Lie superalgebras

Kač, Victor G.

doi:10.1016/0001-8708(77)90017-2

Cited by 2,020 publications

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“…If we restrict our attention to simple Lie superalgebras for a moment our analysis covers three types of infinite series, namely A(m, n) = sl(m|n) (for m = n), A(n, n) = psl(n|n) and C(n + 1) = osp(2|2n) [46]. But, widening Kac's original usage of the qualifiers "basic" and "type I", most of our results also apply to non-(semi)simple Lie superalgebras such as various extended Poincaré superalgebras, the general linear Lie superalgebras gl(m|n) or supersymmetric extensions of Heisenberg algebras.…”

Section: Jhep09(2007)085mentioning

confidence: 99%

“…A Lie superalgebra g = g 0 ⊕ g 1 is a graded generalization of an ordinary Lie algebra [46]. There are even (or bosonic) generators K i which form an ordinary Lie algebra g 0 , i.e.…”

Section: Commutation Relationsmentioning

confidence: 99%

“…superalgebras of type I are Kac modules K µ , µ ∈ Rep(g 0 ) [46,51]. They are induced from irreducible representations V µ of the bosonic subalgebra g 0 .…”

Section: Jhep09(2007)085mentioning

confidence: 99%

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Free fermion resolution of supergroup WZNW models

Quella

Schomerus²

2007

J. High Energy Phys.

132

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Extending our earlier work on PSL(2|2), we explain how to reduce the solution of WZNW models on general type I supergroups to those defined on the bosonic subgroup. The new analysis covers in particular the supergroups GL(M |N ) along with several close relatives such as PSL(N |N ), certain Poincaré supergroups and the series OSP (2|2N ). The technical foundation for this remarkable progress is a special Feigin-Fuchs type representation which allows to keep the bosonic symmetry manifest instead of reducing it to free fields. In preparation for the field theory analysis, we shall exploit a minisuperspace analogue of the resulting free fermion construction to deduce the spectrum of the Laplacian on type I supergroups. The latter is shown to be non-diagonalizable. After lifting these results to the full WZNW model, we address various issues of the field theory, including its modular invariance and the computation of correlation functions. In agreement with previous findings, supergroup WZNW models allow to study chiral and non-chiral aspects of logarithmic conformal field theory within a geometric framework. We shall briefly indicate how insights from WZNW models carry over to non-geometric examples, such as e.g. the W(p) triplet models.

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Section: Jhep09(2007)085mentioning

confidence: 99%

“…A Lie superalgebra g = g 0 ⊕ g 1 is a graded generalization of an ordinary Lie algebra [46]. There are even (or bosonic) generators K i which form an ordinary Lie algebra g 0 , i.e.…”

Section: Commutation Relationsmentioning

confidence: 99%

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Free fermion resolution of supergroup WZNW models

Quella

Schomerus²

2007

J. High Energy Phys.

132

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“…A complete classification of these objects was found by Kac (and others, see e.g. [14]), who split them in two main (disjoint) families: those of "classical" type -still divided into "basic" and "strange" types -and those of "Cartan" type. assumption, and allow G + to be any group-scheme over k .…”

Section: Examples Applications Generalizationsmentioning

confidence: 94%

Global splittings and super Harish-Chandra pairs for affine supergroups

Gavarini

2015

Trans. Amer. Math. Soc.

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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 point-wise. Almost everything should be more or less known, but seems to be not as clear in literature (to the author's knowledge) as it ought to.Second, I present a new contribution to the study of affine supergroups by means of super HarishChandra pairs (a method already introduced by Koszul, and later extended by other authors). Namely, Iprovide a new functorial construction Ψ 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. One knows that there exists a natural functor Φ from affine supergroups to super Harish-Chandra pairs: then I show that the new functor Ψ -which goes the other way round -is indeed a quasi-inverse to Φ , provided we restrict our attention to the subcategory of affine supergroups that are gs-split. Therefore, (the restrictions of) Φ and Ψ 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, via different approaches (see [16], [19], [7]): nevertheless, the novelty in the present paper lies in that I construct a different functor Ψ and thus extend the result to a much larger setup, with a totally different, more geometrical method. In fact, this method (very concrete, indeed) is universal and characteristic-free: I present it here for the algebro-geometric setting, but actually it can be easily adapted to the frameworks of differential or complex-analytic supergeometry.The case of linear supergroups is treated also as an intermediate, inspiring step. Some examples, applications and further generalizations are presented at the end of the paper. 1

