IJPAM: Volume 100, No. 2 (2015)

WALLED SIGNED BRAUER ALGEBRAS AS CENTRALIZER
ALGEBRAS OF THE MIXED TENSOR REPRESENTATION
OF LIE SUPERALGEBRAS

B. Kethesan
Ramanujan Institute for Advanced study in Mathematics
University of Madras
Chepauk, Chennai, 600 005, Tamilnadu, INDIA


Abstract. In this paper, we prove that the walled signed Brauer algebra $\overrightarrow{D}_{r,s}(x)$ is embedded in a canonical fashion as a subalgebra of a walled Brauer algebra $B_{2r,2s}(x)$ and the walled signed Brauer algebra $\overrightarrow{D}_{r,s}(x)$ is the centralizer algebra of $\sigma= (1 \ 2)(3 \ 4)\dots(2r-1 \ 2r)(2r+1 \ 2r+2)\dots(2r+2s-1 \ 2r+2s)$ in the walled Brauer algebra $B_{2r,2s}(x).$ Finally we prove that, if $2r+2s < (m+1)(n+1),$ the walled signed Brauer algebra $\overrightarrow{D}_{r,s}(\delta)$ is isomorphic to the centralizer algebra $End_{\mathfrak{g}^{2r+2s}}(V^{\otimes r}\otimes(V^{\ast})^{\otimes s})^{op}$ of the $\mathfrak{g}^{2r+2s}$- action on the mixed tensor space $(V^{\otimes r}\otimes(V^{\ast})^{\otimes s});$ where $\delta = m-n, \ \mathfrak{g}= gl(m,n)$ is the complex general linear Lie superalgebra, $V= W\otimes W,$ $W= \mathbb{C}^{m\mid n}$ is the natural representation of $\mathfrak{g}$ and $V^{\ast}$ is the dual of $V.$

Received: September 14, 2014

AMS Subject Classification: 17B10

Key Words and Phrases: walled Brauer algebra, signed Brauer algebra, Lie superalgebra

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DOI: 10.12732/ijpam.v100i2.6 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2015
Volume: 100
Issue: 2
Pages: 235 -


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