The Experts below are selected from a list of 318 Experts worldwide ranked by ideXlab platform
A N Zubkov - One of the best experts on this subject based on the ideXlab platform.
-
linkage principle for ortho symplectic Supergroups
Journal of Algebra, 2018Co-Authors: Frantisek Marko, A N ZubkovAbstract:Abstract The purpose of the paper is to derive linkage principle for modular representations of ortho-symplectic Supergroups. We follow the approach of Doty and investigate in detail the representation theory of the ortho-symplectic group S p O ( 2 | 1 ) and that of its Frobenius thickening. Using the description of flags and adjacent Borel supersubgroups we derive first the strong linkage for the Frobenius thickening G r T of the ortho-symplectic supergroup G of type S p O ( 2 m | 2 n + 1 ) and S p O ( 2 m | 2 n ) . Based on this, we derive the linkage principle for ortho-symplectic Supergroups S p O ( 2 m | 2 n + 1 ) and S p O ( 2 m | 2 n ) .
-
solvability and nilpotency for algebraic Supergroups
Journal of Pure and Applied Algebra, 2017Co-Authors: Akira Masuoka, A N ZubkovAbstract:Abstract We study solvability, nilpotency and splitting property for algebraic Supergroups over an arbitrary field K of characteristic char K ≠ 2 . Our first main theorem tells us that an algebraic supergroup G is solvable if the associated algebraic group G e v is trigonalizable. To prove it we determine the algebraic Supergroups G such that dim Lie ( G ) 1 = 1 ; their representations are studied when G e v is diagonalizable. The second main theorem characterizes nilpotent connected algebraic Supergroups. A super-analogue of the Chevalley Decomposition Theorem is proved, though it must be in a weak form. An appendix is given to characterize smooth Noetherian superalgebras as well as smooth Hopf superalgebras.
-
solvable reductive and quasireductive Supergroups
Journal of Algebra, 2016Co-Authors: Alexander Grishkov, A N ZubkovAbstract:It is well known that if the ground field K has characteristic zero and G is a connected algebraic group, defined over K, then the Lie algebra Lie(G′) of the commutant G′ of G coincides with the commutant Lie(G)′ of Lie(G). We show that this result is no longer true in the category of algebraic Supergroups. We also construct a reductive supergroup H=X⋊G, where X and G are connected, reduced and abelian Supergroups, such that Xu≠1 and (Hev)u is non-trivial connected (super)group. Quasi-reductive Supergroups have been introduced in [10]. We prove that a supergroup H is quasi-reductive if and only if the largest even (super)subgroup of the solvable radical R(H) is a torus, H˜=H/R(H) contains a normal supersubgroup U, which is quasi-isomorphic to a direct product of normal supersubgroups Ui, and H˜/U is a triangulizable supergroup with odd unipotent radical. Moreover, for every i, Lie(Ui)=Ui⊗Sym(ni) are such that either ni=0 and Ui is a classical simple Lie superalgebra, or ni=1 and Ui is a simple Lie algebra.
-
solvability and nilpotency for algebraic Supergroups
arXiv: Algebraic Geometry, 2015Co-Authors: Akira Masuoka, A N ZubkovAbstract:We study solvability, nilpotency and splitting property for algebraic Supergroups over an arbitrary field $K$ of characteristic $\mathrm{char}\, K \ne 2$. Our first main theorem tells us that an algebraic supergroup $\mathbb{G}$ is solvable if the associated algebraic group $\mathbb{G}_{ev}$ is trigonalizable. To prove it we determine the algebraic Supergroups $\mathbb{G}$ such that $\dim \mathrm{Lie}(\mathbb{G})_1=1$; their representations are studied when $\mathbb{G}_{ev}$ is diagonalizable. The second main theorem characterizes nilpotent connected algebraic Supergroups. A super-analogue of the Chevalley Decomposition Theorem is proved, though it must be in a weak form. An appendix is given to characterize smooth Noetherian superalgebras as well as smooth Hopf superalgebras.
