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doi: 10.3934/era.2021013

Graded post-Lie algebra structures and homogeneous Rota-Baxter operators on the Schrödinger-Virasoro algebra

School of Mathematical Science, Heilongjiang University, Harbin 150080, China

* Corresponding author: Xiaomin Tang

Received  July 2020 Revised  November 2020 Published  February 2021

Fund Project: This work is supported in part by NNSFC (Grant No. 11771069), NSF of Heilongjiang Province (Grant No. LH2020A020) and the fund of Heilongjiang Provincial Laboratory of the Theory and Computation of Complex Systems

In this paper, we characterize the graded post-Lie algebra structures on the Schrödinger-Virasoro Lie algebra. Furthermore, as an application, we obtain the all homogeneous Rota-Baxter operator of weight $ 1 $ on the Schrödinger-Virasoro Lie algebra.

Citation: Pengliang Xu, Xiaomin Tang. Graded post-Lie algebra structures and homogeneous Rota-Baxter operators on the Schrödinger-Virasoro algebra. Electronic Research Archive, doi: 10.3934/era.2021013
References:
[1]

C. BaiL. Guo and X. Ni, Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras, Commun. Math. Phys., 297 (2010), 553-596.  doi: 10.1007/s00220-010-0998-7.  Google Scholar

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X. GaoM. LiuC. Bai and N. Jing, Rota-Baxter operators on Witt and Virasoro algebras, J. Geom. Phys., 108 (2016), 1-20.  doi: 10.1016/j.geomphys.2016.06.007.  Google Scholar

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Y. Li and X. Tang, Biderivations and commutative post-Lie algebra structure on Schrödinger-Virasoro Lie algebras, Bull. Iranian Math. Soc., 45 (2019), 1743-1754.  doi: 10.1007/s41980-019-00226-2.  Google Scholar

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X. Tang, Post-Lie algebra structures on the Witt algebra, Bull. Malays. Math. Sci. Soc., 42 (2019), 3427-3451.  doi: 10.1007/s40840-019-00730-y.  Google Scholar

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X. TangY. Zhang and Q. Sun, Rota-Baxter operators on $4$-dimensional complex simple associative algebras, Appl. Math. Comput., 229 (2014), 173-186.  doi: 10.1016/j.amc.2013.12.032.  Google Scholar

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X. Tang and Y. Zhong, Graded post-Lie algebra structures, Rota-Baxter operators and Yang-Baxter equations on the W-algebra $W(2, 2)$, Bull. Korean Math. Soc., 55 (2018), 1727-1748.  doi: 10.4134/BKMS.b171021.  Google Scholar

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J. Unterberger, On vertex algrbra representations of the Schrödinger-Virasoro Lie algebra, Nuclear Phys. B, 823 (2009), 320-371.  doi: 10.1016/j.nuclphysb.2009.06.018.  Google Scholar

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B. Vallette, Homology of generalized partition posets, J. Pure. Appl. Algebra, 208 (2007), 699-725.  doi: 10.1016/j.jpaa.2006.03.012.  Google Scholar

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H. YuL. Guo and J.-Y. Thibon, Weak quasi-symmetric functions, Rota-Baxter algebras and Hopf algebras, Adv. Math., 344 (2019), 1-34.  doi: 10.1016/j.aim.2018.12.001.  Google Scholar

show all references

References:
[1]

C. BaiL. Guo and X. Ni, Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras, Commun. Math. Phys., 297 (2010), 553-596.  doi: 10.1007/s00220-010-0998-7.  Google Scholar

[2]

G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math., 10 (1960), 731-742.  doi: 10.2140/pjm.1960.10.731.  Google Scholar

[3]

D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent. Eur. J. Math., 4 (2006), 323-357.  doi: 10.2478/s11533-006-0014-9.  Google Scholar

[4]

D. BurdeK. Dekimpe and K. Vercammen, Affine actions on Lie groups and post-Lie algebra structures, Linear Algebra Appl., 437 (2012), 1250-1263.  doi: 10.1016/j.laa.2012.04.007.  Google Scholar

[5]

D. Burde and W. A. Moens, Commutative post-Lie algebra structures on Lie algebras, J. Algebra, 467 (2016), 183-201.  doi: 10.1016/j.jalgebra.2016.07.030.  Google Scholar

