Abstract
For a couple of associative algebras we define the notion of their double and give a set of examples. Also, we discuss applications of such doubles to representation theory of certain quantum algebras and to a new type of Noncommutative Geometry.
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Notes
Sometimes we will deal with the algebra \(U(gl{{(N)}_{h}})\), where \(h\) is a numerical multiplier introduced in the bracket of the Lie algebra \(gl(N)\). This rescaling of the bracket enables us to treat the algebra \(U(gl{{(N)}_{h}})\) as a quantization of the commutative algebra \({\text{Sym}}(gl(N))\) with respect to the linear Poisson bracket.
By permutation relations we mean equalities \(a \otimes b = \sigma (a \otimes b)\), \(a \in A\), \(b \in B\). All the doubles \((A,B)\) below are defined via relations on generators of each component and the permutation relations.
Note that this extension is not straightforward, since the usual Leibnitz rule for the derivatives on the algebra \(U(u{{(2)}_{h}})\) is not valid.
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Funding
The work of P.S. was partially funded the RFBR grant 19-01-00726.
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Gurevich, D., Saponov, P. Doubles of Associative Algebras and Their Applications. Phys. Part. Nuclei Lett. 17, 774–778 (2020). https://doi.org/10.1134/S1547477120050167
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DOI: https://doi.org/10.1134/S1547477120050167