Odd reflections in the Yangian associated with $$\mathfrak {gl}(m|n)$$ gl ( m | n )

The odd reflections are an effective tool in the Lie superalgebra representation theory, as they relate non-conjugate Borel subalgebras. We introduce analogues of the odd reflections for the Yangian $$ \mathrm{Y}(\mathfrak {gl}_{m|n})$$ Y ( gl m | n ) and use them to produce a transition rule for the parameters of the highest weight modules corresponding to a change of the parity sequence. This leads to a description of the finite-dimensional irreducible representations of the Yangian associated with an arbitrary parity sequence.

[1]  Yung-Ning Peng.  Finite W-Superalgebras and Truncated Super Yangians , 2013, 1304.3913.

[2]  H. Yamane.  Errata to On defining relations of affine lie superalgebras and affine quantized universal enveloping superalgebras , 2001 .

[3]  M. Nazarov.  Quantum Berezinian and the classical capelli identity , 1991 .

[4]  Lucy Gow.  Gauss Decomposition of the Yangian Y (gl M|n ) , 2008 .

[5]  G. Olshanskii,et al.  Yangians and Classical Lie Algebras , 1994, hep-th/9409025.

[6]  A. Tsymbaliuk.  Shuffle algebra realizations of type A super Yangians and quantum affine superalgebras for all Cartan data , 2019, Letters in Mathematical Physics.

[7]  Yung-Ning Peng.  Parabolic Presentations of the Super Yangian , 2011 .

[8]  Parabolic Presentations of the Yangian , 2004, math/0407011.

[9]  V. Drinfeld.  A New realization of Yangians and quantized affine algebras , 1987 .

[10]  Yung-Ning Peng.  Parabolic Presentations of the Super Yangian $${Y(\mathfrak{gl}_{M|N})}$$Y(glM|N) Associated with Arbitrary 01-Sequences , 2015, 1501.07378.

[11]  Lucy Gow.  On the Yangian Y $$(\mathfrak{g}\mathfrak{l}_{m|n} )$$ and its quantum Berezinian , 2005 .

[12]  I. Musson.  Lie Superalgebras and Enveloping Algebras , 2012 .

[13]  Shun-Jen Cheng,et al.  Dualities and Representations of Lie Superalgebras , 2012 .

[14]  V. Stukopin.  Yangians of Lie superalgebras of typeA(m, n) , 1994 .

[15]  Principal W-algebras for GL(m|n) , 2012, 1205.0992.

[16]  V. Stukopin.  Representations of the Yangian of a Lie superalgebra of type , 2013 .