We prove the analog of Kostants Theorem on Lie algebra cohomology in the context of quantum groups. We prove that Kostants cohomology formula holds for quantum groups at a generic parameter $q$, recovering an earlier result of Malikov in the case where the underlying semisimple Lie algebra $mathfrak{g} = mathfrak{sl}(n)$. We also show that Kostants formula holds when $q$ is specialized to an $ell$-th root of unity for odd $ell ge h-1$ (where $h$ is the Coxeter number of $mathfrak{g}$) when the highest weight of the coefficient module lies in the lowest alcove. This can be regarded as an extension of results of Friedlander-Parshall and Polo-Tilouine on the cohomology of Lie algebras of reductive algebraic groups in prime characteristic.
The geometric realizations of Lusztigs symmetries of symmetrizable quantum groups are given in this paper. This construction is a generalization of that in [19].
We investigate a new cohomology of Lie superalgebras, which may be compared to a de Rham cohomology of Lie supergroups involving both differential and integral forms. It is defined by a BRST complex of Lie superalgebra modules, which is formulated in terms of a Weyl superalgebra and incorporates inequivalent representations of the bosonic Weyl subalgebra. The new cohomology includes the standard Lie superalgebra cohomology as a special case. Examples of new cohomology groups are computed.
In this paper, we shall study the structure of the Grothendieck group of the category consisting of Lusztigs perverse sheaves and give a decomposition theorem of it. By using this decomposition theorem and the geometric realizations of Lusztigs symmetries on the positive part of a quantum group, we shall give geometric realizations of Lusztigs symmetries on the whole quantum group.
We construct a basis for a modified quantum group of finite type, extending the PBW bases of positive and negative halves of a quantum group. Generalizing Lusztigs classic results on PBW bases, we show that this basis is orthogonal with respect to its natural bilinear form (and hence called a PBW basis), and moreover, the matrix for the PBW-expansion of the canonical basis is unital triangular. All these follow by a new construction of the modified quantum group of arbitrary type, which is built on limits of sequences of elements in tensor products of lowest and highest weight modules. Explicitly formulas are worked out in the rank one case.
$imath$quantum groups are generalizations of quantum groups which appear as coideal subalgebras of quantum groups in the theory of quantum symmetric pairs. In this paper, we define the notion of classical weight modules over an $imath$quantum group, and study their properties along the lines of the representation theory of weight modules over a quantum group. In several cases, we classify the finite-dimensional irreducible classical weight modules by a highest weight theory.
University of Georgia VIGRE Algebra Group: Irfan Bagci
,Brian D. Boe
,n Leonard Chastkofsky
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(2008)
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"Cohomology of quantum groups: An analog of Kostants Theorem"
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Benjamin Jones
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