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Register a new userChains of twists for symplectic Lie algebras
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Serious difficulties arise in the construction of chains of twists for symplectic Lie algebras. Applying the canonical chains of extended twists to deform the Hopf algebras U(sp(N)) one is forced to deal only with improper chains (induced by the U(sl(N)) subalgebras). In the present paper this problem is solved. For chains of regular injections the sets of maximal extended jordanian twists F_{E,k} are considered. We prove that there exists for U(sp(N)) the twist F_{B,k} composed of the factors F_{E,k}. It is demonstrated that the twisting procedure deforms the space of the primitive subalgebra sp(N-1). The recursive algorithm for such deformation is found. This construction generalizes the results obtained for orthogonal classical Lie algebras and demonstrates the universality of primitivization effect for regular chains of subalgebras. For the chain of maximal length the twists F_{B,k,max} become full, their carriers contain the Borel subalgebra B(sp(N)). Using such twisting procedures one can obtain the explicit quantizations for a wide class of classical r-matrices. As an example the full chain of extended twists for U(sp(3)) is considered.
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We introduce the conception of matched pairs of $(H, beta)$-Lie algebras, construct an $(H, beta)$-Lie algebra through them. We prove that the cocycle twist of a matched pair of $(H, beta)$-Lie algebras can also be matched.
Universal enveloping algebras of braided m-Lie algebras and PBW theorem are obtained by means of combinatorics on words.
Let $p$ be a prime number and $q=p^m$, with $m geq 1$ if $p eq 2,3$ and $m>1$ otherwise. Let $Omega$ be any non-trivial twist for the complex group algebra of $mathbf{PSL}_2(q)$ arising from a $2$-cocycle on an abelian subgroup of $mathbf{PSL}_2(q)$. We show that the twisted Hopf algebra $(mathbb{C} mathbf{PSL}_2(q))_{Omega}$ does not admit a Hopf order over any number ring. The same conclusion is proved for the Suzuki group $^2!B_2(q)$ and $mathbf{SL}_3(p)$ when the twist stems from an abelian $p$-subgroup. This supplies new families of complex semisimple (and simple) Hopf algebras that do not admit a Hopf order over any number ring. The strategy of the proof is formulated in a general framework that includes the finite simple groups of Lie type. As an application, we combine our results with two theorems of Thompson and Barry and Ward on minimal simple groups to establish that for any finite non-abelian simple group $G$ there is a twist $Omega$ for $mathbb{C} G$, arising from a $2$-cocycle on an abelian subgroup of $G$, such that $(mathbb{C} G)_{Omega}$ does not admit a Hopf order over any number ring. This partially answers in the negative a question posed by Meir and the second author.
Let g be a finite dimensional complex semisimple Lie algebra, and let V be a finite dimensional represenation of g. We give a closed formula for the mth Frobenius-Schur indicator, m>1, of V in representation-theoretic terms. We deduce that the indicators take integer values, and that for a large enough m, the mth indicator of V equals the dimension of the zero weight space of V. For the classical Lie algebras sl(n), so(2n), so(2n+1) and sp(2n), this is the case for m greater or equal to 2n-1, 4n-5, 4n-3 and 2n+1, respectively.
It is shown that if $mathfrak B(V) $ is connected Nichols algebra of diagonal type with $dim V>1$, then $dim (mathfrak L^-(V)) = infty$ $($resp. $ dim (mathfrak L(V)) = infty $$)$ $($ resp. $ dim (mathfrak B(V)) = infty $$)$ if and only if $Delta(mathfrak B(V)) $ is an arithmetic root system and the quantum numbers (i.e. the fixed parameters) of generalized Dynkin diagrams of $V$ are of finite order. Sufficient and necessary conditions for $m$-fold adjoint action in $mathfrak B(V)$ equal to zero, viz. $overline{l}_{x_{i}}^{m}[x_{j}]^ -=0$ for $x_i,~x_jin mathfrak B(V)$, are given.
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