Contractions of the Lie algebras d = u(2), f = u(1 ,1) to the oscillator Lie algebra l are realized via the adjoint action of SU(2,2) when d, l, f are viewed as subalgebras of su(2,2). Here D, L, F are the correspondi...Contractions of the Lie algebras d = u(2), f = u(1 ,1) to the oscillator Lie algebra l are realized via the adjoint action of SU(2,2) when d, l, f are viewed as subalgebras of su(2,2). Here D, L, F are the corresponding (four-dimensional) real Lie groups endowed with bi-invariant metrics of Lorentzian signature. Similar contractions of (seven-dimensional) isometry Lie algebras iso(D), iso(F) to iso(L) are determined. The group SU(2,2) acts on each of the D, L, F by conformal transformation which is a core feature of the DLF-theory. Also, d and f are contracted to T, S-abelian subalgebras, generating parallel translations, T, and proper conformal transformations, S (from the decomposition of su(2,2) as a graded algebra T + Ω + S, where Ω is the extended Lorentz Lie algebra of dimension 7).展开更多
Invariant symmetric bilinear forms and derivation algebras of a unitary Lie algebra L over R are characterized： （L） ≌ （R＋/（[R, R] A R＋））＊ and Der（L） = Inn（L） ＋ Der（L）0 = Inn（L） ＋ SDer（R）, which ...Invariant symmetric bilinear forms and derivation algebras of a unitary Lie algebra L over R are characterized： （L） ≌ （R＋/（[R, R] A R＋））＊ and Der（L） = Inn（L） ＋ Der（L）0 = Inn（L） ＋ SDer（R）, which recover what of the special linear Lie algebra and Steinberg Lie algebra over R, where R is a unital involutory associative algebra over a field F.展开更多
We use computer algebra to determine the Lie invariants of degree ≤ 12 in the free Lie algebra on two generators corresponding to the natural representation of the simple 3-dimensional Lie algebra sl2（C）. We then c...We use computer algebra to determine the Lie invariants of degree ≤ 12 in the free Lie algebra on two generators corresponding to the natural representation of the simple 3-dimensional Lie algebra sl2（C）. We then consider the free Lie algebra on three generators, and compute the Lie invariants of degree ≤ 7 corresponding to the adjoint representation of sl2（C）, and the Lie invariants of degree ≤ 9 corresponding to the natural representation of sl3（C）. We represent the action of sl2（C） and sl3（C） on Lie polynomials by computing the coefficient matrix with respect to the basis of Hall words.展开更多
The system of(1+1)-coupled Drinfel’d-Sokolov-Wilson equations describes the surface gravity waves travelling horizontally on the seabed.The objective of the present research is to construct a new variety of analytica...The system of(1+1)-coupled Drinfel’d-Sokolov-Wilson equations describes the surface gravity waves travelling horizontally on the seabed.The objective of the present research is to construct a new variety of analytical solutions for the system.The invariants are derived with the aid of Killing form by using the optimal algebra classification via Lie symmetry approach.The invariant solutions involve time,space variables,and arbitrary constants.Imposing adequate constraints on arbitrary constants,solutions are represented graphically to make them more applicable in designing sea models.The behavior of solutions shows asymptotic,bell-shaped,bright and dark soliton,bright soliton,parabolic,bright and kink,kink,and periodic nature.The constructed results are novel as the reported results[26,28,29,30,33,38,42,49]can be deduced from the results derived in this study.The remaining solutions derived in this study,are absolutely different from the earlier findings.In this study,the physical character of analytical solutions of the system could aid coastal engineers in creating models of beaches and ports.展开更多
In this paper, we study an infinite-dimensional Lie algebra Bq, called the q-analog Klein bottle Lie algebra. We show that Bq is a finitely generated simple Lie algebra with a unique （up to scalars） symmetric invari...In this paper, we study an infinite-dimensional Lie algebra Bq, called the q-analog Klein bottle Lie algebra. We show that Bq is a finitely generated simple Lie algebra with a unique （up to scalars） symmetric invariant bilinear form. The derivation algebra and the universal central extension of Bq are also determined.展开更多
文摘Contractions of the Lie algebras d = u(2), f = u(1 ,1) to the oscillator Lie algebra l are realized via the adjoint action of SU(2,2) when d, l, f are viewed as subalgebras of su(2,2). Here D, L, F are the corresponding (four-dimensional) real Lie groups endowed with bi-invariant metrics of Lorentzian signature. Similar contractions of (seven-dimensional) isometry Lie algebras iso(D), iso(F) to iso(L) are determined. The group SU(2,2) acts on each of the D, L, F by conformal transformation which is a core feature of the DLF-theory. Also, d and f are contracted to T, S-abelian subalgebras, generating parallel translations, T, and proper conformal transformations, S (from the decomposition of su(2,2) as a graded algebra T + Ω + S, where Ω is the extended Lorentz Lie algebra of dimension 7).
文摘Invariant symmetric bilinear forms and derivation algebras of a unitary Lie algebra L over R are characterized： （L） ≌ （R＋/（[R, R] A R＋））＊ and Der（L） = Inn（L） ＋ Der（L）0 = Inn（L） ＋ SDer（R）, which recover what of the special linear Lie algebra and Steinberg Lie algebra over R, where R is a unital involutory associative algebra over a field F.
文摘We use computer algebra to determine the Lie invariants of degree ≤ 12 in the free Lie algebra on two generators corresponding to the natural representation of the simple 3-dimensional Lie algebra sl2（C）. We then consider the free Lie algebra on three generators, and compute the Lie invariants of degree ≤ 7 corresponding to the adjoint representation of sl2（C）, and the Lie invariants of degree ≤ 9 corresponding to the natural representation of sl3（C）. We represent the action of sl2（C） and sl3（C） on Lie polynomials by computing the coefficient matrix with respect to the basis of Hall words.
文摘The system of(1+1)-coupled Drinfel’d-Sokolov-Wilson equations describes the surface gravity waves travelling horizontally on the seabed.The objective of the present research is to construct a new variety of analytical solutions for the system.The invariants are derived with the aid of Killing form by using the optimal algebra classification via Lie symmetry approach.The invariant solutions involve time,space variables,and arbitrary constants.Imposing adequate constraints on arbitrary constants,solutions are represented graphically to make them more applicable in designing sea models.The behavior of solutions shows asymptotic,bell-shaped,bright and dark soliton,bright soliton,parabolic,bright and kink,kink,and periodic nature.The constructed results are novel as the reported results[26,28,29,30,33,38,42,49]can be deduced from the results derived in this study.The remaining solutions derived in this study,are absolutely different from the earlier findings.In this study,the physical character of analytical solutions of the system could aid coastal engineers in creating models of beaches and ports.
基金The first author was supported in part by the NSFC （10931006, 10871125） and the Innovation Program of Shanghai Municipal Education Commission （11ZZ18）. The second author was supported by the NSFC （11326060）. The third author was supported in part by the NSFC （11101285, 11026042, 11071068）, the Shanghai Natural Science Foundation （11ZR1425900）, the Innovation Program of Shanghai Municipal Education Commission （11YZ85）, the Academic Discipline Project of Shanghai Normal University （DZL803） and ZJNSF （Y6100148）.
文摘In this paper, we study an infinite-dimensional Lie algebra Bq, called the q-analog Klein bottle Lie algebra. We show that Bq is a finitely generated simple Lie algebra with a unique （up to scalars） symmetric invariant bilinear form. The derivation algebra and the universal central extension of Bq are also determined.
