Structure of Lie Superalgebras of Block Type Related to Locally Finite Derivations

In this paper, we study the structure theory of two classes of Lie superalgebras [Formula: see text] of Block type, where [Formula: see text] is a positive integer. In particular, the automorphism groups, derivation superalgebras and central extensions of [Formula: see text] are completely determined.

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An Application of Collapsing Levels to the Representation Theory of Affine Vertex Algebras

International Mathematics Research Notices ◽

10.1093/imrn/rny237 ◽

2018 ◽

Vol 2020 (13) ◽

pp. 4103-4143 ◽

Cited By ~ 2

Author(s):

Dražen Adamović ◽

Victor G Kac ◽

Pierluigi Möseneder Frajria ◽

Paolo Papi ◽

Ozren Perše

Keyword(s):

Maximal Ideal ◽

Vertex Algebra ◽

Lie Superalgebras ◽

Vertex Algebras ◽

Locally Finite ◽

Complete Reducibility ◽

Finite Module ◽

Nontrivial Ideal ◽

Conformal Embeddings ◽

Reducibility Result

Abstract We discover a large class of simple affine vertex algebras $V_{k} ({\mathfrak{g}})$, associated to basic Lie superalgebras ${\mathfrak{g}}$ at non-admissible collapsing levels $k$, having exactly one irreducible ${\mathfrak{g}}$-locally finite module in the category ${\mathcal O}$. In the case when ${\mathfrak{g}}$ is a Lie algebra, we prove a complete reducibility result for $V_k({\mathfrak{g}})$-modules at an arbitrary collapsing level. We also determine the generators of the maximal ideal in the universal affine vertex algebra $V^k ({\mathfrak{g}})$ at certain negative integer levels. Considering some conformal embeddings in the simple affine vertex algebras $V_{-1/2} (C_n)$ and $V_{-4}(E_7)$, we surprisingly obtain the realization of non-simple affine vertex algebras of types $B$ and $D$ having exactly one nontrivial ideal.

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On classification of quasifinite representations of two classes of Lie superalgebras of block type

Communications in Algebra ◽

10.1080/00927872.2016.1172620 ◽

2016 ◽

Vol 45 (2) ◽

pp. 502-513

Author(s):

Chunguang Xia

Keyword(s):

Lie Superalgebras ◽

Block Type

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REPRESENTATIONS OF THE EXCEPTIONAL LIE SUPERALGEBRA E(3,6) III: CLASSIFICATION OF SINGULAR VECTORS

Journal of Algebra and Its Applications ◽

10.1142/s0219498805001095 ◽

2005 ◽

Vol 04 (01) ◽

pp. 15-57 ◽

Cited By ~ 3

Author(s):

VICTOR G. KAC ◽

ALEXEI RUDAKOV

Keyword(s):

Vector Fields ◽

Lie Superalgebra ◽

Lie Superalgebras ◽

Irreducible Representations ◽

Verma Modules ◽

Locally Finite ◽

Infinite Dimensional ◽

Singular Vectors ◽

Zero Degree

We continue the study of irreducible representations of the exceptional Lie superalgebra E(3,6). This is one of the two simple infinite-dimensional Lie superalgebras of vector fields which have a Lie algebra sℓ(3) × sℓ(2) × gℓ(1) as the zero degree component of its consistent ℤ-grading. We provide the classification of the singular vectors in the degenerate Verma modules over E(3,6), completing thereby the classification and construction of all irreducible E(3,6)-modules that are L0-locally finite.

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