{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T09:02:02Z","timestamp":1771578122397,"version":"3.50.1"},"reference-count":26,"publisher":"Wiley","license":[{"start":{"date-parts":[[2022,9,19]],"date-time":"2022-09-19T00:00:00Z","timestamp":1663545600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2022,9,19]]},"abstract":"<jats:p>Consider the <jats:inline-formula>\n                     <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\">\n                        <a:mi>q<\/a:mi>\n                     <\/a:math>\n                  <\/jats:inline-formula>-deformed Lie algebra, <jats:inline-formula>\n                     <c:math xmlns:c=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\">\n                        <c:msub>\n                           <c:mrow>\n                              <c:mi mathvariant=\"fraktur\">t<\/c:mi>\n                           <\/c:mrow>\n                           <c:mrow>\n                              <c:mi>q<\/c:mi>\n                           <\/c:mrow>\n                        <\/c:msub>\n                        <c:mo>:<\/c:mo>\n                        <c:msub>\n                           <c:mrow>\n                              <c:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                                 <c:mrow>\n                                    <c:msub>\n                                       <c:mrow>\n                                          <c:mover accent=\"true\">\n                                             <c:mi>K<\/c:mi>\n                                             <c:mo>^<\/c:mo>\n                                          <\/c:mover>\n                                       <\/c:mrow>\n                                       <c:mrow>\n                                          <c:mn>1<\/c:mn>\n                                       <\/c:mrow>\n                                    <\/c:msub>\n                                    <c:mo>,<\/c:mo>\n                                    <c:msub>\n                                       <c:mrow>\n                                          <c:mover accent=\"true\">\n                                             <c:mi>K<\/c:mi>\n                                             <c:mo>^<\/c:mo>\n                                          <\/c:mover>\n                                       <\/c:mrow>\n                                       <c:mrow>\n                                          <c:mn>2<\/c:mn>\n                                       <\/c:mrow>\n                                    <\/c:msub>\n                                 <\/c:mrow>\n                              <\/c:mfenced>\n                           <\/c:mrow>\n                           <c:mrow>\n                              <c:mi>q<\/c:mi>\n                           <\/c:mrow>\n                        <\/c:msub>\n                        <c:mo>=<\/c:mo>\n                        <c:mfenced open=\"(\" close=\")\" separators=\"|\">\n                           <c:mrow>\n                              <c:mn>1<\/c:mn>\n                              <c:mo>\u2212<\/c:mo>\n                              <c:mi>q<\/c:mi>\n                           <\/c:mrow>\n                        <\/c:mfenced>\n                        <c:msub>\n                           <c:mrow>\n                              <c:mover accent=\"true\">\n                                 <c:mi>K<\/c:mi>\n                                 <c:mo>^<\/c:mo>\n                              <\/c:mover>\n                           <\/c:mrow>\n                           <c:mrow>\n                              <c:mn>1<\/c:mn>\n                           <\/c:mrow>\n                        <\/c:msub>\n                        <c:msub>\n                           <c:mrow>\n                              <c:mover accent=\"true\">\n                                 <c:mi>K<\/c:mi>\n                                 <c:mo>^<\/c:mo>\n                              <\/c:mover>\n                           <\/c:mrow>\n                           <c:mrow>\n                              <c:mn>2<\/c:mn>\n                           <\/c:mrow>\n                        <\/c:msub>\n                        <c:mo>,<\/c:mo>\n                        <c:msub>\n                           <c:mrow>\n                              <c:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                                 <c:mrow>\n                                    <c:msub>\n                                       <c:mrow>\n                                          <c:mover accent=\"true\">\n                                             <c:mi>K<\/c:mi>\n                                             <c:mo>^<\/c:mo>\n                                          <\/c:mover>\n                                       <\/c:mrow>\n                                       <c:mrow>\n                                          <c:mn>3<\/c:mn>\n                                       <\/c:mrow>\n                                    <\/c:msub>\n                                    <c:mo>,<\/c:mo>\n                                    <c:msub>\n                                       <c:mrow>\n                                          <c:mover accent=\"true\">\n                                             <c:mi>K<\/c:mi>\n                                             <c:mo>^<\/c:mo>\n                                          <\/c:mover>\n                                       <\/c:mrow>\n                                       <c:mrow>\n                                          <c:mn>1<\/c:mn>\n                                       <\/c:mrow>\n                                    <\/c:msub>\n                                 <\/c:mrow>\n                              <\/c:mfenced>\n                           <\/c:mrow>\n                           <c:mrow>\n                              <c:mi>q<\/c:mi>\n                           <\/c:mrow>\n                        <\/c:msub>\n                        <c:mo>=<\/c:mo>\n                        <c:mi>s<\/c:mi>\n                        <c:msub>\n                           <c:mrow>\n                              <c:mover accent=\"true\">\n                                 <c:mi>K<\/c:mi>\n                                 <c:mo>^<\/c:mo>\n                              <\/c:mover>\n                           <\/c:mrow>\n                           <c:mrow>\n                              <c:mn>3<\/c:mn>\n                           <\/c:mrow>\n                        <\/c:msub>\n                     <\/c:math>\n                  <\/jats:inline-formula>, <jats:inline-formula>\n                     <v:math xmlns:v=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\">\n                        <v:msub>\n                           <v:mrow>\n                              <v:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                                 <v:mrow>\n                                    <v:msub>\n                                       <v:mrow>\n                                          <v:mover accent=\"true\">\n                                             <v:mi>K<\/v:mi>\n                                             <v:mo>^<\/v:mo>\n                                          <\/v:mover>\n                                       <\/v:mrow>\n                                       <v:mrow>\n                                          <v:mn>1<\/v:mn>\n                                       <\/v:mrow>\n                                    <\/v:msub>\n                                    <v:mo>,<\/v:mo>\n                                    <v:msub>\n                                       <v:mrow>\n                                          <v:mover accent=\"true\">\n                                             <v:mi>K<\/v:mi>\n                                             <v:mo>^<\/v:mo>\n                                          <\/v:mover>\n                                       <\/v:mrow>\n                                       <v:mrow>\n                                          <v:mn>4<\/v:mn>\n                                       <\/v:mrow>\n                                    <\/v:msub>\n                                 <\/v:mrow>\n                              <\/v:mfenced>\n                           <\/v:mrow>\n                           <v:mrow>\n                              <v:mi>q<\/v:mi>\n                           <\/v:mrow>\n                        <\/v:msub>\n                        <v:mo>=<\/v:mo>\n                        <v:mi>s<\/v:mi>\n                        <v:msub>\n                           <v:mrow>\n                              <v:mover accent=\"true\">\n                                 <v:mi>K<\/v:mi>\n                                 <v:mo>^<\/v:mo>\n                              <\/v:mover>\n                           <\/v:mrow>\n                           <v:mrow>\n                              <v:mn>4<\/v:mn>\n                           <\/v:mrow>\n                        <\/v:msub>\n                        <v:mo>,<\/v:mo>\n                        <v:msub>\n                           <v:mrow>\n                              <v:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                                 <v:mrow>\n                                    <v:msub>\n                                       <v:mrow>\n                                          <v:mover accent=\"true\">\n                                             <v:mi>K<\/v:mi>\n                                             <v:mo>^<\/v:mo>\n                                          <\/v:mover>\n                                       <\/v:mrow>\n                                       <v:mrow>\n                                          <v:mn>3<\/v:mn>\n                                       <\/v:mrow>\n                                    <\/v:msub>\n                                    <v:mo>,<\/v:mo>\n                                    <v:msub>\n                                       <v:mrow>\n                                          <v:mover accent=\"true\">\n                                             <v:mi>K<\/v:mi>\n                                             <v:mo>^<\/v:mo>\n                                          <\/v:mover>\n                                       <\/v:mrow>\n                                       <v:mrow>\n                                          <v:mn>2<\/v:mn>\n                                       <\/v:mrow>\n                                    <\/v:msub>\n                                 <\/v:mrow>\n                              <\/v:mfenced>\n                           <\/v:mrow>\n                           <v:mrow>\n                              <v:mi>q<\/v:mi>\n                           <\/v:mrow>\n                        <\/v:msub>\n                        <v:mo>=<\/v:mo>\n                        <v:mi>t<\/v:mi>\n                        <v:msub>\n                           <v:mrow>\n                              <v:mover accent=\"true\">\n                                 <v:mi>K<\/v:mi>\n                                 <v:mo>^<\/v:mo>\n                              <\/v:mover>\n                           <\/v:mrow>\n                           <v:mrow>\n                              <v:mn>3<\/v:mn>\n                           <\/v:mrow>\n                        <\/v:msub>\n                        <v:mo>,<\/v:mo>\n                        <v:msub>\n                           <v:mrow>\n                              <v:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                                 <v:mrow>\n                                    <v:msub>\n                                       <v:mrow>\n                                          <v:mover accent=\"true\">\n                                             <v:mi>K<\/v:mi>\n                                             <v:mo>^<\/v:mo>\n                                          <\/v:mover>\n                                       <\/v:mrow>\n                                       <v:mrow>\n                                          <v:mn>2<\/v:mn>\n                                       <\/v:mrow>\n                                    <\/v:msub>\n                                    <v:mo>,<\/v:mo>\n                                    <v:msub>\n                                       <v:mrow>\n                                          <v:mover accent=\"true\">\n                                             <v:mi>K<\/v:mi>\n                                             <v:mo>^<\/v:mo>\n                                          <\/v:mover>\n                                       <\/v:mrow>\n                                       <v:mrow>\n                                          <v:mn>4<\/v:mn>\n                                       <\/v:mrow>\n                                    <\/v:msub>\n                                 <\/v:mrow>\n                              <\/v:mfenced>\n                           <\/v:mrow>\n                           <v:mrow>\n                              <v:mi>q<\/v:mi>\n                           <\/v:mrow>\n                        <\/v:msub>\n                        <v:mo>=<\/v:mo>\n                        <v:mi>t<\/v:mi>\n                        <v:msub>\n                           <v:mrow>\n                              <v:mover accent=\"true\">\n                                 <v:mi>K<\/v:mi>\n                                 <v:mo>^<\/v:mo>\n                              <\/v:mover>\n                           <\/v:mrow>\n                           <v:mrow>\n                              <v:mn>4<\/v:mn>\n                           <\/v:mrow>\n                        <\/v:msub>\n                     <\/v:math>\n                  <\/jats:inline-formula>, and <jats:inline-formula>\n                     <pb:math xmlns:pb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\">\n                        <pb:msub>\n                           <pb:mrow>\n                              <pb:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                                 <pb:mrow>\n                                    <pb:msub>\n                                       <pb:mrow>\n                                          <pb:mover accent=\"true\">\n                                             <pb:mi>K<\/pb:mi>\n                                             <pb:mo>^<\/pb:mo>\n                                          <\/pb:mover>\n                                       <\/pb:mrow>\n                                       <pb:mrow>\n                                          <pb:mn>4<\/pb:mn>\n                                       <\/pb:mrow>\n                                    <\/pb:msub>\n                                    <pb:mo>,<\/pb:mo>\n                                    <pb:msub>\n                                       <pb:mrow>\n                                          <pb:mover accent=\"true\">\n                                             <pb:mi>K<\/pb:mi>\n                                             <pb:mo>^<\/pb:mo>\n                                          <\/pb:mover>\n                                       <\/pb:mrow>\n                                       <pb:mrow>\n                                          <pb:mn>3<\/pb:mn>\n                                       <\/pb:mrow>\n                                    <\/pb:msub>\n                                 <\/pb:mrow>\n                              <\/pb:mfenced>\n                           <\/pb:mrow>\n                           <pb:mrow>\n                              <pb:mi>q<\/pb:mi>\n                           <\/pb:mrow>\n                        <\/pb:msub>\n                        <pb:mo>=<\/pb:mo>\n                        <pb:mi>r<\/pb:mi>\n                        <pb:msub>\n                           <pb:mrow>\n                              <pb:mover accent=\"true\">\n                                 <pb:mi>K<\/pb:mi>\n                                 <pb:mo>^<\/pb:mo>\n                              <\/pb:mover>\n                           <\/pb:mrow>\n                           <pb:mrow>\n                              <pb:mn>1<\/pb:mn>\n                           <\/pb:mrow>\n                        <\/pb:msub>\n                     <\/pb:math>\n                  <\/jats:inline-formula>, where <jats:inline-formula>\n                     <xb:math xmlns:xb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\">\n                        <xb:mi>r<\/xb:mi>\n                        <xb:mo>,<\/xb:mo>\n                        <xb:mi>s<\/xb:mi>\n                        <xb:mo>,<\/xb:mo>\n                        <xb:mi>t<\/xb:mi>\n                        <xb:mo>\u2208<\/xb:mo>\n                        <xb:mi>\u211d<\/xb:mi>\n                        <xb:mo>\u2212<\/xb:mo>\n                        <xb:mfenced open=\"{\" close=\"}\" separators=\"|\">\n                           <xb:mrow>\n                              <xb:mn>0<\/xb:mn>\n                           <\/xb:mrow>\n                        <\/xb:mfenced>\n                     <\/xb:math>\n                  <\/jats:inline-formula>, subject to the physical properties: <jats:inline-formula>\n                     <cc:math xmlns:cc=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\">\n                        <cc:msub>\n                           <cc:mrow>\n                              <cc:mover accent=\"true\">\n                                 <cc:mi>K<\/cc:mi>\n                                 <cc:mo>^<\/cc:mo>\n                              <\/cc:mover>\n                           <\/cc:mrow>\n                           <cc:mrow>\n                              <cc:mn>1<\/cc:mn>\n                           <\/cc:mrow>\n                        <\/cc:msub>\n                     <\/cc:math>\n                  <\/jats:inline-formula> and <jats:inline-formula>\n                     <fc:math xmlns:fc=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\">\n                        <fc:msub>\n                           <fc:mrow>\n                              <fc:mover accent=\"true\">\n                                 <fc:mi>K<\/fc:mi>\n                                 <fc:mo>^<\/fc:mo>\n                              <\/fc:mover>\n                           <\/fc:mrow>\n                           <fc:mrow>\n                              <fc:mn>2<\/fc:mn>\n                           <\/fc:mrow>\n                        <\/fc:msub>\n                     <\/fc:math>\n                  <\/jats:inline-formula> are real diagonal operators, and <jats:inline-formula>\n                     <ic:math xmlns:ic=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\">\n                        <ic:msub>\n                           <ic:mrow>\n                              <ic:mover accent=\"true\">\n                                 <ic:mi>K<\/ic:mi>\n                                 <ic:mo>^<\/ic:mo>\n                              <\/ic:mover>\n                           <\/ic:mrow>\n                           <ic:mrow>\n                              <ic:mn>3<\/ic:mn>\n                           <\/ic:mrow>\n                        <\/ic:msub>\n                        <ic:mo>=<\/ic:mo>\n                        <ic:msubsup>\n                           <ic:mrow>\n                              <ic:mover accent=\"true\">\n                                 <ic:mi>K<\/ic:mi>\n                                 <ic:mo>^<\/ic:mo>\n                              <\/ic:mover>\n                           <\/ic:mrow>\n                           <ic:mn>4<\/ic:mn>\n                           <ic:mo>\u2020<\/ic:mo>\n                        <\/ic:msubsup>\n                     <\/ic:math>\n                  <\/jats:inline-formula>, (<jats:inline-formula>\n                     <mc:math xmlns:mc=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\">\n                        <mc:mo>\u2020<\/mc:mo>\n                     <\/mc:math>\n                  <\/jats:inline-formula> is for Hermitian conjugation). The <jats:inline-formula>\n                     <oc:math xmlns:oc=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\">\n                        <oc:mi>q<\/oc:mi>\n                     <\/oc:math>\n                  <\/jats:inline-formula>-deformed Lie algebra, <jats:inline-formula>\n                     <qc:math xmlns:qc=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\">\n                        <qc:msub>\n                           <qc:mrow>\n                              <qc:mi mathvariant=\"fraktur\">t<\/qc:mi>\n                           <\/qc:mrow>\n                           <qc:mrow>\n                              <qc:mi>q<\/qc:mi>\n                           <\/qc:mrow>\n                        <\/qc:msub>\n                     <\/qc:math>\n                  <\/jats:inline-formula> is introduced as a generalized model of the Tavis\u2013Cummings model (Tavis and Cummings 1968, Bashir and Sebawe Abdalla 1995), namely, <jats:inline-formula>\n                     <tc:math xmlns:tc=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\">\n                        <tc:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                           <tc:mrow>\n                              <tc:msub>\n                                 <tc:mrow>\n                                    <tc:mover accent=\"true\">\n                                       <tc:mi>K<\/tc:mi>\n                                       <tc:mo>^<\/tc:mo>\n                                    <\/tc:mover>\n                                 <\/tc:mrow>\n                                 <tc:mrow>\n                                    <tc:mn>1<\/tc:mn>\n                                 <\/tc:mrow>\n                              <\/tc:msub>\n                              <tc:mo>,<\/tc:mo>\n                              <tc:msub>\n                                 <tc:mrow>\n                                    <tc:mover accent=\"true\">\n                                       <tc:mi>K<\/tc:mi>\n                                       <tc:mo>^<\/tc:mo>\n                                    <\/tc:mover>\n                                 <\/tc:mrow>\n                                 <tc:mrow>\n                                    <tc:mn>2<\/tc:mn>\n                                 <\/tc:mrow>\n                              <\/tc:msub>\n                           <\/tc:mrow>\n                        <\/tc:mfenced>\n                        <tc:mo>=<\/tc:mo>\n                        <tc:mn>0<\/tc:mn>\n                        <tc:mo>,<\/tc:mo>\n                        <tc:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                           <tc:mrow>\n                              <tc:msub>\n                                 <tc:mrow>\n                                    <tc:mover accent=\"true\">\n                                       <tc:mi>K<\/tc:mi>\n                                       <tc:mo>^<\/tc:mo>\n                                    <\/tc:mover>\n                                 <\/tc:mrow>\n                                 <tc:mrow>\n                                    <tc:mn>1<\/tc:mn>\n                                 <\/tc:mrow>\n                              <\/tc:msub>\n                              <tc:mo>,<\/tc:mo>\n                              <tc:msub>\n                                 <tc:mrow>\n                                    <tc:mover accent=\"true\">\n                                       <tc:mi>K<\/tc:mi>\n                                       <tc:mo>^<\/tc:mo>\n                                    <\/tc:mover>\n                                 <\/tc:mrow>\n                                 <tc:mrow>\n                                    <tc:mn>3<\/tc:mn>\n                                 <\/tc:mrow>\n                              <\/tc:msub>\n                           <\/tc:mrow>\n                        <\/tc:mfenced>\n                        <tc:mo>=<\/tc:mo>\n                        <tc:mo>\u2212<\/tc:mo>\n                        <tc:mn>2<\/tc:mn>\n                        <tc:msub>\n                           <tc:mrow>\n                              <tc:mover accent=\"true\">\n                                 <tc:mi>K<\/tc:mi>\n                                 <tc:mo>^<\/tc:mo>\n                              <\/tc:mover>\n                           <\/tc:mrow>\n                           <tc:mrow>\n                              <tc:mn>3<\/tc:mn>\n                           <\/tc:mrow>\n                        <\/tc:msub>\n                        <tc:mo>,<\/tc:mo>\n                        <tc:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                           <tc:mrow>\n                              <tc:msub>\n                                 <tc:mrow>\n                                    <tc:mover accent=\"true\">\n                                       <tc:mi>K<\/tc:mi>\n                                       <tc:mo>^<\/tc:mo>\n                                    <\/tc:mover>\n                                 <\/tc:mrow>\n                                 <tc:mrow>\n                                    <tc:mn>1<\/tc:mn>\n                                 <\/tc:mrow>\n                              <\/tc:msub>\n                              <tc:mo>,<\/tc:mo>\n                              <tc:msub>\n                                 <tc:mrow>\n                                    <tc:mover accent=\"true\">\n                                       <tc:mi>K<\/tc:mi>\n                                       <tc:mo>^<\/tc:mo>\n                                    <\/tc:mover>\n                                 <\/tc:mrow>\n                                 <tc:mrow>\n                                    <tc:mn>4<\/tc:mn>\n                                 <\/tc:mrow>\n                              <\/tc:msub>\n                           <\/tc:mrow>\n                        <\/tc:mfenced>\n                        <tc:mo>=<\/tc:mo>\n                        <tc:mn>2<\/tc:mn>\n                        <tc:msub>\n                           <tc:mrow>\n                              <tc:mover accent=\"true\">\n                                 <tc:mi>K<\/tc:mi>\n                                 <tc:mo>^<\/tc:mo>\n                              <\/tc:mover>\n                           <\/tc:mrow>\n                           <tc:mrow>\n                              <tc:mn>4<\/tc:mn>\n                           <\/tc:mrow>\n                        <\/tc:msub>\n                        <tc:mo>,<\/tc:mo>\n                        <tc:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                           <tc:mrow>\n                              <tc:msub>\n                                 <tc:mrow>\n                                    <tc:mover accent=\"true\">\n                                       <tc:mi>K<\/tc:mi>\n                                       <tc:mo>^<\/tc:mo>\n                                    <\/tc:mover>\n                                 <\/tc:mrow>\n                                 <tc:mrow>\n                                    <tc:mn>2<\/tc:mn>\n                                 <\/tc:mrow>\n                              <\/tc:msub>\n                              <tc:mo>,<\/tc:mo>\n                              <tc:msub>\n                                 <tc:mrow>\n                                    <tc:mover accent=\"true\">\n                                       <tc:mi>K<\/tc:mi>\n                                       <tc:mo>^<\/tc:mo>\n                                    <\/tc:mover>\n                                 <\/tc:mrow>\n                                 <tc:mrow>\n                                    <tc:mn>3<\/tc:mn>\n                                 <\/tc:mrow>\n                              <\/tc:msub>\n                           <\/tc:mrow>\n                        <\/tc:mfenced>\n                        <tc:mo>=<\/tc:mo>\n                        <tc:msub>\n                           <tc:mrow>\n                              <tc:mover accent=\"true\">\n                                 <tc:mi>K<\/tc:mi>\n                                 <tc:mo>^<\/tc:mo>\n                              <\/tc:mover>\n                           <\/tc:mrow>\n                           <tc:mrow>\n                              <tc:mn>3<\/tc:mn>\n                           <\/tc:mrow>\n                        <\/tc:msub>\n                        <tc:mo>,<\/tc:mo>\n                        <tc:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                           <tc:mrow>\n                              <tc:msub>\n                                 <tc:mrow>\n                                    <tc:mover accent=\"true\">\n                                       <tc:mi>K<\/tc:mi>\n                                       <tc:mo>^<\/tc:mo>\n                                    <\/tc:mover>\n                                 <\/tc:mrow>\n                                 <tc:mrow>\n                                    <tc:mn>2<\/tc:mn>\n                                 <\/tc:mrow>\n                              <\/tc:msub>\n                              <tc:mo>,<\/tc:mo>\n                              <tc:msub>\n                                 <tc:mrow>\n                                    <tc:mover accent=\"true\">\n                                       <tc:mi>K<\/tc:mi>\n                                       <tc:mo>^<\/tc:mo>\n                                    <\/tc:mover>\n                                 <\/tc:mrow>\n                                 <tc:mrow>\n                                    <tc:mn>4<\/tc:mn>\n                                 <\/tc:mrow>\n                              <\/tc:msub>\n                           <\/tc:mrow>\n                        <\/tc:mfenced>\n                        <tc:mo>=<\/tc:mo>\n                        <tc:msub>\n                           <tc:mrow>\n                              <tc:mover accent=\"true\">\n                                 <tc:mi>K<\/tc:mi>\n                                 <tc:mo>^<\/tc:mo>\n                              <\/tc:mover>\n                           <\/tc:mrow>\n                           <tc:mrow>\n                              <tc:mn>4<\/tc:mn>\n                           <\/tc:mrow>\n                        <\/tc:msub>\n                     <\/tc:math>\n                  <\/jats:inline-formula>, and <jats:inline-formula>\n                     <yd:math