-
solvable reductive and quasireductive Supergroups
arXiv: Representation Theory, 2013Co-Authors: Alexander Grishkov, A N ZubkovAbstract:This work was inspired by two natural questions. The first question is when Lie(G')=Lie(G)', where G is a connected algebraic supergroup defined over a field of characteristic zero. The second question is whether the unipotent radical of any normal supersubgroup H of G coincides with the intersection of H and G_u, where G_u is the unipotent radical of G. Both questions have affirmative answers in the category of algebraic groups (in the second case one has to assume additionally that G and H are reduced whenever char K >0). Surprisingly, using the technique of Harish-Chandra superpairs and a complete description of an action of an algebraic supergroup on an abelian Supergroups by supergroup automorphisms we found out rather simple counterexamples to both questions. Besides, the second counterexample shows that the reductivity of G does not imply that G_{ev} has even finite unipotent radical. On the other hand, if G_{ev} is reductive, then it is easy to see that G_u is finite (odd) supergroup. In other words, the reductivity of an algebraic supergroup does not correspond to the reductivity of its largest even subgroup in contrast to such properties as unipotency or solvability. In the last section of our article we describe reductive algebraic Supergroups in terms of sandwich pairs and give necessary and sufficient conditions for an algebraic supergroup to be quasireductive. The last result complements the recent Serganova's description of quasireductive Supergroups in terms of structural properties of their Lie superalgebras. Our approach is focused on the normal subgroup structure.
Hadi Salmasian - One of the best experts on this subject based on the ideXlab platform.
-
crossed product algebras and direct integral decomposition for lie Supergroups
Pacific Journal of Mathematics, 2016Co-Authors: Karlhermann Neeb, Hadi SalmasianAbstract:For every nite dimensional Lie supergroup ( G;g), we dene a C -algebraA :=A(G;g), and show that there exists a canonical bijective correspondence between unitary representations of (G;g) and nondegenerate -representations ofA. The proof of existence of such a correspondence relies on a subtle characterization of smoothing operators of unitary representations from [NSZ]. For a broad class of Lie Supergroups, which includes nilpotent as well as classical simple ones, we prove that the associated C -algebra is CCR. In particular, we obtain the uniqueness of direct integral decomposition for unitary representations of these Lie Supergroups.
-
categories of unitary representations of banach lie Supergroups and restriction functors
Pacific Journal of Mathematics, 2012Co-Authors: Stephane Merigon, Karlhermann Neeb, Hadi SalmasianAbstract:We prove two results which show that the categories of smooth and analytic unitary representations of a Banach‐Lie supergroup are well-behaved. The first result states that the restriction functor corresponding to any homomorphism of Banach‐Lie Supergroups is well-defined. The second result shows that the category of analytic representations is isomorphic to a subcategory of the category of smooth representations. These facts are needed as a crucial first step to a rigorous treatment of the analytic theory of unitary representations of Banach‐Lie Supergroups. They extend the known results for finite-dimensional Lie Supergroups. In the infinite-dimensional case the proofs require several new ideas. As an application, we give an analytic realization of the oscillator representation of the restricted orthosymplectic Banach‐Lie supergroup.
-
categories of unitary representations of banach lie Supergroups and restriction functors
arXiv: Representation Theory, 2011Co-Authors: Stephane Merigon, Karlhermann Neeb, Hadi SalmasianAbstract:We prove that the categories of smooth and analytic unitary representations of Banach--Lie Supergroups are well-behaved under restriction functors, in the sense that the restriction of a representation to an integral subsupergroup is well-defined. We also prove that the category of analytic representations is isomorphic to a subcategory of the category of smooth representations. These facts are needed as a crucial first step to a rigorous treatment of the analytic theory of unitary representations of Banach--Lie Supergroups. They extend the known results for finite dimensional Lie Supergroups. In the infinite dimensional case the proofs require several new ideas. As an application, we give an analytic realization of the oscillator representation of the restricted orthosymplectic Banach--Lie supergroup.
-
lie Supergroups unitary representations and invariant cones
arXiv: Representation Theory, 2011Co-Authors: Karlhermann Neeb, Hadi SalmasianAbstract:The goal of this article is twofold. First, it presents an application of the theory of invariant convex cones of Lie algebras to the study of unitary representations of Lie Supergroups. Second, it provides an exposition of recent results of the second author on the classification of irreducible unitary representations of nilpotent Lie Supergroups using the method of orbits.