[6]

X. GaoM. LiuC. Bai and N. Jing, Rota-Baxter operators on Witt and Virasoro algebras, J. Geom. Phys., 108 (2016), 1-20.  doi: 10.1016/j.geomphys.2016.06.007.  Google Scholar

[7] L. Guo, An Introduction to Rota-Baxter Algebra, Somerville: International Press, 2012.   Google Scholar
[8]

I. Z. Golubschik and V. V. Sokolov, Generalized operator Yang-Baxter equations, integrable ODES and nonassociative algebras, J. Nonlinear Math. Phys., 7 (2000), 184-197.  doi: 10.2991/jnmp.2000.7.2.5.  Google Scholar

[9]

J. Han, J. Li and Y. Su, Lie bialgebra structures on the Schrödinger-Virasoro Lie algebras, J. Math. Phys., 50 (2009), 083504, 12 pp.  Google Scholar

[10]

M. Henkel, Schrödinger invariance and stringly anisotropic critical systems, J. Stat. Phys., 75 (1994), 1023-1061.  doi: 10.1007/BF02186756.  Google Scholar

[11]

Y. Li and X. Tang, Biderivations and commutative post-Lie algebra structure on Schrödinger-Virasoro Lie algebras, Bull. Iranian Math. Soc., 45 (2019), 1743-1754.  doi: 10.1007/s41980-019-00226-2.  Google Scholar

[12]

H. Z. Munthe-Kaas and A. Lundervold, On post-Lie algebras, Lie-Butcher series and moving frames, Found. Comput. Math., 13 (2013), 583-613.  doi: 10.1007/s10208-013-9167-7.  Google Scholar

[13]

Y. PanQ. LiuC. Bai and L. Guo, Post-Lie algebra structures on the Lie algebra SL (2, $\mathbb{C} $), Electron. J. Linear Algebra, 23 (2012), 180-197.  doi: 10.13001/1081-3810.1514.  Google Scholar

[14]

Y. Pei and C. Bai, Novikov algebras and Schrödinger-Virasoro Lie algebras, J. Phys., 44 (2011), 045201, 18 pp. doi: 10.1088/1751-8113/44/4/045201.  Google Scholar

[15]

C. Roger and J. Unterberger, The Schrödinger-Virasoro Lie group and algebra: Representation theory and cohomological study, Ann. Henri Poincaré, 7 (2006), 1477-1529.  doi: 10.1007/s00023-006-0289-1.  Google Scholar

[16]

G.-C. Rota, Baxter algebras and combinatorial identities I, Bull. Amer. Math. Soc., 75 (1969), 325–329. doi: 10.1090/S0002-9904-1969-12158-0.  Google Scholar

[17]

G.-C. Rota, Baxter operators, an introduction, in "Gian-Carlo Rota on combinatorics, introductory papers and commentaries", Joesph PS Kung, Editor, J., (1995), 504–512.  Google Scholar

[18]

X. Tang, Post-Lie algebra structures on the Witt algebra, Bull. Malays. Math. Sci. Soc., 42 (2019), 3427-3451.  doi: 10.1007/s40840-019-00730-y.  Google Scholar

[19]

X. TangY. Zhang and Q. Sun, Rota-Baxter operators on $4$-dimensional complex simple associative algebras, Appl. Math. Comput., 229 (2014), 173-186.  doi: 10.1016/j.amc.2013.12.032.  Google Scholar

[20]

X. Tang and Y. Zhong, Graded post-Lie algebra structures, Rota-Baxter operators and Yang-Baxter equations on the W-algebra $W(2, 2)$, Bull. Korean Math. Soc., 55 (2018), 1727-1748.  doi: 10.4134/BKMS.b171021.  Google Scholar

[21]

J. Unterberger, On vertex algrbra representations of the Schrödinger-Virasoro Lie algebra, Nuclear Phys. B, 823 (2009), 320-371.  doi: 10.1016/j.nuclphysb.2009.06.018.  Google Scholar

[22]

B. Vallette, Homology of generalized partition posets, J. Pure. Appl. Algebra, 208 (2007), 699-725.  doi: 10.1016/j.jpaa.2006.03.012.  Google Scholar

[23]

