The question of how the category of entwined modules can be made into a braided monoidal category is studied. First, the sufficient and necessary conditions making the category into a monoidal category are obtained by...The question of how the category of entwined modules can be made into a braided monoidal category is studied. First, the sufficient and necessary conditions making the category into a monoidal category are obtained by using the fact that if （A, C, ψ） is an entwining structure, then A × C can be made into an entwined module. The conditions are that the algebra and coalgebra in question are both bialgebras with some extra compatibility relations. Then given a monodial category of entwined modules, the braiding is constructed by means of a twisted convolution invertible map Q, and the conditions making the category form into a braided monoidal category are obtained similarly. Finally, the construction is applied to the category of Doi-Hopf modules and （α, β ）-Yetter-Drinfeld modules as examples.展开更多
In this paper, we categorify a Hom-associative algebra by imposing the Homassociative law up to some isomorphisms on the multiplication map and requiring that these isomorphisms satisfy the Pentagon axiom, and obtain ...In this paper, we categorify a Hom-associative algebra by imposing the Homassociative law up to some isomorphisms on the multiplication map and requiring that these isomorphisms satisfy the Pentagon axiom, and obtain a 2-Hom-associative algebra. On the other hand, we introduce the dual Hom-quasi-Hopf algebra and show that any dual Homquasi-Hopf algebras can be viewed as a 2-Hom-associative algebra.展开更多
We define the right regular dual of an object X in a monoidal category l, and give several results regarding the weak rigid monoidal category. Based on the definition of the right regular dual, we construct a weak Hop...We define the right regular dual of an object X in a monoidal category l, and give several results regarding the weak rigid monoidal category. Based on the definition of the right regular dual, we construct a weak Hopf algebra structure of H = End（F） whenever （F, J） is a fiber functor from category l to Vec and every X ∈ l has a right regular dual. To conclude, we give a weak reconstruction theorem for a kind of weak Hopf algebra.展开更多
Let A and H be Hopf algebra, T-smash product AT H generalizes twisted smash product A ＊ H. This paper shows a necessary and sufficient condition for T-smash product moduie category AT HM to be braided monoidal category.
In this paper, we first give the definitions of a crossed left π-H-comodules over a crossed weak Hopf π-algebra H, and show that the category of crossed left π-H-comodules is a monoidal category. Finally, we show t...In this paper, we first give the definitions of a crossed left π-H-comodules over a crossed weak Hopf π-algebra H, and show that the category of crossed left π-H-comodules is a monoidal category. Finally, we show that a family σ = {σα,β： Hα Hβ→ k}α,β∈πof k-linear maps is a coquasitriangular structure of a crossed weak Hopf π-algebra H if and only if the category of crossed left π-H-comodules over H is a braided monoidal category with braiding defined by σ.展开更多
The problem of embedding the Tsallis, Rényi and generalized Rényi entropies in the framework of category theory and their axiomatic foundation is studied. To this end, we construct a special category MES rel...The problem of embedding the Tsallis, Rényi and generalized Rényi entropies in the framework of category theory and their axiomatic foundation is studied. To this end, we construct a special category MES related to measured spaces. We prove that both of the Rényi and Tsallis entropies can be imbedded in the formalism of category theory by proving that the same basic partition functional that appears in their definitions, as well as in the associated Lebesgue space norms, has good algebraic compatibility properties. We prove that this functional is both additive and multiplicative with respect to the direct product and the disjoint sum (the coproduct) in the category MES, so it is a natural candidate for the measure of information or uncertainty. We prove that the category MES can be extended to monoidal category, both with respect to the direct product as well as to the coproduct. The basic axioms of the original Rényi entropy theory are generalized and reformulated in the framework of category MES and we prove that these axioms foresee the existence of an universal exponent having the same values for all the objects of the category MES. In addition, this universal exponent is the parameter, which appears in the definition of the Tsallis and Rényi entropies. It is proved that in a similar manner, the partition functional that appears in the definition of the Generalized Rényi entropy is a multiplicative functional with respect to direct product and additive with respect to the disjoint sum, but its symmetry group is reduced compared to the case of classical Rényi entropy.展开更多