A N Manashov - One of the best experts on this subject based on the ideXlab platform.
-
the baxter q operator for the graded sl 2 1 spin chain
Journal of Statistical Mechanics: Theory and Experiment, 2007Co-Authors: Andrei Belitsky, S E Derkachov, G P Korchemsky, A N ManashovAbstract:We study an integrable non-compact superspin chain model that emerged in recent studies of the dilatation operator in the super-Yang–Mills theory. It was found that the latter can be mapped into a homogeneous Heisenberg magnet with the quantum space in all sites corresponding to infinite dimensional representations of the SL(2|1) group. We extend the method of the Baxter Q-operator to spin chains with supergroup symmetry and apply it to determine the eigenspectrum of the model. Our analysis relies on a factorization property of the -operators acting on the tensor product of two generic infinite dimensional SL(2|1) representations. It allows us to factorize an arbitrary transfer matrix into a product of three 'elementary' transfer matrices which we identify as Baxter Q-operators. We establish functional relations between transfer matrices and use them to derive the T–Q relations for the Q-operators. The proposed construction can be generalized to integrable models based on Supergroups of higher rank and, as distinct from the Bethe ansatz, it is not sensitive to the existence of the pseudovacuum state in the quantum space of the model.
-
baxter q operator for graded sl 2 1 spin chain
arXiv: High Energy Physics - Theory, 2006Co-Authors: Andrei Belitsky, S E Derkachov, G P Korchemsky, A N ManashovAbstract:We study an integrable noncompact superspin chain model that emerged in recent studies of the dilatation operator in the N=1 super-Yang-Mills theory. It was found that the latter can be mapped into a homogeneous Heisenberg magnet with the quantum space in all sites corresponding to infinite-dimensional representations of the SL(2|1) group. We extend the method of the Baxter Q-operator to spin chains with supergroup symmetry and apply it to determine the eigenspectrum of the model. Our analysis relies on a factorization property of the R-operators acting on the tensor product of two generic infinite-dimensional SL(2|1) representations. It allows us to factorize an arbitrary transfer matrix into a product of three `elementary' transfer matrices which we identify as Baxter Q-operators. We establish functional relations between transfer matrices and use them to derive the TQ-relations for the Q-operators. The proposed construction can be generalized to integrable models based on Supergroups of higher rank and, in distinction to the Bethe Ansatz, it is not sensitive to the existence of the pseudovacuum state in the quantum space of the model.
Fabio Gavarini - One of the best experts on this subject based on the ideXlab platform.
-
Erratum to \Algebraic Supergroups of Cartan type"
2020Co-Authors: Fabio GavariniAbstract:In this note I x a mistake in my previous paper [6]: namely, the result concerning the uniqueness (up to isomorphisms) of such Supergroups needs a new formulation and proof. By the same occasion, I explain more in detail the existence result which comes out of the construction of Chevalley Supergroups.
-
real forms of complex lie superalgebras and Supergroups
arXiv: Rings and Algebras, 2020Co-Authors: R Fioresi, Fabio GavariniAbstract:We investigate the notion of real form of complex Lie superalgebras and Supergroups, both in the standard and graded version. Our functorial approach allows most naturally to go from the superalgebra to the supergroup and retrieve the real forms as fixed points, as in the ordinary setting. We also introduce a more general notion of compact real form for Lie superalgebras and Supergroups, and we prove some existence results for Lie superalgebras that are simple contragredient and their associated connected simply connected Supergroups.