H. YuL. Guo and J.-Y. Thibon, Weak quasi-symmetric functions, Rota-Baxter algebras and Hopf algebras, Adv. Math., 344 (2019), 1-34.  doi: 10.1016/j.aim.2018.12.001.  Google Scholar

Table 2.  Values of $f,g,h$ satisfying (16)-(22), where $a,\mu\in\mathbb{C}$, $k\in \{-2,-1,1,2,3\}$, $t\in \mathbb{Z}\setminus\{0,1\}$ and $s\in \{2t-2,2t-1,2t,2t+1,2t+2\}$
$Cases$ $f(n)$ from Table 1 $a$, $\mu$ $h(n),g(n)$
$\mathcal{W}_{1}^{\mathcal{P}_{1}}$ $\mathcal{P}_{1}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=0 $
$\mathcal{W}_{2}^{\mathcal{P}_{1}}$ $\mathcal{P}_{1}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1 $
$\mathcal{W}_{1}^{\mathcal{P}_{2}}$ $\mathcal{P}_{2}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=0 $
$\mathcal{W}_{2}^{\mathcal{P}_{2}}$ $\mathcal{P}_{2}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1 $
$\mathcal{W}_{1,\mu}^{\mathcal{P}_{3}^c}$ $\mathcal{P}^{c}_{3}$ $a=0$ and $\forall \mu$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=0 $
$\mathcal{W}_{2,\mu}^{\mathcal{P}_{3}^c}$ $\mathcal{P}^{c}_{3}$ $a=0$ and $\forall \mu$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1 $
$\mathcal{W}_{3,\mu}^{\mathcal{P}^{c}_{3},k}$ $\mathcal{P}^{c}_{3}$ $a=0$ and $\forall \mu$ $h{(\mathbb{Z}_{> 0})}=-1$, $h{(\mathbb{Z}_{< 0})}=0$ and
$ g{(\mathbb{Z}^*_{\geqslant k})}=-1$, $g{(\mathbb{Z}^*_{\leqslant k-1})}=0$
$\mathcal{W}_{4,a,\mu}^{\mathcal{P}_{3}^c,k=1}$ $\mathcal{P}^{c}_{3}$ $\forall a$ and $\forall \mu$ $ h{(\mathbb{Z}_{> 0})}=-1$, $h{(\mathbb{Z}_{< 0})}=0$ and
$g{(\mathbb{Z}_{> 0})}=-1$, $g{(\mathbb{Z}_{< 0})}=0$
$\mathcal{W}_{5,\mu}^{\mathcal{P}_{3}^c,s,t}$ $\mathcal{P}^{c}_{3}$ $a=0$ and $\forall \mu$ $h{(\mathbb{Z}_{\geqslant t})}=-1$, $h{(\mathbb{Z}_{\leqslant t-1})}=0$ and
$ g{(\mathbb{Z}_{\geqslant s})}=-1$, $g{(\mathbb{Z}_{\leqslant s-1})}=0$
$\mathcal{W}_{1,\mu}^{\mathcal{P}_{4}^c}$ $\mathcal{P}^{c}_{4}$ $a=0$ and $\forall \mu$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=0 $
$\mathcal{W}_{2,\mu}^{\mathcal{P}_{4}^c}$ $\mathcal{P}^{c}_{4}$ $a=0$ and $\forall \mu$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1 $
$\mathcal{W}_{3,\mu}^{\mathcal{P}^{c}_{4},k}$ $\mathcal{P}^{c}_{4}$ $a=0$ and $\forall \mu$ $h{(\mathbb{Z}_{> 0})}=0$, $h{(\mathbb{Z}_{< 0})}=-1$ and
$ g{(\mathbb{Z}^*_{\geqslant k})}=0$, $g{(\mathbb{Z}^*_{\leqslant k-1})}=-1$
$\mathcal{W}_{4,a,\mu}^{\mathcal{P}^{c}_{4},k=1}$ $\mathcal{P}^{c}_{4}$ $\forall a$ and $\forall \mu$ $ h{(\mathbb{Z}_{> 0})}=0$, $h{(\mathbb{Z}_{< 0})}=-1$ and
$g{(\mathbb{Z}_{> 0})}=0$, $g{(\mathbb{Z}_{< 0})}=-1$