Let （C, C） be a braided monoidal category. The relationship between the braided Lie algebra and the left Jacobi braided Lie algebra in the category （C, C） is investigated. First, a braided C2-commutative algebra i...Let （C, C） be a braided monoidal category. The relationship between the braided Lie algebra and the left Jacobi braided Lie algebra in the category （C, C） is investigated. First, a braided C2-commutative algebra in the category （C, C） is defined and three equations on the braiding in the category （C, C） are proved. Secondly, it is verified that （A, [, ] ） is a left （strict） Jacobi braided Lie algebra if and only if （A, [, ] ） is a braided Lie algebra, where A is an associative algebra in the category （C, C）. Finally, as an application, the structures of braided Lie algebras are given in the category of Yetter-Drinfel＇d modules and the category of Hopf bimodules.展开更多
We investigate how the category of comodules of bimonads can be made into a monoidal category.It suffices that the monad and comonad in question are bimonads,with some extra compatibility relation.On a monoidal catego...We investigate how the category of comodules of bimonads can be made into a monoidal category.It suffices that the monad and comonad in question are bimonads,with some extra compatibility relation.On a monoidal category of comodules of bimonads,we cons true t a braiding and get the necessary and sufficien t conditions making it a braided monoidal category.As an application,we consider the category of comodules of corings and the category of entwined modules.展开更多
Firstly,the notion of the left-left Yetter-Drinfeld quasicomodule M=(M,·,ρ)over a Hopf coquasigroup H is given,which generalizes the left-left Yetter-Drinfeld module over Hopf algebras.Secondly,the braided monoi...Firstly,the notion of the left-left Yetter-Drinfeld quasicomodule M=(M,·,ρ)over a Hopf coquasigroup H is given,which generalizes the left-left Yetter-Drinfeld module over Hopf algebras.Secondly,the braided monoidal category HHYDQCM is introduced and the specific structure maps are given.Thirdly,Sweedler's dual of infinite-dimensional Hopf algebras in HHYDQCM is discussed.It proves that if(B,mB,μB,ΔB,εB)is a Hopf algebra in HHYDQCM with antipode SB,then(B^0,(mB0)^op,εB^*,(ΔB0)^op,μB^*)is a Hopf algebra in HHYDQCM with antipode SB^*,which generalizes the corresponding results over Hopf algebras.展开更多
In this paper, we introduce the dual Hom-quasi-Hopf algebra and prove that the comodules category of a （braided） dual Hom-quasi-bialgebra is a monoidal category. Finally, we give a categorical realization of dual Ho...In this paper, we introduce the dual Hom-quasi-Hopf algebra and prove that the comodules category of a （braided） dual Hom-quasi-bialgebra is a monoidal category. Finally, we give a categorical realization of dual Hom-quasi-Hopf algebras.展开更多
In this paper, we give a necessary and sufficient condition for a comodule algebra over a weak Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the Hopf algebra setting. A...In this paper, we give a necessary and sufficient condition for a comodule algebra over a weak Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the Hopf algebra setting. Also, from these results, we deduce a version of Maschke＇s Theorem for （H, B）-Hopf modules associated with a weak Hopf algebra H and a right H-comodule algebra B.展开更多
In this paper, we give a sufficient condition for double crossproduct X A to be X A for some skew pairing T if X A is a 2-cocycle deformation of X A. Then we give a sufficient and necessary condition for X A to b...In this paper, we give a sufficient condition for double crossproduct X A to be X A for some skew pairing T if X A is a 2-cocycle deformation of X A. Then we give a sufficient and necessary condition for X A to be X A by using natural isomorphism terminology.展开更多
We introduce the notions of differential graded(DG) Poisson algebra and DG Poisson module. Let A be any DG Poisson algebra. We construct the universal enveloping algebra of A explicitly, which is denoted by A^(ue). We...We introduce the notions of differential graded(DG) Poisson algebra and DG Poisson module. Let A be any DG Poisson algebra. We construct the universal enveloping algebra of A explicitly, which is denoted by A^(ue). We show that A^(ue) has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over A is isomorphic to the category of DG modules over A^(ue). Furthermore, we prove that the notion of universal enveloping algebra A^(ue) is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.展开更多
In this paper we obtain a criterion under which the bijectivity of the canonical morphism of a weak Galois extension associated to a weak invertible entwining structure is equivalent to the existence of a strong conne...In this paper we obtain a criterion under which the bijectivity of the canonical morphism of a weak Galois extension associated to a weak invertible entwining structure is equivalent to the existence of a strong connection form. Also we obtain an explicit formula for a strong connection under equivariant projective conditions or under coseparability conditions.展开更多
We give a monoidal category approach to Hom-coassociative coalgebra by imposing the Hom-coassociative law up to some isomorphisms on the comultiplication map and requiring that these isomorphisms satisfy the copentago...We give a monoidal category approach to Hom-coassociative coalgebra by imposing the Hom-coassociative law up to some isomorphisms on the comultiplication map and requiring that these isomorphisms satisfy the copentagon axiom and obtain a Hom-coassociative 2-coalgebra, which is a 2- category. Second, we characterize Hom-bialgebras in terms of their categories of modules. Finally, we give a categorical realization of Hom-quasi-Hopf algebras using Hom-coassociative 2-coalgebra.展开更多
