain of solid varieties of any fixed type.

Affine Type A Crystal Structure on Tensor Products of Rectangles, Demazure Characters, and Nilpotent Varieties

Answering a question of Kuniba, Misra, Okado, Takagi, and Uchiyama, it is shown that certain higher level Demazure characters of affine type A, coincide with the graded characters of coordinate rings of closures of conjugacy classes of nilpotent matrices.

Two results on equations of nilpotent orbits

We prove two results on the defining ideals of certain varieties of matrices. Let us fix two positive integers r, e. Let M(r) be the set of r x r matrices over a field K. We consider the closed subscheme of the nilpotent variety of M(r) over K defined by the conditions char_A(T)=T^r, A^e=0. We prove that when the characteristic of K is zero this scheme is reduced. Also for e=2 we prove that this scheme is reduced over a field K of arbitrary characteristic. These results were motivated by the questions of G. Pappas and M. Rapoport (compare their paper ''Local models in the ramified case I. The EL-case, math.AG/0006222) and give answers to some of their conjectures.