64 research outputs found

    Dual canonical bases for the quantum general linear supergroup

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    Dual canonical bases of the quantum general linear supergroup are constructed which are invariant under the multiplication of the quantum Berezinian. By setting the quantum Berezinian to identity, we obtain dual canonical bases of the quantum special linear supergroup {\s O}_q(SL_{m\mid n}). We apply the canonical bases to study invariant subalgebras of the quantum supergroups under left and right translations. In the case n=1n=1, it is shown that each invariant subalgebra is spanned by a part of the dual canonical bases. This in turn leads to dual canonical bases for any Kac module constructed by using an analogue of Borel-Weil theorem.Comment: 32 page

    Quantized Heisenberg Space

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    We investigate the algebra Fq(N)F_q(N) introduced by Faddeev, Reshetikhin and Takhadjian. In case qq is a primitive root of unity the degree, the center, and the set of irreducible representations are found. The Poisson structure is determined and the De Concini-Kac-Procesi Conjecture is proved for this case. In the case of qq generic, the primitive ideals are described. A related algebra studied by Oh is also treated.Comment: 20 pages LaTeX documen

    Derivation-Simple Algebras and the Structures of Generalized Lie Algebras of Witt Type

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    We classify all the pairs of a commutative associative algebra with an identity element and its finite-dimensional commutative locally-finite derivation subalgebra such that the commutative associative algebra is derivation-simple with respect to the derivation subalgebra over an algebraically closed field with characteristic 0. Such pairs are the fundamental ingredients for constructing generalized simple Lie algebras of Cartan type. Moreover, we determine the isomorphic classes of the generalized simple Lie algebras of Witt Type. The structure space of these algebras is given explicitly.Comment: 20pages, Latex file; To appear in Journal of Algebr

    Lie algebras and related topics

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    On dual canonical bases

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    The dual basis of the canonical basis of the modified quantized enveloping algebra is studied, in particular for type AA. The construction of a basis for the coordinate algebra of the n×nn\times n quantum matrices is appropriate for the study the multiplicative property. It is shown that this basis is invariant under multiplication by certain quantum minors including the quantum determinant. Then a basis of quantum SL(n) is obtained by setting the quantum determinant to one. This basis turns out to be equivalent to the dual canonical basis
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