$\mathcal{W}_{5,\mu}^{\mathcal{P}_{4}^{c},s,t}$ $\mathcal{P}^{c}_{4}$ $a=0$ and $\forall \mu$ $h{(\mathbb{Z}_{\geqslant t})}=0$, $h{(\mathbb{Z}_{\leqslant t-1})}=-1$ and
$ g{(\mathbb{Z}_{\geqslant s})}=0$, $g{(\mathbb{Z}_{\leqslant s-1})}=-1$
$\mathcal{W}_{1}^{\mathcal{P}_{5}}$ $\mathcal{P}_{5}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=0 $
$\mathcal{W}_{2}^{\mathcal{P}_{5}}$ $\mathcal{P}_{5}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1 $
$\mathcal{W}_{3}^{\mathcal{P}_{5}}$ $\mathcal{P}_{5}$ $a=\mu=0$ $h{(\mathbb{Z}_{\leqslant 0})}=0$, $h{(\mathbb{Z}_{\geqslant 1})}=-1$ and
$ g{(\mathbb{Z}_{\leqslant -1})}=0$, $g{(\mathbb{Z}_{\geqslant 1})}=-1 $
$\mathcal{W}_{1}^{\mathcal{P}_{6}}$ $\mathcal{P}_{6}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=0 $
$\mathcal{W}_{2}^{\mathcal{P}_{6}}$ $\mathcal{P}_{6}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1 $
$\mathcal{W}_{3}^{\mathcal{P}_{6}}$ $\mathcal{P}_{6}$ $a=\mu=0$ $h{(\mathbb{Z}_{\leqslant 0})}=-1$, $h{(\mathbb{Z}_{\geqslant 1})}=0$ and
$ g{(\mathbb{Z}_{\leqslant -1})}=-1$, $g{(\mathbb{Z}_{\geqslant 1})}=0 $
$\mathcal{W}_{1}^{\mathcal{P}_{7}}$ $\mathcal{P}_{7}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=0 $
$\mathcal{W}_{2}^{\mathcal{P}_{7}}$ $\mathcal{P}_{7}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1 $
$\mathcal{W}_{3}^{\mathcal{P}_{7}}$ $\mathcal{P}_{7}$ $a=\mu=0$ $h{(\mathbb{Z}_{\leqslant 0})}=-1$, $h{(\mathbb{Z}_{\geqslant 1})}=0$ and
$ g{(\mathbb{Z}_{\leqslant -1})}=-1$, $g{(\mathbb{Z}_{\geqslant 1})}=0 $
$\mathcal{W}_{1}^{\mathcal{P}_{8}}$ $\mathcal{P}_{8}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=0 $
$\mathcal{W}_{2}^{\mathcal{P}_{8}}$ $\mathcal{P}_{8}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1 $
$\mathcal{W}_{3}^{\mathcal{P}_{8}}$ $\mathcal{P}_{8}$ $a=\mu=0$ $h{(\mathbb{Z}_{\leqslant 0})}=0$, $h{(\mathbb{Z}_{\geqslant 1})}=-1$ and
$ g{(\mathbb{Z}_{\leqslant -1})}=0$, $g{(\mathbb{Z}_{\geqslant 1})}=-1 $
$Cases$ $f(n)$ from Table 1 $a$, $\mu$ $h(n),g(n)$
$\mathcal{W}_{1}^{\mathcal{P}_{1}}$ $\mathcal{P}_{1}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=0 $
$\mathcal{W}_{2}^{\mathcal{P}_{1}}$ $\mathcal{P}_{1}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1 $
$\mathcal{W}_{1}^{\mathcal{P}_{2}}$ $\mathcal{P}_{2}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=0 $
$\mathcal{W}_{2}^{\mathcal{P}_{2}}$ $\mathcal{P}_{2}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1 $
