Let $$(X,r_X)$$ ( X , r X ) and $$(Y,r_Y)$$ ( Y , r Y ) be finite nondegenerate involutive set-th... more Let $$(X,r_X)$$ ( X , r X ) and $$(Y,r_Y)$$ ( Y , r Y ) be finite nondegenerate involutive set-theoretic solutions of the Yang–Baxter equation, and let $$A_X = \mathcal {A}({{\textbf {k}}}, X, r_X)$$ A X = A ( k , X , r X ) and $$A_Y= \mathcal {A}({{\textbf {k}}}, Y, r_Y)$$ A Y = A ( k , Y , r Y ) be their quadratic Yang–Baxter algebras over a field $${{\textbf {k}}}$$ k . We find an explicit presentation of the Segre product $$A_X\circ A_Y$$ A X ∘ A Y in terms of one-generators and quadratic relations. We introduce analogues of Segre maps in the class of Yang–Baxter algebras and find their images and their kernels. The results agree with their classical analogues in the commutative case.
We study noninvolutive set-theoretic solutions (X, r) of the YangBaxter equations on a set X in t... more We study noninvolutive set-theoretic solutions (X, r) of the YangBaxter equations on a set X in terms of the induced left and right actions of X on itself and in terms of the combinatorial properties of the canonically associated algebraic objects-the braided monoid S(X, r) and the graded quadratic k-algebra A = A(k, X, r) over a field k. We investigate the class of (noninvolutive) square-free solutions (X, r). It contains the particular class of self distributive solutions (i.e. quandles). We show that, similarly to the involutive case, every square-free braided set (possibly infinite, and not involutive) satisfies the cyclicity condions. We make a detailed characterization in terms of various algebraic and combinatorial properties each of which shows the contrast between involutive and noninvolutive square-free solutions. We study an interesting class of finite square-free braided sets (X, r) of order n ≥ 3 which satisfy the minimality condition M, that is dimk A2 = 2n − 1 (equiva...
Transactions of the American Mathematical Society, Series B
Let X = { x 1 , x 2 , ⋯ , x n } X= \{x_1, x_2, \cdots , x_n\} be a finite alphabet, and let K K b... more Let X = { x 1 , x 2 , ⋯ , x n } X= \{x_1, x_2, \cdots , x_n\} be a finite alphabet, and let K K be a field. We study classes C ( X , W ) \mathfrak {C}(X, W) of graded K K -algebras A = K ⟨ X ⟩ / I A = K\langle X\rangle / I , generated by X X and with a fixed set of obstructions W W . Initially we do not impose restrictions on W W and investigate the case when the algebras in C ( X , W ) \mathfrak {C} (X, W) have polynomial growth and finite global dimension d d . Next we consider classes C ( X , W ) \mathfrak {C} (X, W) of algebras whose sets of obstructions W W are antichains of Lyndon words. The central question is “when a class C ( X , W ) \mathfrak {C} (X, W) contains Artin-Schelter regular algebras?” Each class C ( X , W ) \mathfrak {C} (X, W) defines a Lyndon pair ( N , W ) (N,W) , which, if N N is finite, determines uniquely the global dimension, g l d i m A gl\,dimA , and the Gelfand-Kirillov dimension, G K d i m A GK dimA , for every A ∈ C ( X , W ) A \in \mathfrak {C}(X, W...
We study Veronese and Segre morphisms between non-commutative projective spaces. We compute finit... more We study Veronese and Segre morphisms between non-commutative projective spaces. We compute finite reduced Gröbner bases for their kernels, and compare them with their analogues in the commutative case.
We study set-theoretic solutions (X; r) of the Yang-Baxter equations on a set X in terms of the i... more We study set-theoretic solutions (X; r) of the Yang-Baxter equations on a set X in terms of the induced left and right actions of X on itself. We give a characterization of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of an induced matched pair of unital semigroups S(X; r) and construct (S; rS) from the matched pair. Finally, we study extensions of solutions in terms of matched pairs of their associated semigroups. We also prove several general results about matched pairs of unital semigroups of the required type, including iterated products S ./ S ./ S underlying the proof that rS is a solution, and extensions (S ./ T; rS./T ). Examples include a general ‘double’ construction (S ./ S; rS./S) and some concrete extensions, their actions and graphs based on small sets.
Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physic... more Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution $(X,r)$ consists of a set $X$ and a bijective map $r:X\times X\to X\times X$ which satisfies the braid relations. In this work we study the braided group $G=G(X,r)$ of an involutive square-free solution $(X,r)$ of finite order $n$ and cyclic index $p=p(X,r)$ and the group algebra $\textbf{k} [G]$ over a field $\textbf{k}$. We show that $G$ contains a $G$-invariant normal subgroup $\mathcal{F}_p$ of finite index $p^n$, $\mathcal{F}_p$ is isomorphic to the free abelian group of rank $n$. We describe explicitly the quotient braided group $\widetilde{G}=G/\mathcal{F}_p$ of order $p^n$ and show that $X$ is embedded in $\widetilde{G}$. We prove that the group algebra $\textbf{k} [G]$ is a free left (resp. right) module of finite rank $p^n$ over its commutative subalgebra $\textbf{k}[\mathcal{F}_p]$ and give an explicit free basis. The center of $\t...
We study set-theoretic solutions (X, r) of the Yang-Baxter equations on a set X in terms of the i... more We study set-theoretic solutions (X, r) of the Yang-Baxter equations on a set X in terms of the induced left and right actions of X on itself. We give a characterization of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of an induced matched pair of unital semigroups S(X, r) and construct (S, rS) from the matched pair. Finally, we study extensions of solutions in terms of matched pairs of their associated semigroups. We also prove several general results about matched pairs of unital semigroups of the required type, including iterated products S ./ S ./ S underlying the proof that rS is a solution, and extensions (S ./ T, rS./T ). Examples include a general ‘double’ construction (S ./ S, rS./S) and some concrete extensions, their actions and graphs based on small sets. MIRAMARE – TRIESTE August 2005 tatianagateva@yahoo.com, tatyana@aubg.bgs.majid@qmul.ac.uk
Let $$(X,r_X)$$ ( X , r X ) and $$(Y,r_Y)$$ ( Y , r Y ) be finite nondegenerate involutive set-th... more Let $$(X,r_X)$$ ( X , r X ) and $$(Y,r_Y)$$ ( Y , r Y ) be finite nondegenerate involutive set-theoretic solutions of the Yang–Baxter equation, and let $$A_X = \mathcal {A}({{\textbf {k}}}, X, r_X)$$ A X = A ( k , X , r X ) and $$A_Y= \mathcal {A}({{\textbf {k}}}, Y, r_Y)$$ A Y = A ( k , Y , r Y ) be their quadratic Yang–Baxter algebras over a field $${{\textbf {k}}}$$ k . We find an explicit presentation of the Segre product $$A_X\circ A_Y$$ A X ∘ A Y in terms of one-generators and quadratic relations. We introduce analogues of Segre maps in the class of Yang–Baxter algebras and find their images and their kernels. The results agree with their classical analogues in the commutative case.
We study noninvolutive set-theoretic solutions (X, r) of the YangBaxter equations on a set X in t... more We study noninvolutive set-theoretic solutions (X, r) of the YangBaxter equations on a set X in terms of the induced left and right actions of X on itself and in terms of the combinatorial properties of the canonically associated algebraic objects-the braided monoid S(X, r) and the graded quadratic k-algebra A = A(k, X, r) over a field k. We investigate the class of (noninvolutive) square-free solutions (X, r). It contains the particular class of self distributive solutions (i.e. quandles). We show that, similarly to the involutive case, every square-free braided set (possibly infinite, and not involutive) satisfies the cyclicity condions. We make a detailed characterization in terms of various algebraic and combinatorial properties each of which shows the contrast between involutive and noninvolutive square-free solutions. We study an interesting class of finite square-free braided sets (X, r) of order n ≥ 3 which satisfy the minimality condition M, that is dimk A2 = 2n − 1 (equiva...
