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Geometric representation theory of restricted Lie algebras of classical type
 BEZRUKAVNIKOV, IVAN MIRKOVIĆ, AND DMITRIY RUMYNIN
"... Abstract. We modify the Hochschild ϕmap to construct central extensions of a restricted Lie algebra. Such central extension gives rise to a group scheme which leads to a geometric construction of unrestricted representations. For a classical semisimple Lie algebra, we construct equivariant line bun ..."
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Abstract. We modify the Hochschild ϕmap to construct central extensions of a restricted Lie algebra. Such central extension gives rise to a group scheme which leads to a geometric construction of unrestricted representations. For a classical semisimple Lie algebra, we construct equivariant line bundles whose global sections afford representations with a nilpotent pcharacter. Let G be a connected simply connected semisimple algebraic group over an algebraically closed field K of characteristic p and g be its Lie algebra. The representation theory of g is connected with the coadjoint orbits through the notion of a pcharacter [27, 3, 14, 10]. An irreducible representation ρ is finitedimensional and determines a pcharacter χ ∈ g ∗ by χ(x) p Id = ρ(x) p −ρ(x [p] ) for each x ∈ g [27]. There are indications that a geometry stands behind this representation theory, for instance, the KacWeisfeiler conjecture proved by Premet [21]. This work has been motivated by an idea of Humphreys that the representations affording χ should be related to the Springer fiber B χ. Some of our intuition comes from algebraic calculations of Jantzen [12, 13]. The most interesting evidence for the relation between Springer fibers and representations of g is now given by Lusztig [17]. The main goal of this paper is to introduce a method for constructing unrestricted representations of g by taking global sections of line bundles on infinitesimal neighborhoods of certain subvarieties of B χ. A more general approach implementing twisted sheaves of crystalline differential operators will be explained elsewhere. An attempt to study representations of g with a single pcharacter χ has led to the notion of a reduced enveloping algebra. We modify this approach by considering a set of p different pcharacters {0, χ, 2χ,...,
RESTRICTED SIMPLE LIE ALGEBRAS AND THEIR INFINITESIMAL DEFORMATIONS
, 2007
"... Abstract. In the first two sections, we review the BlockWilsonPremetStrade classification of restricted simple Lie algebras. In the third section, we compute their infinitesimal deformations. In the last section, we indicate some possible generalizations by formulating some open problems. 1. Rest ..."
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Abstract. In the first two sections, we review the BlockWilsonPremetStrade classification of restricted simple Lie algebras. In the third section, we compute their infinitesimal deformations. In the last section, we indicate some possible generalizations by formulating some open problems. 1. Restricted Lie algebras We fix a field F of characteristic p> 0 and we denote with Fp the prime field with p elements. All the Lie algebras that we will consider are of finite dimension over F. We are interested in particular class of Lie algebras, called restricted (or pLie algebras). Definition 1.1 (Jacobson [JAC37]). A Lie algebra L over F is said to be restricted (or a pLie algebra) if there exits a map (called pmap), [p] : L → L, x ↦ → x [p], which verifies the following conditions: (1) ad(x [p]) = ad(x) [p] for every x ∈ L. (2) (αx)[p] = αpx [p] for every x ∈ L and every α ∈ F. (3) (x0 + x1) [p] = x [p] 0 + x[p] 1 + ∑ p−1 i=1 si(x0, x1) for every x, y ∈ L, where the element si(x0, x1) ∈ L is defined by si(x0, x1) = − 1 ∑ adxu(1) ◦ adxu(2) ◦ · · · ◦ adxu(p−1)(x1), r u the summation being over all the maps u: [1, · · · , p − 1] → {0, 1} taking rtimes the value 0. Example. (1) Let A an associative Falgebra. Then the Lie algebra DerFA of Fderivations of A is a restricted Lie algebra with respect to the pmap D ↦ → Dp: = D ◦ · · · ◦ D. (2) Let G a group scheme over F. Then the Lie algebra Lie(G) associated to G is a restricted Lie algebra with respect to the pmap given by the differential of the homomorphism G → G, x ↦ → xp: = x ◦ · · · ◦ x. One can naturally ask when a FLie algebra can acquire the structure of a restricted Lie algebra and how many such structures there can be. The following criterion of Jacobson answers to that question. Proposition 1.2 (Jacobson). Let L be a Lie algebra over F. Then (1) It is possible to define a pmap on L if and only if, for every element x ∈ L, the pth iterate of ad(x) is still an inner derivation. (2) Two such pmaps differ by a semilinear map from L to the center Z(L) of L, that is a map f: L → Z(L) such that f(αx) = α p f(x) for every x ∈ L and α ∈ F.
On The Distribution Of ARComponents Of Restricted Lie Algebras
 Contemp. Math
"... In this paper we study the distribution of indecomposable modules of the reduced enveloping algebra u(L; ) associated to a finite dimensional restricted Lie algebra (L; [p]). Each component of the stable AuslanderReiten quiver possesses only finitely modules of a given dimension. We combine this fa ..."
