scholarly journals Irreducible tensor representations of general linear Lie superalgebras

2009 ◽  
Vol 114 (1) ◽  
pp. 15-32
Author(s):  
Tadeusz Józefiak
Keyword(s):  
Lie Superalgebras ◽  
General Linear ◽  
Irreducible Tensor ◽  
Tensor Representations

Double Derivations of n-Lie Superalgebras

2018 ◽  
Vol 25 (01) ◽  
pp. 161-180
Author(s):  
Bing Sun ◽  
Liangyun Chen ◽  
Xin Zhou
Keyword(s):  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
General Linear ◽  
Base Field ◽  
Inner Derivation ◽  
Derivation Algebra ◽  
General Linear Lie Superalgebra

Let 𝔤 be an n-Lie superalgebra. We study the double derivation algebra [Formula: see text] and describe the relation between [Formula: see text] and the usual derivation Lie superalgebra Der(𝔤). We show that the set [Formula: see text] of all double derivations is a subalgebra of the general linear Lie superalgebra gl(𝔤) and the inner derivation algebra ad(𝔤) is an ideal of [Formula: see text]. We also show that if 𝔤 is a perfect n-Lie superalgebra with certain constraints on the base field, then the centralizer of ad(𝔤) in [Formula: see text] is trivial. Finally, we give that for every perfect n-Lie superalgebra 𝔤, the triple derivations of the derivation algebra Der(𝔤) are exactly the derivations of Der(𝔤).

On the Dimension of an Irreducible Tensor Representation of the General Linear Group GL(d)

1970 ◽  
Vol 13 (3) ◽  
pp. 389-390
Author(s):  
J. A. J. Matthews ◽  
G. de B. Robinson
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Tensor Representation ◽  
Irreducible Tensor ◽  
Tensor Representations ◽  
Identity Representation ◽  
Raising Operator ◽  
Image Position

As has long been known, the irreducible tensor representations of GL(d) of rank n may be labeled by means of the irreducible representations of Sn, i.e., by means of the Young diagrams [λ], where λ1 + λ2 + … λr = n. We denote such a tensor representation by 〈λ〉. Using Young's raising operator Rij we can write [1, p. 42]1.1where the dot denotes the inducing process. For example, [3] . [2] is that representation of S5 induced by the identity representation of its subgroup S3 × S2.

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