While the Yoneda embedding and its generalizations have been studied extensively in the literature,the so-called tensor embedding has only received a little attention.In this paper,we study the tensor embedding for cl...While the Yoneda embedding and its generalizations have been studied extensively in the literature,the so-called tensor embedding has only received a little attention.In this paper,we study the tensor embedding for closed symmetric monoidal categories and show how it is connected to the notion of geometrically purity,which has recently been investigated in the works of Enochs et al.(2016)and Estrada et al.(2017).More precisely,for a Grothendieck cosmos,i.e.,a bicomplete Grothendieck category V with a closed symmetric monoidal structure,we prove that the geometrically pure exact category(V,ε■)has enough relative injectives;in fact,every object has a geometrically pure injective envelope.We also show that for some regular cardinalλ,the tensor embedding yields an exact equivalence between(V,ε■)and the category ofλ-cocontinuous V-functors from Presλ(V)to V,where the former is the full V-subcategory ofλ-presentable objects in V.In many cases of interest,λcan be chosen to be■0 and the tensor embedding identifies the geometrically pure injective objects in V with the(categorically)injective objects in the abelian category of V-functors from fp(V)to V.As we explain,the developed theory applies,e.g.,to the category Ch(R)of chain complexes of modules over a commutative ring R and to the category Qcoh(X)of quasi-coherent sheaves over a(suitably nice)scheme X.展开更多
In this paper, we present the general theory and universal properties of weak crossed biproducts. We prove that every weak projection of weak bialgebras induces one of these weak crossed structures. Finally, we comput...In this paper, we present the general theory and universal properties of weak crossed biproducts. We prove that every weak projection of weak bialgebras induces one of these weak crossed structures. Finally, we compute explicitly the weak crossed biproduct associated with a groupoid that admits an exact factorization.展开更多
The authors present the general theory of cleft extensions for a cocommutative weak Hopf algebra H. For a right H-comodule algebra, they obtain a bijective corre- spondence between the isomorphisms classes of H-cleft ...The authors present the general theory of cleft extensions for a cocommutative weak Hopf algebra H. For a right H-comodule algebra, they obtain a bijective corre- spondence between the isomorphisms classes of H-cleft extensions AH → A, where AH is the subalgebra of coinvariants, and the equivalence classes of crossed systems for H over AH. Finally, they establish a bijection between the set of equivalence classes of crossed systems with a fixed weak H-module algebra structure and the second cohomology group H2φZ（AH） （H, Z（AH））, where Z（AH） is the center of AH.展开更多
We introduce the notions of a four-angle Hopf quasimodule and an adjoint quasiaction over a Hopf quasigroup H in a,symmetric monoidal category C.li H possesses an adjoint quasiaction,we show that symmetric Yetter-Drin...We introduce the notions of a four-angle Hopf quasimodule and an adjoint quasiaction over a Hopf quasigroup H in a,symmetric monoidal category C.li H possesses an adjoint quasiaction,we show that symmetric Yetter-Drinfeld categories are trivial,and hence we obtain a braided monoidal category equivalence between the category of right Yetter-Drinfeld modules over H and the category of four-angle Hopf modules over H under some suitable conditions.展开更多
In this paper we define the notion of Brauer Clifford group for(S,■)-Azumaya algebras when S is a commutative algebra and■is a(k,S)-Lie algebra over a commutative ring k.This is the situation that arises in applicat...In this paper we define the notion of Brauer Clifford group for(S,■)-Azumaya algebras when S is a commutative algebra and■is a(k,S)-Lie algebra over a commutative ring k.This is the situation that arises in applications having connections to differential geometry.This Brauer-Clifford group turns out to be an example of a Brauer group of a.symmetric monoidal category.展开更多
基金Specialized Research Fund for the Doctoral Program of Higher Education(No.20060286006)the National Natural Science Founda-tion of China(No.10571026)
文摘The question of how the category of entwined modules can be made into a braided monoidal category is studied. First, the sufficient and necessary conditions making the category into a monoidal category are obtained by using the fact that if （A, C, ψ） is an entwining structure, then A × C can be made into an entwined module. The conditions are that the algebra and coalgebra in question are both bialgebras with some extra compatibility relations. Then given a monodial category of entwined modules, the braiding is constructed by means of a twisted convolution invertible map Q, and the conditions making the category form into a braided monoidal category are obtained similarly. Finally, the construction is applied to the category of Doi-Hopf modules and （α, β ）-Yetter-Drinfeld modules as examples.
