# Finite-dimensional representations of the special linear Lie superalgebra sl(1,n). II. Nontypical representations

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In a series of two papers all finite-dimensional irreducible representations of the special linear Lie superalgebra sl(1,3) are constructed. Explicit formulas for the generators in an orthonormal Gel'fand-Zetlin basis of the even subalgebra gl(3) are given. This paper develops a background for...

- Irreducible representations of the exceptional Lie superalgebras D(2,1;Î±). Van der Jeugt, J. // Journal of Mathematical Physics;May85, Vol. 26 Issue 5, p913
The shift operator technique is used to give a complete analysis of all finite- and infinite-dimensional irreducible representations of the exceptional Lie superalgebras D(2,1;Î±). For all cases, the star or grade star conditions for the algebra are investigated. Among the finite-dimensional...

- Principal five-dimensional subalgebras of Lie superalgebras. Van der Jeugt, Joris // Journal of Mathematical Physics;Dec86, Vol. 27 Issue 12, p2842
The analog of sl(2) for Lie superalgebras is osp(1,2), a five-dimensional superalgebra. All basic classical Lie superalgebras L that contain a principal five-dimensional osp(1,2) subalgebra are classified. Moreover, the decomposition of the standard representation and of the adjoint...

- Irreducible *-representations of Lie superalgebras B(0,n) with finite-degenerated vacuum. Blank, J.; Havlícˇek, M. // Journal of Mathematical Physics;Dec86, Vol. 27 Issue 12, p2823
The problem of getting irreducible *-representations Ï€ of Lie superalgebras B(0,n), n=1,2, is studied, starting with a recently constructed family of linear representations in terms of differential operators on the space CâˆžN of CN -valued Câˆž -functions. Equivalent formulation via...

- Invariant supersymmetric multilinear forms and the Casimir elements of P-type Lie superalgebras. Scheunert, Manfred // Journal of Mathematical Physics;May87, Vol. 28 Issue 5, p1180
The Casimir elements of the P-type Lie superalgebras are investigated. Depending on the class of algebras under consideration either there do not exist any nontrivial Casimir elements at all or else the Casimir elements are highly degenerate. Basic to the investigation is a lemma about invariant...

- Irreducible finite-dimensional representations of the Lie superalgebra gl(n/1) in a Gelâ€™fandâ€“Zetlin basis. Palev, Tchavdar D. // Journal of Mathematical Physics;Jul89, Vol. 30 Issue 7, p1433
All finite-dimensional irreducible representations of the general linear Lie superalgebra gl(n/1) are studied. For each representation, a concept of a Gelâ€™fandâ€“Zetlin basis is defined. Expressions for the transformation of the basis under the action of the generators are written down.

- On chains of orthosymplectic Lie superalgebras and the n-dimensional quantum harmonic oscillator. Beckers, J.; Cornwell, J. F. // Journal of Mathematical Physics;Aug89, Vol. 30 Issue 8, p1655
Triplets of embedded orthosymplectic Lie superalgebras are singled out and analyzed in terms of their respective even and odd root systems. The corresponding chains of embeddings are considered for arbitrary integers m and n (for both mâ‰ n and m=n). It is shown that the second member inside...

- Classification of star and grade-star representations of C(n+1). Zhang, R. B.; Gould, M. D. // Journal of Mathematical Physics;Aug90, Vol. 31 Issue 8, p1889
The two types of * and grade-* representations of the Lie superalgebra C(n+1) are classified. A type (1) irreducible *-representation is characterized by the single condition (Î›,Î±s)â‰¥0, Î› and Î±s being the highest weight and the odd simple root, respectively, while the type (2)...

- Classification of all star irreps of gl(m|n). Gould, M. D.; Zhang, R. B. // Journal of Mathematical Physics;Nov90, Vol. 31 Issue 11, p2552
All the finite-dimensional star irreps of gl(m|n) are classified in terms of their highest weights, thereby completing the classification of all finite-dimensional star irreps of the basic classical Lie superalgebras. The lowest weights of such irreps are determined explicitly and it is shown...