Transactions of the American Mathematical Society, Series B
Let X = { x 1 , x 2 , ⋯ , x n } X= \{x_1, x_2, \cdots , x_n\} be a finite alphabet, and let K K b... more Let X = { x 1 , x 2 , ⋯ , x n } X= \{x_1, x_2, \cdots , x_n\} be a finite alphabet, and let K K be a field. We study classes C ( X , W ) \mathfrak {C}(X, W) of graded K K -algebras A = K ⟨ X ⟩ / I A = K\langle X\rangle / I , generated by X X and with a fixed set of obstructions W W . Initially we do not impose restrictions on W W and investigate the case when the algebras in C ( X , W ) \mathfrak {C} (X, W) have polynomial growth and finite global dimension d d . Next we consider classes C ( X , W ) \mathfrak {C} (X, W) of algebras whose sets of obstructions W W are antichains of Lyndon words. The central question is “when a class C ( X , W ) \mathfrak {C} (X, W) contains Artin-Schelter regular algebras?” Each class C ( X , W ) \mathfrak {C} (X, W) defines a Lyndon pair ( N , W ) (N,W) , which, if N N is finite, determines uniquely the global dimension, g l d i m A gl\,dimA , and the Gelfand-Kirillov dimension, G K d i m A GK dimA , for every A ∈ C ( X , W ) A \in \mathfrak {C}(X, W...
We study Veronese and Segre morphisms between non-commutative projective spaces. We compute finit... more We study Veronese and Segre morphisms between non-commutative projective spaces. We compute finite reduced Gröbner bases for their kernels, and compare them with their analogues in the commutative case.
We study set-theoretic solutions (X; r) of the Yang-Baxter equations on a set X in terms of the i... more We study set-theoretic solutions (X; r) of the Yang-Baxter equations on a set X in terms of the induced left and right actions of X on itself. We give a characterization of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of an induced matched pair of unital semigroups S(X; r) and construct (S; rS) from the matched pair. Finally, we study extensions of solutions in terms of matched pairs of their associated semigroups. We also prove several general results about matched pairs of unital semigroups of the required type, including iterated products S ./ S ./ S underlying the proof that rS is a solution, and extensions (S ./ T; rS./T ). Examples include a general ‘double’ construction (S ./ S; rS./S) and some concrete extensions, their actions and graphs based on small sets.
Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physic... more Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution $(X,r)$ consists of a set $X$ and a bijective map $r:X\times X\to X\times X$ which satisfies the braid relations. In this work we study the braided group $G=G(X,r)$ of an involutive square-free solution $(X,r)$ of finite order $n$ and cyclic index $p=p(X,r)$ and the group algebra $\textbf{k} [G]$ over a field $\textbf{k}$. We show that $G$ contains a $G$-invariant normal subgroup $\mathcal{F}_p$ of finite index $p^n$, $\mathcal{F}_p$ is isomorphic to the free abelian group of rank $n$. We describe explicitly the quotient braided group $\widetilde{G}=G/\mathcal{F}_p$ of order $p^n$ and show that $X$ is embedded in $\widetilde{G}$. We prove that the group algebra $\textbf{k} [G]$ is a free left (resp. right) module of finite rank $p^n$ over its commutative subalgebra $\textbf{k}[\mathcal{F}_p]$ and give an explicit free basis. The center of $\t...
We study set-theoretic solutions (X, r) of the Yang-Baxter equations on a set X in terms of the i... more We study set-theoretic solutions (X, r) of the Yang-Baxter equations on a set X in terms of the induced left and right actions of X on itself. We give a characterization of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of an induced matched pair of unital semigroups S(X, r) and construct (S, rS) from the matched pair. Finally, we study extensions of solutions in terms of matched pairs of their associated semigroups. We also prove several general results about matched pairs of unital semigroups of the required type, including iterated products S ./ S ./ S underlying the proof that rS is a solution, and extensions (S ./ T, rS./T ). Examples include a general ‘double’ construction (S ./ S, rS./S) and some concrete extensions, their actions and graphs based on small sets. MIRAMARE – TRIESTE August 2005 tatianagateva@yahoo.com, tatyana@aubg.bgs.majid@qmul.ac.uk
Uploads
Papers by Tatiana Gateva-Ivanova