$\mathcal{W}_{1,\mu}^{\mathcal{P}_{3}^c}$ $\mathcal{P}^{c}_{3}$ $a=0$ and $\forall \mu$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=0 $
$\mathcal{W}_{2,\mu}^{\mathcal{P}_{3}^c}$ $\mathcal{P}^{c}_{3}$ $a=0$ and $\forall \mu$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1 $
$\mathcal{W}_{3,\mu}^{\mathcal{P}^{c}_{3},k}$ $\mathcal{P}^{c}_{3}$ $a=0$ and $\forall \mu$ $h{(\mathbb{Z}_{> 0})}=-1$, $h{(\mathbb{Z}_{< 0})}=0$ and
$ g{(\mathbb{Z}^*_{\geqslant k})}=-1$, $g{(\mathbb{Z}^*_{\leqslant k-1})}=0$
$\mathcal{W}_{4,a,\mu}^{\mathcal{P}_{3}^c,k=1}$ $\mathcal{P}^{c}_{3}$ $\forall a$ and $\forall \mu$ $ h{(\mathbb{Z}_{> 0})}=-1$, $h{(\mathbb{Z}_{< 0})}=0$ and
$g{(\mathbb{Z}_{> 0})}=-1$, $g{(\mathbb{Z}_{< 0})}=0$
$\mathcal{W}_{5,\mu}^{\mathcal{P}_{3}^c,s,t}$ $\mathcal{P}^{c}_{3}$ $a=0$ and $\forall \mu$ $h{(\mathbb{Z}_{\geqslant t})}=-1$, $h{(\mathbb{Z}_{\leqslant t-1})}=0$ and
$ g{(\mathbb{Z}_{\geqslant s})}=-1$, $g{(\mathbb{Z}_{\leqslant s-1})}=0$
$\mathcal{W}_{1,\mu}^{\mathcal{P}_{4}^c}$ $\mathcal{P}^{c}_{4}$ $a=0$ and $\forall \mu$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=0 $
$\mathcal{W}_{2,\mu}^{\mathcal{P}_{4}^c}$ $\mathcal{P}^{c}_{4}$ $a=0$ and $\forall \mu$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1 $
$\mathcal{W}_{3,\mu}^{\mathcal{P}^{c}_{4},k}$ $\mathcal{P}^{c}_{4}$ $a=0$ and $\forall \mu$ $h{(\mathbb{Z}_{> 0})}=0$, $h{(\mathbb{Z}_{< 0})}=-1$ and
$ g{(\mathbb{Z}^*_{\geqslant k})}=0$, $g{(\mathbb{Z}^*_{\leqslant k-1})}=-1$
$\mathcal{W}_{4,a,\mu}^{\mathcal{P}^{c}_{4},k=1}$ $\mathcal{P}^{c}_{4}$ $\forall a$ and $\forall \mu$ $ h{(\mathbb{Z}_{> 0})}=0$, $h{(\mathbb{Z}_{< 0})}=-1$ and
$g{(\mathbb{Z}_{> 0})}=0$, $g{(\mathbb{Z}_{< 0})}=-1$
$\mathcal{W}_{5,\mu}^{\mathcal{P}_{4}^{c},s,t}$ $\mathcal{P}^{c}_{4}$ $a=0$ and $\forall \mu$ $h{(\mathbb{Z}_{\geqslant t})}=0$, $h{(\mathbb{Z}_{\leqslant t-1})}=-1$ and
$ g{(\mathbb{Z}_{\geqslant s})}=0$, $g{(\mathbb{Z}_{\leqslant s-1})}=-1$
$\mathcal{W}_{1}^{\mathcal{P}_{5}}$ $\mathcal{P}_{5}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=0 $
$\mathcal{W}_{2}^{\mathcal{P}_{5}}$ $\mathcal{P}_{5}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1 $
$\mathcal{W}_{3}^{\mathcal{P}_{5}}$ $\mathcal{P}_{5}$ $a=\mu=0$ $h{(\mathbb{Z}_{\leqslant 0})}=0$, $h{(\mathbb{Z}_{\geqslant 1})}=-1$ and
$ g{(\mathbb{Z}_{\leqslant -1})}=0$, $g{(\mathbb{Z}_{\geqslant 1})}=-1 $
$\mathcal{W}_{1}^{\mathcal{P}_{6}}$ $\mathcal{P}_{6}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=0 $
$\mathcal{W}_{2}^{\mathcal{P}_{6}}$ $\mathcal{P}_{6}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1 $