xmlns:yd=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\">\n                        <yd:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                           <yd:mrow>\n                              <yd:msub>\n                                 <yd:mrow>\n                                    <yd:mover accent=\"true\">\n                                       <yd:mi>K<\/yd:mi>\n                                       <yd:mo>^<\/yd:mo>\n                                    <\/yd:mover>\n                                 <\/yd:mrow>\n                                 <yd:mrow>\n                                    <yd:mn>4<\/yd:mn>\n                                 <\/yd:mrow>\n                              <\/yd:msub>\n                              <yd:mo>,<\/yd:mo>\n                              <yd:msub>\n                                 <yd:mrow>\n                                    <yd:mover accent=\"true\">\n                                       <yd:mi>K<\/yd:mi>\n                                       <yd:mo>^<\/yd:mo>\n                                    <\/yd:mover>\n                                 <\/yd:mrow>\n                                 <yd:mrow>\n                                    <yd:mn>3<\/yd:mn>\n                                 <\/yd:mrow>\n                              <\/yd:msub>\n                           <\/yd:mrow>\n                        <\/yd:mfenced>\n                        <yd:mo>=<\/yd:mo>\n                        <yd:msub>\n                           <yd:mrow>\n                              <yd:mover accent=\"true\">\n                                 <yd:mi>K<\/yd:mi>\n                                 <yd:mo>^<\/yd:mo>\n                              <\/yd:mover>\n                           <\/yd:mrow>\n                           <yd:mrow>\n                              <yd:mn>1<\/yd:mn>\n                           <\/yd:mrow>\n                        <\/yd:msub>\n                     <\/yd:math>\n                  <\/jats:inline-formula>, which is subject to the physical properties <jats:inline-formula>\n                     <ge:math xmlns:ge=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\">\n                        <ge:msub>\n                           <ge:mrow>\n                              <ge:mover accent=\"true\">\n                                 <ge:mi>K<\/ge:mi>\n                                 <ge:mo>^<\/ge:mo>\n                              <\/ge:mover>\n                           <\/ge:mrow>\n                           <ge:mrow>\n                              <ge:mn>1<\/ge:mn>\n                           <\/ge:mrow>\n                        <\/ge:msub>\n                     <\/ge:math>\n                  <\/jats:inline-formula> and <jats:inline-formula>\n                     <je:math xmlns:je=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\">\n                        <je:msub>\n                           <je:mrow>\n                              <je:mover accent=\"true\">\n                                 <je:mi>K<\/je:mi>\n                                 <je:mo>^<\/je:mo>\n                              <\/je:mover>\n                           <\/je:mrow>\n                           <je:mrow>\n                              <je:mn>2<\/je:mn>\n                           <\/je:mrow>\n                        <\/je:msub>\n                     <\/je:math>\n                  <\/jats:inline-formula> are real diagonal operators, and <jats:inline-formula>\n                     <me:math xmlns:me=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\">\n                        <me:msub>\n                           <me:mrow>\n                              <me:mover accent=\"true\">\n                                 <me:mi>K<\/me:mi>\n                                 <me:mo>^<\/me:mo>\n                              <\/me:mover>\n                           <\/me:mrow>\n                           <me:mrow>\n                              <me:mn>3<\/me:mn>\n                           <\/me:mrow>\n                        <\/me:msub>\n                        <me:mo>=<\/me:mo>\n                        <me:msubsup>\n                           <me:mrow>\n                              <me:mover accent=\"true\">\n                                 <me:mi>K<\/me:mi>\n                                 <me:mo>^<\/me:mo>\n                              <\/me:mover>\n                           <\/me:mrow>\n                           <me:mn>4<\/me:mn>\n                           <me:mo>\u2020<\/me:mo>\n                        <\/me:msubsup>\n                     <\/me:math>\n                  <\/jats:inline-formula>. Faithful matrix representations of the least degree of <jats:inline-formula>\n                     <qe:math xmlns:qe=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\">\n                        <qe:msub>\n                           <qe:mrow>\n                              <qe:mi mathvariant=\"fraktur\">t<\/qe:mi>\n                           <\/qe:mrow>\n                           <qe:mrow>\n                              <qe:mi>q<\/qe:mi>\n                           <\/qe:mrow>\n                        <\/qe:msub>\n                     <\/qe:math>\n                  <\/jats:inline-formula> are discussed, and conditions are given to guarantee the existence of the faithful representations.