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Register a new userUnitary Harish-Chandra representations of real supergroups
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We give conditions for unitarizability of Harish-Chandra super modules for Lie supergroups and superalgebras.
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Mathematical physicists have studied degenerations of Lie groups and their representations, which they call contractions. In this paper we study these contractions, and also other families, within the framework of algebraic families of Harish-Chandra modules. We construct a family that incorporates both a real reductive group and its compact form, separate parts of which have been studied individually as contractions. We give a complete classification of generically irreducible families of Harish-Chandra modules in the case of the family associated to SL(2, R).
In this paper, we classify all indecomposable Harish-Chandra modules of the intermediate series over the twisted Heisenberg-Virasoro algebra. Meanwhile, some bosonic modules are also studied.
In this paper we discuss the highest weight $frak k_r$-finite representations of the pair $(frak g_r,frak k_r)$ consisting of $frak g_r$, a real form of a complex basic Lie superalgebra of classical type $frak g$ (${frak g} eq A(n,n)$), and the maximal compact subalgebra $frak k_r$ of $frak g_{r,0}$, together with their geometric global realizations. These representations occur, as in the ordinary setting, in the superspaces of sections of holomorphic super vector bundles on the associated Hermitian superspaces $G_r/K_r$.
We study the quotient of $mathcal{T}_n = Rep(GL(n|n))$ by the tensor ideal of negligible morphisms. If we consider the full subcategory $mathcal{T}_n^+$ of $mathcal{T}_n$ of indecomposable summands in iterated tensor products of irreducible representations up to parity shifts, its quotient is a semisimple tannakian category $Rep(H_n)$ where $H_n$ is a pro-reductive algebraic group. We determine the connected derived subgroup $G_n subset H_n$ and the groups $G_{lambda} = (H_{lambda}^0)_{der}$ corresponding to the tannakian subcategory in $Rep(H_n)$ generated by an irreducible representation $L(lambda)$. This gives structural information about the tensor category $Rep(GL(n|n))$, including the decomposition law of a tensor product of irreducible representations up to summands of superdimension zero. Some results are conditional on a hypothesis on $2$-torsion in $pi_0(H_n)$.
We give a complete description of the finite-dimensional irreducible representations of the Yangian associated with the orthosymplectic Lie superalgebra $frak{osp}_{1|2}$. The representations are parameterized by monic polynomials in one variable, they are classified in terms of highest weights. We give explicit constructions of a family of elementary modules of the Yangian and show that a wide class of irreducible representations can be produced by taking tensor products of the elementary modules.
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