-
global splittings and super harish chandra pairs for affine Supergroups
Transactions of the American Mathematical Society, 2015Co-Authors: Fabio GavariniAbstract:This paper dwells upon two aspects of affine supergroup theory, investigating the links among them. First, I discuss the "splitting" properties of affine Supergroups, i.e. special kinds of factorizations they may admit - either globally, or pointwise. Second, I present a new contribution to the study of affine Supergroups by means of super Harish-Chandra pairs (a method already introduced by Koszul, and later extended by other authors). Namely, I provide an explicit, functorial construction \Psi which, with each super Harish-Chandra pair, associates an affine supergroup that is always globally strongly split (in short, gs-split) - thus setting a link with the first part of the paper. On the other hand, there exists a natural functor \Phi from affine Supergroups to super Harish-Chandra pairs: then I show that the new functor \Psi - which goes the other way round - is indeed a quasi-inverse to \Phi, provided we restrict our attention to the subcategory of affine Supergroups that are gs-split. Therefore, (the restrictions of) \Phi and \Psi are equivalences between the categories of gs-split affine Supergroups and of super Harish-Chandra pairs. Such a result was known in other contexts, such as the smooth differential or the complex analytic one, or in some special cases, via different approaches: the novelty in the present paper lies in that I construct a different functor \Psi and thus extend the result to a much larger setup, with a totally different, more geometrical method (very concrete indeed, and characteristic free). The case of linear algebraic groups is treated also as an intermediate, inspiring step. Some examples, applications and further generalizations are presented at the end of the paper.
-
algebraic Supergroups of cartan type
Forum Mathematicum, 2014Co-Authors: Fabio GavariniAbstract:I present a construction of connected affine algebraic Supergroups G_V associated with simple Lie superalgebras g of Cartan type and with g-modules V. Conversely, I prove that every connected affine algebraic supergroup whose tangent Lie superalgebra is of Cartan type is necessarily isomorphic to one of the Supergroups G_V that I introduced. In particular, the supergroup associated in this way with g = W(n) and its standard representation is described.
-
chevalley Supergroups of type d 2 1 a
Proceedings of the Edinburgh Mathematical Society (Series 2), 2014Co-Authors: Fabio GavariniAbstract:We present a construction “`a la Chevalley” of affine Supergroups associated to Lie superalgebras of type D(2,1;a), for any possible value of the parameter a. This extends the similar work performed in [4] which associates an affine supergroup to any other simple Lie superalgebras of classical type.
S E Derkachov - One of the best experts on this subject based on the ideXlab platform.
-
the baxter q operator for the graded sl 2 1 spin chain
Journal of Statistical Mechanics: Theory and Experiment, 2007Co-Authors: Andrei Belitsky, S E Derkachov, G P Korchemsky, A N ManashovAbstract:We study an integrable non-compact superspin chain model that emerged in recent studies of the dilatation operator in the super-Yang–Mills theory. It was found that the latter can be mapped into a homogeneous Heisenberg magnet with the quantum space in all sites corresponding to infinite dimensional representations of the SL(2|1) group. We extend the method of the Baxter Q-operator to spin chains with supergroup symmetry and apply it to determine the eigenspectrum of the model. Our analysis relies on a factorization property of the -operators acting on the tensor product of two generic infinite dimensional SL(2|1) representations. It allows us to factorize an arbitrary transfer matrix into a product of three 'elementary' transfer matrices which we identify as Baxter Q-operators. We establish functional relations between transfer matrices and use them to derive the T–Q relations for the Q-operators. The proposed construction can be generalized to integrable models based on Supergroups of higher rank and, as distinct from the Bethe ansatz, it is not sensitive to the existence of the pseudovacuum state in the quantum space of the model.
-
baxter q operator for graded sl 2 1 spin chain
arXiv: High Energy Physics - Theory, 2006Co-Authors: Andrei Belitsky, S E Derkachov, G P Korchemsky, A N ManashovAbstract:We study an integrable noncompact superspin chain model that emerged in recent studies of the dilatation operator in the N=1 super-Yang-Mills theory. It was found that the latter can be mapped into a homogeneous Heisenberg magnet with the quantum space in all sites corresponding to infinite-dimensional representations of the SL(2|1) group. We extend the method of the Baxter Q-operator to spin chains with supergroup symmetry and apply it to determine the eigenspectrum of the model. Our analysis relies on a factorization property of the R-operators acting on the tensor product of two generic infinite-dimensional SL(2|1) representations. It allows us to factorize an arbitrary transfer matrix into a product of three `elementary' transfer matrices which we identify as Baxter Q-operators. We establish functional relations between transfer matrices and use them to derive the TQ-relations for the Q-operators. The proposed construction can be generalized to integrable models based on Supergroups of higher rank and, in distinction to the Bethe Ansatz, it is not sensitive to the existence of the pseudovacuum state in the quantum space of the model.
