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F. Stoll - One of the best experts on this subject based on the ideXlab platform.
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The Quantized Walled Brauer Algebra and Mixed Tensor Space
Algebras and Representation Theory, 2014Co-Authors: R. Dipper, S. Doty, F. StollAbstract:In this paper we investigate a multi-parameter deformation $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ of the walled Brauer algebra which was previously introduced by Leduc ( 1994 ). We construct an integral basis of $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ consisting of oriented tangles which is in bijection with walled Brauer diagrams. Moreover, we study a natural action of $\mathfrak{B}_{r,s}^n(q)= \mathfrak{B}_{r,s}^n(q^{-1}-q,q^n,[n]_q)$ on mixed tensor space and prove that the kernel is free over the ground ring R of rank independent of R . As an application, we prove one side of Schur–Weyl duality for mixed tensor space: the image of $\mathfrak{B}_{r,s}^n(q)$ in the R -endomorphism ring of mixed tensor space is, for all choices of R and the parameter q , the endomorphism algebra of the action of the (specialized via the Lusztig integral form) quantized enveloping algebra U of the general Linear Lie algebra $\mathfrak{gl}_n$ on mixed tensor space. Thus, the U -invariants in the ring of R -Linear Endomorphisms of mixed tensor space are generated by the action of $\mathfrak{B}_{r,s}^n(q)$ .
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The quantized walled Brauer algebra and mixed tensor space
arXiv: Quantum Algebra, 2008Co-Authors: R. Dipper, S. Doty, F. StollAbstract:In this paper we investigate a multi-parameter deformation $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ of the walled Brauer algebra which was previously introduced by Leduc (\cite{leduc}). We construct an integral basis of $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ consisting of oriented tangles which is in bijection with walled Brauer diagrams. Moreover, we study a natural action of $\mathfrak{B}_{r,s}^n(q)= \mathfrak{B}_{r,s}^n(q^{-1}-q,q^n,[n]_q)$ on mixed tensor space and prove that the kernel is free over the ground ring $R$ of rank independent of $R$. As an application, we prove one side of Schur--Weyl duality for mixed tensor space: the image of $\mathfrak{B}_{r,s}^n(q)$ in the $R$-endomorphism ring of mixed tensor space is, for all choices of $R$ and the parameter $q$, the endomorphism algebra of the action of the (specialized via the Lusztig integral form) quantized enveloping algebra $\mathbf{U}$ of the general Linear Lie algebra $\mathfrak{gl}_n$ on mixed tensor space. Thus, the $\mathbf{U}$-invariants in the ring of $R$-Linear Endomorphisms of mixed tensor space are generated by the action of $\mathfrak{B}_{r,s}^n(q)$.
R. Dipper - One of the best experts on this subject based on the ideXlab platform.
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The Quantized Walled Brauer Algebra and Mixed Tensor Space
Algebras and Representation Theory, 2014Co-Authors: R. Dipper, S. Doty, F. StollAbstract:In this paper we investigate a multi-parameter deformation $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ of the walled Brauer algebra which was previously introduced by Leduc ( 1994 ). We construct an integral basis of $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ consisting of oriented tangles which is in bijection with walled Brauer diagrams. Moreover, we study a natural action of $\mathfrak{B}_{r,s}^n(q)= \mathfrak{B}_{r,s}^n(q^{-1}-q,q^n,[n]_q)$ on mixed tensor space and prove that the kernel is free over the ground ring R of rank independent of R . As an application, we prove one side of Schur–Weyl duality for mixed tensor space: the image of $\mathfrak{B}_{r,s}^n(q)$ in the R -endomorphism ring of mixed tensor space is, for all choices of R and the parameter q , the endomorphism algebra of the action of the (specialized via the Lusztig integral form) quantized enveloping algebra U of the general Linear Lie algebra $\mathfrak{gl}_n$ on mixed tensor space. Thus, the U -invariants in the ring of R -Linear Endomorphisms of mixed tensor space are generated by the action of $\mathfrak{B}_{r,s}^n(q)$ .
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The quantized walled Brauer algebra and mixed tensor space
arXiv: Quantum Algebra, 2008Co-Authors: R. Dipper, S. Doty, F. StollAbstract:In this paper we investigate a multi-parameter deformation $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ of the walled Brauer algebra which was previously introduced by Leduc (\cite{leduc}). We construct an integral basis of $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ consisting of oriented tangles which is in bijection with walled Brauer diagrams. Moreover, we study a natural action of $\mathfrak{B}_{r,s}^n(q)= \mathfrak{B}_{r,s}^n(q^{-1}-q,q^n,[n]_q)$ on mixed tensor space and prove that the kernel is free over the ground ring $R$ of rank independent of $R$. As an application, we prove one side of Schur--Weyl duality for mixed tensor space: the image of $\mathfrak{B}_{r,s}^n(q)$ in the $R$-endomorphism ring of mixed tensor space is, for all choices of $R$ and the parameter $q$, the endomorphism algebra of the action of the (specialized via the Lusztig integral form) quantized enveloping algebra $\mathbf{U}$ of the general Linear Lie algebra $\mathfrak{gl}_n$ on mixed tensor space. Thus, the $\mathbf{U}$-invariants in the ring of $R$-Linear Endomorphisms of mixed tensor space are generated by the action of $\mathfrak{B}_{r,s}^n(q)$.
Walter Moreira - One of the best experts on this subject based on the ideXlab platform.
