scholarly journals Representation Theory of the Affine Lie Superalgebra $\hslc$ at Fractional Level

1997 ◽  
Vol 185 (2) ◽  
pp. 467-493 ◽  
Author(s):  
P. Bowcock ◽  
A. Taormina
Keyword(s):  
Representation Theory ◽  
Lie Superalgebra

scholarly journals Schramm–Loewner evolution with Lie superalgebra symmetry

2018 ◽  
Vol 33 (20) ◽  
pp. 1850117 ◽  
Author(s):  
Shinji Koshida
Keyword(s):  
Differential Equations ◽  
Representation Theory ◽  
Stochastic Differential Equations ◽  
Degrees Of Freedom ◽  
Lie Superalgebra ◽  
Classical Type ◽  
Finite Dimensional ◽  
Loewner Evolution ◽  
Internal Degrees Of Freedom

We propose a generalization of Schramm–Loewner evolution (SLE) that has internal degrees of freedom described by an affine Lie superalgebra. We give a general formulation of SLE corresponding to representation theory of an affine Lie superalgebra whose underlying finite-dimensional Lie superalgebra is basic classical type, and write down stochastic differential equations on internal degrees of freedom in case that the corresponding affine Lie superalgebra is [Formula: see text]. We also demonstrate computation of local martingales associated with the solution from a representation of [Formula: see text].

The Primitive Spectrum and Category  for the Periplectic Lie Superalgebra

2018 ◽  
Vol 72 (3) ◽  
pp. 625-655 ◽  
Author(s):  
Chih-Whi Chen ◽  
Kevin Coulembier
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Braid Group ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Left Action ◽  
Right Action ◽  
New Type

AbstractWe solve two problems in representation theory for the periplectic Lie superalgebra $\mathfrak{p}\mathfrak{e}(n)$, namely, the description of the primitive spectrum in terms of functorial realisations of the braid group and the decomposition of category ${\mathcal{O}}$ into indecomposable blocks.To solve the first problem, we establish a new type of equivalence between category ${\mathcal{O}}$ for all (not just simple or basic) classical Lie superalgebras and a category of Harish-Chandra bimodules. The latter bimodules have a left action of the Lie superalgebra but a right action of the underlying Lie algebra. To solve the second problem, we establish a BGG reciprocity result for the periplectic Lie superalgebra.

scholarly journals A Basis Theorem for the Degenerate Affine Oriented Brauer–Clifford Supercategory

2019 ◽  
Vol 71 (5) ◽  
pp. 1061-1101 ◽  
Author(s):  
Jonathan Brundan ◽  
Jonathan Comes ◽  
Jonathan Robert Kujawa
Keyword(s):  
Representation Theory ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Basis Theorem

AbstractWe introduce the oriented Brauer–Clifford and degenerate affine oriented Brauer–Clifford supercategories. These are diagrammatically defined monoidal supercategories that provide combinatorial models for certain natural monoidal supercategories of supermodules and endosuperfunctors, respectively, for the Lie superalgebras of type Q. Our main results are basis theorems for these diagram supercategories. We also discuss connections and applications to the representation theory of the Lie superalgebra of type Q.

scholarly journals On Low-Dimensional Complex ω-Lie Superalgebras

2021 ◽  
Author(s):  
Jia Zhou ◽  
Liangyun Chen
Keyword(s):  
Automorphism Group ◽  
Representation Theory ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
3 Dimensional ◽  
Finite Dimensional ◽  
Low Dimensional ◽  
Derivation Superalgebra

Abstract Let (g, [−, −], ω) be a finite-dimensional complex ω-Lie superalgebra. In this paper, we introduce the notions of derivation superalgebra Der(g) and the automorphism group Aut(g) of (g, [−, −], ω). We study Derω (g) and Autω (g), which are superalgebra of Der(g) and subgroup of Aut(g), respectively. For any 3-dimensional or 4-dimensional complex ω-Lie superalgebra g, we explicitly calculate Der(g) and Aut(g), and obtain Jordan standard forms of elements in the two sets. We also study representation theory of ω-Lie superalgebras and give a conclusion that all nontrivial non-ω-Lie 3-dimensional and 4-dimensional ω-Lie superalgebras are multiplicative, as well as we show that any irreducible respresentation of the 4-dimensional ω-Lie superalgebra P2,k(k 6= 0, −1) is 1-dimensional.

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