system: [SYSTEM]

user: 3. BiHom-pre-Jordan代数 定义2.1 设J是线性空间，J上有双线性的代数运算 ( x ， y ) → x ⋅ y ， α ， β ∈ E n d ( J ) 且 α ， β 均可逆，若满足下面的条件 α β = β α ， (2.1) { ( β 2 ( x ) ⋅ α β ( y ) + β 2 ( y ) ⋅ α β ( x ) ) ⋅ α 2 β ( u ) } ⋅ β α 3 ( z ) + { α β 2 ( u ) ⋅ α β − 1 ( β 2 ( x ) ⋅ α β ( y )   + β 2 ( y ) ⋅ α β ( x ) ) } ⋅ β α 3 ( z ) + α 2 β 2 ( y ) ⋅ α β − 1 { α β 2 ( u ) ⋅ α β − 1 ( β 2 ( x ) ⋅ α β ( z ) ) }   + α 2 β 2 ( x ) ⋅ α β − 1 { α β 2 ( u ) ⋅ α β − 1 ( β 2 ( y ) ⋅ α β ( z ) ) } − α ( β 2 ( x ) ⋅ α β ( y ) + β 2 ( y ) ⋅ α β ( x ) ) ⋅ ( α 2 β ( u ) ⋅ α 3 ( z ) ) - α − 1 β ( α 2 β ( u ) ⋅ α 3 ( y ) + α 2 β ( y ) ⋅ α 3 ( u ) ) ⋅ α 2 β − 1 ( β 2 ( x ) ⋅ α β ( z ) )   − α − 1 β ( α 2 β ( u ) ⋅ α 3 ( x ) + α 2 β ( x ) ⋅ α 3 ( u ) ) ⋅ α 2 β − 1 ( β 2 ( y ) ⋅ α β ( z ) ) = 0 ， (2.2) α 2 β 2 ( z ) ⋅ α β − 1 { [ β 2 ( x ) ⋅ α β ( y ) + β 2 ( y ) ⋅ α β ( x ) ] ⋅ α 2 β ( u ) } + α 2 β 2 ( y ) ⋅ α β − 1 { [ β 2 ( x ) ⋅ α β ( z ) + β 2 ( z ) ⋅ α β ( x ) ] ⋅ α 2 β ( u ) } + α 2 β 2 ( x ) ⋅ α β − 1 { [ β 2 ( y ) ⋅ α β ( z ) + β 2 ( z ) ⋅ α β ( y ) ] ⋅ α 2 β ( u ) } − α ( β 2 ( x ) ⋅ α β ( y ) + β 2 ( y ) ⋅ α β ( x ) ) ⋅ ( α 2 β ( z ) ⋅ α 3 ( u ) ) − α ( β 2 ( x ) ⋅ α β ( z ) + β 2 ( z ) ⋅ α β ( x ) ) ⋅ ( α 2 β ( y ) ⋅ α 3 ( u ) ) − α ( β 2 ( y ) ⋅ α β ( z ) + β 2 ( z ) ⋅ α β ( y ) ) ⋅ ( α 2 β ( x ) ⋅ α 3 ( u ) ) = 0 ， (2.3) 其中 ∀ x ， y ， z ， u ∈ J ，则称 ( J ， ⋅ ， α ， β ) 是BiHom-pre-Jordan代数。 定理2.2 设 ( J ， ⋅ ， α ， β ) 是BiHom-pre-Jordan代数，在J上定义 x ∗ y = x ⋅ y + α − 1 β ( y ) ⋅ α β − 1 ( x ) ，     ∀ x ， y ∈ J ， (2.4) 则 ( J ， ∗ ， α ， β ) 是BiHom-Jordan代数。 证明：由(2.4)可知 β ( x ) ∗ α ( y ) − β ( y ) ∗ α ( x ) = β ( x ) ⋅ α ( y ) + β ( y ) ⋅ α ( x ) − β ( y ) ⋅ α ( x ) − β ( x ) ⋅ α ( y ) = 0 ， 经直接计算，可得 { ( β 2 ( x ) ∗ α β ( y ) ) ∗ α 2 β ( u ) } ∗ β α 3 ( z ) + { ( β 2 ( x ) ∗ α β ( z ) ) ∗ α 2 β ( u ) } ∗ β α 3 ( y )   + { ( β 2 ( y ) ∗ α β ( z ) ) ∗ α 2 β ( u ) } ∗ β α 3 ( x ) − α ( β 2 ( x ) ∗ α β ( y ) ) ∗ ( α 2 β ( u ) ∗ α 3 ( z ) )   − α ( β 2 ( x ) ∗ α β ( z ) ) ∗ ( α 2 β ( u ) ∗ α 3 ( y ) ) − α ( β 2 ( y ) ∗ α β ( z ) ) ∗ ( α 2 β ( u ) ∗ α 3 ( x ) ) = { ( β 2 ( x ) ⋅ α β ( y ) + β 2 ( y ) ⋅ α β ( x ) ) ⋅ α 2 β ( u ) } ⋅ β α 3 ( z ) + { α β 2 ( u ) ⋅ α β − 1 ( β 2 ( x )     ⋅ α β ( y ) + β 2 ( y ) ⋅ α β ( x ) ) } ⋅ β α 3 ( z ) + α 2 β 2 ( y ) ⋅ α β − 1 { α β 2 ( u ) ⋅ α β − 1 ( β 2 ( x ) ⋅ α β ( z ) ) } ⋅ α 2 β ( u ) } − α ( β 2 ( x ) ⋅ α β ( y ) + β 2 ( y ) ⋅ α β ( x ) ) ⋅ ( α 2 β ( z ) ⋅ α 3 ( u ) ) − α ( β 2 ( x ) ⋅ α β ( z )   + β 2 ( z ) ⋅ α β ( x ) ) ⋅ ( α 2 β ( y ) ⋅ α 3 ( u ) ) − α ( β 2 ( y ) ⋅ α β ( z ) + β 2 ( z ) ⋅ α β ( y ) ) ⋅ ( α 2 β ( u )   ⋅ α 3 ( x ) ) − α ( β 2 ( y ) ⋅ α β ( z ) + β 2 ( z ) ⋅ α β ( y ) ) ⋅ ( α 2 β ( x ) ⋅ α 3 ( u ) ) ) + { ( β 2 ( x ) ⋅ α β ( z ) +   β 2 ( z ) ⋅ α β ( x ) ) ⋅ α 2 β ( u ) } ⋅ β α 3 ( y ) + { α β 2 ( u ) ⋅ α β − 1 ( β 2 ( x ) ⋅ α β ( z ) + β 2 ( z ) ⋅ α β ( x ) ) }   ⋅ β α 3 ( y ) + α 2 β 2 ( z ) ⋅ α β − 1 { α β 2 ( u ) ⋅ α β − 1 ( β 2 ( x ) ⋅ α β ( y ) ) } + α 2 β 2 ( x ) ⋅ α β − 1 { α β 2 ( u )   ⋅ α β − 1 ( β 2 ( z ) ⋅ α β ( y ) ) } − α ( β 2 ( x ) ⋅ α β ( z ) + β 2 ( z ) ⋅ α β ( x ) ) ⋅ ( α 2 β ( u ) ⋅ α 3 ( y ) ) − α − 1 β ( α 2 β ( u ) ⋅ α 3 ( z ) + α 2 β ( z ) ⋅ α 3 ( u ) ) ⋅ α 2 β − 1 ( β 2 ( x ) ⋅ α β ( y ) ) − α − 1 β ( α 2 β ( u ) ⋅ α 3 ( x )   + α 2 β ( x ) ⋅ α 3 ( u ) ) ⋅ α 2 β − 1 ( β 2 ( z ) ⋅ α β ( y ) ) ) + { ( β 2 ( y ) ⋅ α β ( z ) + β 2 ( z ) ⋅ α β ( y ) ) ⋅ α 2 β ( u ) }   ⋅ β α 3 ( x ) + { α β 2 ( u ) ⋅ α β − 1 ( β 2 ( y ) ⋅ α β ( z ) + β 2 ( z ) ⋅ α β ( y ) ) } ⋅ β α 3 ( x ) + α 2 β 2 ( z ) ⋅ α β − 1 { α β 2 ( u ) ⋅ α β − 1 ( β 2 ( y ) ⋅ α β ( x ) ) } + α 2 β 2 ( y ) ⋅ α β − 1 { α β 2 ( u ) ⋅ α β − 1 ( β 2 ( z ) ⋅ α β ( x ) ) } − α ( β 2 ( y ) ⋅ α β ( z ) + β 2 ( z ) ⋅ α β ( y ) ) ⋅ ( α 2 β ( u ) ⋅ α 3 ( x ) ) − α − 1 β ( α 2 β ( u ) ⋅ α 3 ( z )   + α 2 β ( z ) ⋅ α 3 ( u ) ) ⋅ α 2 β − 1 ( β 2 ( y ) ⋅ α β ( x ) ) − α − 1 β ( α 2 β ( u ) ⋅ α 3 ( y ) + α 2 β ( y ) ⋅ α 3 ( u ) )   ⋅ α 2 β − 1 ( β 2 ( z ) ⋅ α β ( x ) ) = 0 ， 所以， ( J ， ∗ ， α ， β ) 是BiHom-Jordan代数。