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THE SMASH PRODUCT OF SYMMETRIC FUNCTIONS. EXTENDED ABSTRACT
2011Co-Authors: Marcelo Aguiar, Walter Ferrer, Walter MoreiraAbstract:Abstract. We construct a new operation among representations of the symmetric group that interpolates between the classical internal and external products, which are defined in terms of tensor product and induction of representations. Following Malvenuto and Reutenauer, we pass from symmetric functions to non-commutative symmetric functions and from there to the algebra of permutations in order to relate the internal and external products to the composition and convolution of Linear Endomorphisms of the tensor algebra. The new product we construct corresponds to the smash product of Endomorphisms of the tensor algebra. For symmetric functions, the smash product is given by a construction which combines induction and restriction of representations. For non-commutative symmetric functions, the structure constants of the smash product are given by an explicit combinatorial rule which extends a wellknown result of Garsia, Remmel, Reutenauer, and Solomon for the descent algebra. We describe the dual operation among quasi-symmetric functions in terms of alphabets. Résumé. Nous construisons une nouvelle opération parmi les représentations du groupe symétrique qui interpole entre les produits interne et externe. Ces derniers sont définis en termes du produit tensoriel et de l’induction des représentations. D’après Malvenuto et Reutenauer, nous passons des fonctions symétriques aux fonctions symétriques non commutatives et à l’algèbre des permutations afin de rapporter les produits internes et externes à la composition et à la convolution d’endomorphismes linéaires de l’algèbre tensorielle. Le nouveau produit correspond au produit smash d’endomorphismes de l’algèbre tensorielle. Pour les fonctions symétriques, le produit smash est donné par une construction qui combine l’induction et la restriction de représentations. Pour les fonctions symétriques non commutatives, les constantes de structure du produit smash sont données par une règle combinatoire explicite qui prolonge un résultat bien connu de Garsia, Remmel, Reutenauer et Solomon pour l’algèbre de descentes. Nous décrivons l’opération duale au niveau des fonctions quasi-symétriques en termes d’alphabets
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The smash product of symmetric functions. Extended abstract
arXiv: Combinatorics, 2004Co-Authors: Marcelo Aguiar, Walter Ferrer, Walter MoreiraAbstract:We construct a new operation among representations of the symmetric group that interpolates between the classical internal and external products, which are defined in terms of tensor product and induction of representations. Following Malvenuto and Reutenauer, we pass from symmetric functions to non-commutative symmetric functions and from there to the algebra of permutations in order to relate the internal and external products to the composition and convolution of Linear Endomorphisms of the tensor algebra. The new product we construct corresponds to the smash product of Endomorphisms of the tensor algebra. For symmetric functions, the smash product is given by a construction which combines induction and restriction of representations. For non-commutative symmetric functions, the structure constants of the smash product are given by an explicit combinatorial rule which extends a well-known result of Garsia, Remmel, Reutenauer, and Solomon for the descent algebra. We describe the dual operation among quasi-symmetric functions in terms of alphabets.
Marco Manetti - One of the best experts on this subject based on the ideXlab platform.
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Cohomological constraint on deformations of compact Kähler manifolds
Advances in Mathematics, 2004Co-Authors: Marco ManettiAbstract:Abstract We prove that for every compact Kahler manifold X the cup product H ∗ (X,T X )⊗H ∗ (X,Ω X ∗ )→H ∗ (X,Ω X ∗−1 ) can be lifted to an L∞-morphism from the Kodaira–Spencer differential graded Lie algebra to the suspension of the space of Linear Endomorphisms of the singular cohomology of X. As a consequence we get an algebraic proof of the principle “obstructions to deformations of compact Kahler manifolds annihilate ambient cohomology”.
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Cohomological constraint to deformations of compact Kaehler manifolds
arXiv: Algebraic Geometry, 2001Co-Authors: Marco ManettiAbstract:We prove that for every compact K\"ahler manifold $X$ there exists an $L$-infinity morphism, lifting the usual cup product in cohomology, from the Kodaira-Spencer differential graded Lie algebra to the suspension of the space of Linear Endomorphisms of the singular cohomology of $X$. As a consequence we get an algebraic proof of the principle ``obstructions to deformations of compact Kaehler manifolds annihilate ambient cohomology''.
Sergei Silvestrov - One of the best experts on this subject based on the ideXlab platform.
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Hom-Lie structures on 3-dimensional skew symmetric algebras
Journal of Physics: Conference Series, 2019Co-Authors: Elvice Ongong'a, Johan Richter, Sergei SilvestrovAbstract:We describe the dimension of the space of possible Linear Endomorphisms that turn skew-symmetric three-dimensional algebras into Hom-Lie algebras. We find a correspondence between the rank of a mat ...
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Classification of 3-dimensional Hom-Lie algebras
Journal of Physics: Conference Series, 2019Co-Authors: Elvice Ongong'a, Johan Richter, Sergei SilvestrovAbstract:We derive conditions for an arbitrary n-dimensional algebra to be a Hom-Lie algebra, in the form of a system of polynomial equations, containing both structure constants of the skew-symmetric biLinear map and constants describing the twisting Linear endomorphism. The equations are Linear in the constants representing the endomorphism and non-Linear in the structure constants. When the algebra is 3 or 4-dimensional we describe the space of possible Endomorphisms with minimum dimension. For the 3-dimensional case we give families of 3-dimensional Hom-Lie algebras arising from a general nilpotent Linear endomorphism constructed upto isomorphism together with non-isomorphic canonical representatives for all the families in that case. We further give a list of 4-dimensional Hom-Lie algebras arising from a general nilpotent Linear Endomorphisms.