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“…Very few of them are generalized Kac-Moody superalgebras: apart from the affine Lie algebras, only those of type A (2) (0, 2n − 1), n ≥ 3, A (2) (0, 3), C (2) (n + 1), n ≥ 2, C (2) (2), A (4) (0, 2n), n ≥ 2, and A (4) (0, 2) [Kac4], [Leu], [Ray3] are. Also Borcherds' above characterization in terms of an almost positive definite bilinear form has little meaning for superalgebras, as A(1, n), B(0, n) and B (1) (0, n) [Kac3], [Kac4] are the only Lie superalgebras which are not Lie algebras satisfying this characterization [Ray3].…”

Section: 1mentioning

confidence: 99%

Generalized Kac-Moody algebras and some related topics

Ray¹

2000

Bull. Amer. Math. Soc.

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To the memory of my father IntroductionIn this review article, we shall give an introduction to the field of generalized KacMoody algebras. This subject gained much interest when Borcherds first proved the remarkable Moonshine Conjecture, which connects two areas apparently far apart: on the one hand, the Monster simple group and on the other elliptic modular functions. Any connection found between an object which has as yet played a limited abstract role and a more fundamental concept is always very fascinating. Also it is not surprising that ideas on which the proof of such a result are based would give rise to many new questions, thus opening up different research directions and finding applications in a wide variety of areas.We shall start by explaining the Moonshine Conjecture, giving at the same time a brief history of it. We will then give a brief outline of its proof. In the following chapters, we will take one by one the important concepts used in the proof (generalized Kac-Moody algebras, Vertex algebras, automorphic forms), explain them in some detail and mention recent developments connected to them. We shall finish by briefly discussing two interesting topics related to the field of generalized Kac-Moody algebras: Lie superalgebras and lattices.The author would like to offer her apologies to anyone whose contribution was omitted or not extensively covered. This is not intentional.Remark. Throughout this paper, the base field will be C. ContentsThe modular invariant j is a complex valued holomorphic function on the upper half plane H = {x + iy|y > 0}, which is fixed by the discrete group SL(2, Z):for all τ ∈ H, and g ∈ SL(2, Z), j(gτ ) = j(τ ). GENERALIZED KAC-MOODY ALGEBRAS AND SOME RELATED TOPICS 3We remind the reader that the group Γ = SL(2, Z) acts on H in the following way.Let a, b, c, d ∈ Z be integers such that g = a b c d ∈ Γ. Then for any point τ ∈ H,The modular invariant j is meromorphic at infinity and is a bijection from the compactification of H/Γ, H ∪ {i∞}/Γ to the compact Riemann sphere C ∪ {∞} such that j(i∞) = ∞. Hence any meromorphic function on H fixed by Γ is a rational function of j. Since j(τ + 1) = j(τ ), we can write j as a function of q = e 2πiτ . Since J is holomorphic on H, as a function of q, J has a Laurent series in the punctured disc of radius 1 centered at 0: would be an interesting function. When c(n) was a known linear combination, the M -modules V (n) were to be taken to be the sums of the irreducible representations whose dimensions appeared in the linear combination. As in the case of V (1), the head representations are obviously not just sums of the trivial module! Thus their work suggested these series, now known as the McKay-Thompson series, were worth investigating. For g = 1, as already noted, we get the function J. Furthermore, they conjectured that M has a natural infinite dimensional representationwith graded character T g . It came as a surprise that the most natural module of a finite group was infinite dimensional. Moonshine Conjecture. For all g ∈ M ,...

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Lie superalgebras

Cited by 2,020 publications

References 16 publications

Free fermion resolution of supergroup WZNW models

Free fermion resolution of supergroup WZNW models

Global splittings and super Harish-Chandra pairs for affine supergroups

Generalized Kac-Moody algebras and some related topics

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