$\mathcal{W}_{3}^{\mathcal{P}_{6}}$ $\mathcal{P}_{6}$ $a=\mu=0$ $h{(\mathbb{Z}_{\leqslant 0})}=-1$, $h{(\mathbb{Z}_{\geqslant 1})}=0$ and
$ g{(\mathbb{Z}_{\leqslant -1})}=-1$, $g{(\mathbb{Z}_{\geqslant 1})}=0 $
$\mathcal{W}_{1}^{\mathcal{P}_{7}}$ $\mathcal{P}_{7}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=0 $
$\mathcal{W}_{2}^{\mathcal{P}_{7}}$ $\mathcal{P}_{7}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1 $
$\mathcal{W}_{3}^{\mathcal{P}_{7}}$ $\mathcal{P}_{7}$ $a=\mu=0$ $h{(\mathbb{Z}_{\leqslant 0})}=-1$, $h{(\mathbb{Z}_{\geqslant 1})}=0$ and
$ g{(\mathbb{Z}_{\leqslant -1})}=-1$, $g{(\mathbb{Z}_{\geqslant 1})}=0 $
$\mathcal{W}_{1}^{\mathcal{P}_{8}}$ $\mathcal{P}_{8}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=0 $
$\mathcal{W}_{2}^{\mathcal{P}_{8}}$ $\mathcal{P}_{8}$ $a=\mu=0$ $ h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1 $
$\mathcal{W}_{3}^{\mathcal{P}_{8}}$ $\mathcal{P}_{8}$ $a=\mu=0$ $h{(\mathbb{Z}_{\leqslant 0})}=0$, $h{(\mathbb{Z}_{\geqslant 1})}=-1$ and
$ g{(\mathbb{Z}_{\leqslant -1})}=0$, $g{(\mathbb{Z}_{\geqslant 1})}=-1 $
Table 1.  Values of f satisfying (16), where c$\mathbb{C}$
$Cases$ $f(n) $
$\mathcal{P}_{1} $ $f{(\mathbb{Z})}=0 $
$\mathcal{P}_{2} $ $f{(\mathbb{Z})}=-1 $
$\mathcal{P}^{c}_{3} $ $f{(\mathbb{Z}_{>0})}=-1,f{(\mathbb{Z}_{<0})}=0 \,$and$\, f{(0)}=c $
$\mathcal{P}^{c}_{4} $ $f{(\mathbb{Z}_{>0})}=0,f{(\mathbb{Z}_{<0})}=-1 \,$and$\, f{(0)}=c $
$\mathcal{P}_{5} $ $f{(\mathbb{Z}_{\geqslant2})}=-1 \,$and$\, f{(\mathbb{Z}_{\leqslant1})}=0 $
$\mathcal{P}_{6} $ $f{(\mathbb{Z}_{\geqslant2})}=0 \,$and$\, f{(\mathbb{Z}_{\leqslant1})}=-1 $
$\mathcal{P}_{7} $ $f{(\mathbb{Z}_{\geqslant-1})}=0 \,$and$\, f{(\mathbb{Z}_{\leqslant-2})}=-1 $
$\mathcal{P}_{8} $ $f{(\mathbb{Z}_{\geqslant-1})}=-1 \,$and$\, f{(\mathbb{Z}_{\leqslant-2})}=0 $
$Cases$ $f(n) $
$\mathcal{P}_{1} $ $f{(\mathbb{Z})}=0 $
$\mathcal{P}_{2} $ $f{(\mathbb{Z})}=-1 $
$\mathcal{P}^{c}_{3} $ $f{(\mathbb{Z}_{>0})}=-1,f{(\mathbb{Z}_{<0})}=0 \,$and$\, f{(0)}=c $
$\mathcal{P}^{c}_{4} $ $f{(\mathbb{Z}_{>0})}=0,f{(\mathbb{Z}_{<0})}=-1 \,$and$\, f{(0)}=c $
$\mathcal{P}_{5} $ $f{(\mathbb{Z}_{\geqslant2})}=-1 \,$and$\, f{(\mathbb{Z}_{\leqslant1})}=0 $
$\mathcal{P}_{6} $ $f{(\mathbb{Z}_{\geqslant2})}=0 \,$and$\, f{(\mathbb{Z}_{\leqslant1})}=-1 $
$\mathcal{P}_{7} $ $f{(\mathbb{Z}_{\geqslant-1})}=0 \,$and$\, f{(\mathbb{Z}_{\leqslant-2})}=-1 $
$\mathcal{P}_{8} $ $f{(\mathbb{Z}_{\geqslant-1})}=-1 \,$and$\, f{(\mathbb{Z}_{\leqslant-2})}=0 $
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