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In this paper we study the distribution of indecomposable modules of the reduced enveloping algebra u(L; ) associated to a finite dimensional restricted Lie algebra (L; [p]). Each component of the stable AuslanderReiten quiver possesses only finitely modules of a given dimension. We combine this fact with geometric techniques in order to produce families of components of type ZZ[A 1 ]. 0. Introduction and Preliminaries Let F be an algebraically closed field, a finite dimensional F algebra of infinite representation type. If is tame, it follows from Brauer Thrall II and the work of CrawleyBoevey [5, Corollary E] that there exists a number d 2 IN, such that for each multiple `d there are infinitely many nonisomorphic indecomposablemodules of dimension `d. In case is wild, this conclusion continues to hold (with d even) since there is a representation embedding from the module category of the Kronecker algebra F [X; Y ]=(X 2 ; Y 2 ) into the module category of . A related questi...
Schemes of Tori and the Structure of Tame Restricted Lie Algebras
"... This paper was written while the second author was visiting the Sonderforschungsbereich 343 at the University of Bielefeld. He gratefully acknowledges the hospitality and support he has received. 1. Schemes of Tori ..."
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This paper was written while the second author was visiting the Sonderforschungsbereich 343 at the University of Bielefeld. He gratefully acknowledges the hospitality and support he has received. 1. Schemes of Tori
ON RESTRICTED LEIBNIZ ALGEBRAS
"... Abstract. In this paper we prove that in prime characteristic there is a functor ..."
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Abstract. In this paper we prove that in prime characteristic there is a functor
On Infinitesimal Groups of Tame Representation Type
"... this paper innitesimal groups whose distribution algebras admit a principal block of tame representation type. This problem has two interrelated aspects concerning the isomorphism types of the underlying groups and the Morita equivalence classes of their Hopf algebras. The corresponding program for ..."
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this paper innitesimal groups whose distribution algebras admit a principal block of tame representation type. This problem has two interrelated aspects concerning the isomorphism types of the underlying groups and the Morita equivalence classes of their Hopf algebras. The corresponding program for representation nite innitesimal groups was carried out in [11, 12], and we will occasionally invoke the results of these papers in order to deal with tame innitesimal groups. Our approach combines basic results on dimensions of rank varieties [34, 27, 16] with recent work on schemes of tori of restricted Lie algebras [13]. Tame innitesimal groups are known to possess twodimensional rank varieties, a property that is inherited by all subgroups. On the other hand, factor groups of tame groups are tame or representation nite, so that the dimensions of their varieties are also bounded by 2. Schemes of tori of maximal dimension enable us, via the construction of certain characteristic ideals, to link properties of rank varieties to the structure of the Lie algebras of innitesimal groups. Roughly speaking, the geometric methods usually aord a reduction to the consideration of relatively small groups, whose Gabriel quivers are well enough understood to determine their representation type. Our paper is organized as follows. In section 1 we collect basic results on the solvable radical and the center of innitesimal groups with a tame principal block. We then turn
RESTRICTED INFINITESIMAL DEFORMATIONS OF RESTRICTED SIMPLE LIE ALGEBRAS
, 705
"... Abstract. We compute the restricted infinitesimal deformations of the restricted simple Lie algebras over an algebraically closed field of characteristic p ≥ 5. 1. ..."
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Abstract. We compute the restricted infinitesimal deformations of the restricted simple Lie algebras over an algebraically closed field of characteristic p ≥ 5. 1.
SCHUNCK CLASSES OF SOLUBLE RESTRICTED LIE ALGEBRAS
, 2006
"... Abstract. I set out the theory of Schunck classes and projectors for soluble restricted Lie algebras and investigate its links to the corresponding theory for ordinary soluble Lie algebras over a field F of characteristic p ̸ = 0. 1. ..."
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Abstract. I set out the theory of Schunck classes and projectors for soluble restricted Lie algebras and investigate its links to the corresponding theory for ordinary soluble Lie algebras over a field F of characteristic p ̸ = 0. 1.
On the Structure of Cohomology of Hamiltonian pAlgebras
, 2004
"... Abstract. We demonstrate advantages of nonstandard grading for computing cohomology of restricted Hamiltonian and Poisson algebras. These algebras contain the inner grading element in the properly defined symmetric grading compatible with the symplectic structure. Using modulo p analog of the theor ..."
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Abstract. We demonstrate advantages of nonstandard grading for computing cohomology of restricted Hamiltonian and Poisson algebras. These algebras contain the inner grading element in the properly defined symmetric grading compatible with the symplectic structure. Using modulo p analog of the theorem on the structure of cohomology of Lie algebra with inner grading element, we show that all nontrivial cohomology classes are located in the grades which are the multiples of the characteristic p. Besides, this grading implies another symmetries in the structure of cohomology. These symmetries are based on the Poincaré duality and symmetry with respect to transpositions of conjugate variables of the symplectic space. Some results obtained by computer program utilizing these peculiarities in the cohomology structure are presented. 1
On the Cohomology of Modular Lie Algebras
, 2006
"... Dedicated to Robert L. Wilson and James Lepowsky on the occasion of their 60 th birthdays In this paper we establish a connection between the cohomology of a modular Lie algebra and its penvelopes. We also compute the cohomology of Zassenhaus algebras and their minimal penvelopes with coefficients ..."
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Dedicated to Robert L. Wilson and James Lepowsky on the occasion of their 60 th birthdays In this paper we establish a connection between the cohomology of a modular Lie algebra and its penvelopes. We also compute the cohomology of Zassenhaus algebras and their minimal penvelopes with coefficients in generalized baby Verma modules and in simple modules over fields of characteristic p> 2.