<\/jats:p>","DOI":"10.1155\/2022\/6737287","type":"journal-article","created":{"date-parts":[[2022,9,19]],"date-time":"2022-09-19T22:50:14Z","timestamp":1663627814000},"page":"1-8","source":"Crossref","is-referenced-by-count":1,"title":["On Faithful Matrix Representations of q-Deformed Models in Quantum Optics"],"prefix":"10.1155","volume":"2022","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8227-7123","authenticated-orcid":true,"given":"Latif A -M.","family":"Hanna","sequence":"first","affiliation":[{"name":"Kuwait University, Faculty of Science, Department of Mathematics, P.O. Box 5969, Safat 13060, Kuwait City, Kuwait"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9383-3121","authenticated-orcid":true,"given":"Abdullah","family":"Alazemi","sequence":"additional","affiliation":[{"name":"Kuwait University, Faculty of Science, Department of Mathematics, P.O. Box 5969, Safat 13060, Kuwait City, Kuwait"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4079-3642","authenticated-orcid":true,"given":"Anwar","family":"Al-Dhafeeri","sequence":"additional","affiliation":[{"name":"Kuwait University, Faculty of Science, Department of Mathematics, P.O. Box 5969, Safat 13060, Kuwait City, Kuwait"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1103\/physrev.170.379"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1109\/proc.1963.1664"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(77)90088-2"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1063\/1.528937"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1007\/bf02724503"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/0375-9601(95)00469-j"},{"key":"7","first-page":"229","volume-title":"Quantum Theorey of Angular Momentum","author":"Schwinger","year":"1965"},{"key":"8","first-page":"1351","article-title":"Phys. Rev. A 34","volume":"40","author":"M. S. Abdalla","year":"1993","journal-title":"Journal of Modern Optics"},{"key":"9","first-page":"131","article-title":"Physica A","volume":"179","author":"M. S. Abdalla","year":"1991","journal-title":"Journal of Modern Optics"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1103\/physrev.58.1098"},{"key":"11","first-page":"49","article-title":"A deformed Tavis-Cummings model and its matrix representation","volume":"35","author":"L. A.-M. Hanna","year":"2014","journal-title":"JP Journal of Algebra, Number Theory and Applications"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1007\/bf01121107"},{"key":"13","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1088\/0305-4470\/27\/1\/009","article-title":"-invariant polynomial extensions of Lie algebras in quantum many-body physics","volume":"27","author":"V. P. Karassiov","year":"1994","journal-title":"Journal of Physics A: Mathematical and General"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1016\/0375-9601(94)90816-8"},{"key":"15","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1142\/S0217979206033498","article-title":"Interpolating statistics and q-deformed oscillator algebras","volume":"20","author":"P. Narayana Swamy","year":"2006","journal-title":"International Journal of Modern Physics B"},{"key":"16","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/S0079-6638(08)70112-4","article-title":"Optical Atoms","volume":"31","author":"J. C. Spreeuw","year":"1993","journal-title":"Progress in Optics"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1016\/s0024-3795(03)00453-1"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1016\/s0024-3795(96)00631-3"},{"key":"19","first-page":"137","article-title":"On matrix representations of deformed Lie algebras","volume":"13","author":"L. A.-M. Hanna","year":"2010","journal-title":"Int. J. Theol. Phys., Group Th. and Nonlinear Opt."},{"key":"20","first-page":"45","article-title":"Lie series, Lie transformations, and their applications, Lie methods in optics","volume-title":"Lecture Notes in Physics","author":"S. Steinberg","year":"1986"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.37394\/23206.2020.19.43"},{"key":"22","article-title":"-deformed quantum mechanics based on the q-addition","volume-title":"Advanced Science News, Fortschr. Phys. 1800111","author":"W. S. Chung","year":"2019"},{"key":"23","doi-asserted-by":"crossref","first-page":"1554","DOI":"10.1007\/s10773-011-0666-5","article-title":"A comparative study on q-deformed Fermion oscillators","volume":"50","author":"A. Algin","year":"2011","journal-title":"International Journal of Theoretical Physics"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1155\/s1110757x03202047"},{"issue":"6","key":"25","first-page":"1083","article-title":"On faithful matrix representations of q-deformed Lie algebra for coupled quantized oscillators","volume":"33","author":"L. A.-M. 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