Twisted Tensor Products of Kn with Km

Abstract

We find three families of twisting maps of Km with Kn, where K is a field, and we make a detailed study of its properties. One of them is related to truncated quiver algebras, the second one consists of deformations of the first and the third one requires m = n and yields algebras isomorphic to Mn(K).

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Correspondence to Juan J. Guccione.

Additional information

Christian Valqui was supported by PUCP-DGI-CAP-2016-1-0045.

Jorge A. Guccione and Juan J. Guccione were supported by UBACyT 20020150100153BA (UBA) and PIP 11220110100800CO (CONICET)

Presented by: Sarah Witherspoon

Appendix: Quasi-Standard Twisting Maps of K 3 with K 3

Appendix: Quasi-Standard Twisting Maps of K 3 with K 3

There is a bijection between the set of quivers that satisfy Conditions (1), (2) and (3) of Remark 10.5 with n = m =?3 and the set Stm of standard twisting maps of K3 with K3. Moreover, by Remark 7.8 we know that Stm splits in classes of isomorphic twisting maps. The first, second, fourth and fifth columns of the following table are self explanatory. In the third column we list a quiver for each one of the equivalent classes in Stm. In the sixth column we list the number of the standard twisting maps equivalent to the one determined by the quiver listed in the third column. Finally, in the last column we list all the quasi-standard twisting maps (which are not standard) associated with the standard twisting map determined in the third column. For this we use recursively the construction in Definition 10.7, verifying in each step that the conditions in item (2) of Proposition 10.11 are satisfied.

Table 1 Quasi-standard twisting maps of K3 with K3

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Arce, J., Guccione, J.A., Guccione, J.J. et al. Twisted Tensor Products of Kn with Km. Algebr Represent Theor 22, 1599–1651 (2019). https://doi.org/10.1007/s10468-018-9833-1

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Keywords

  • Twisted tensor products
  • Quivers

Mathematics Subject Classification (2010)

  • Primary 16S35
  • Secondary 16S38