基金Supported by the National Natural Science Foundation of China（11047030, 11171055） Supported by the Grant from China Scholarship Counci1（2011841026）
文摘In this paper, we categorify a Hom-associative algebra by imposing the Homassociative law up to some isomorphisms on the multiplication map and requiring that these isomorphisms satisfy the Pentagon axiom, and obtain a 2-Hom-associative algebra. On the other hand, we introduce the dual Hom-quasi-Hopf algebra and show that any dual Homquasi-Hopf algebras can be viewed as a 2-Hom-associative algebra.
文摘We define the right regular dual of an object X in a monoidal category l, and give several results regarding the weak rigid monoidal category. Based on the definition of the right regular dual, we construct a weak Hopf algebra structure of H = End（F） whenever （F, J） is a fiber functor from category l to Vec and every X ∈ l has a right regular dual. To conclude, we give a weak reconstruction theorem for a kind of weak Hopf algebra.
文摘Let A and H be Hopf algebra, T-smash product AT H generalizes twisted smash product A ＊ H. This paper shows a necessary and sufficient condition for T-smash product moduie category AT HM to be braided monoidal category.
基金Supported by the Natural Science Foundation of Jiangsu Province(Grant No.BK2012736)the Fund of Science and Technology Department of Guizhou Province(Grant No.2014GZ81365)
文摘In this paper, we first give the definitions of a crossed left π-H-comodules over a crossed weak Hopf π-algebra H, and show that the category of crossed left π-H-comodules is a monoidal category. Finally, we show that a family σ = {σα,β： Hα Hβ→ k}α,β∈πof k-linear maps is a coquasitriangular structure of a crossed weak Hopf π-algebra H if and only if the category of crossed left π-H-comodules over H is a braided monoidal category with braiding defined by σ.
文摘The problem of embedding the Tsallis, Rényi and generalized Rényi entropies in the framework of category theory and their axiomatic foundation is studied. To this end, we construct a special category MES related to measured spaces. We prove that both of the Rényi and Tsallis entropies can be imbedded in the formalism of category theory by proving that the same basic partition functional that appears in their definitions, as well as in the associated Lebesgue space norms, has good algebraic compatibility properties. We prove that this functional is both additive and multiplicative with respect to the direct product and the disjoint sum (the coproduct) in the category MES, so it is a natural candidate for the measure of information or uncertainty. We prove that the category MES can be extended to monoidal category, both with respect to the direct product as well as to the coproduct. The basic axioms of the original Rényi entropy theory are generalized and reformulated in the framework of category MES and we prove that these axioms foresee the existence of an universal exponent having the same values for all the objects of the category MES. In addition, this universal exponent is the parameter, which appears in the definition of the Tsallis and Rényi entropies. It is proved that in a similar manner, the partition functional that appears in the definition of the Generalized Rényi entropy is a multiplicative functional with respect to direct product and additive with respect to the disjoint sum, but its symmetry group is reduced compared to the case of classical Rényi entropy.
基金The National Natural Science Foundation of China(No.10871042)
文摘Let （C, C） be a braided monoidal category. The relationship between the braided Lie algebra and the left Jacobi braided Lie algebra in the category （C, C） is investigated. First, a braided C2-commutative algebra in the category （C, C） is defined and three equations on the braiding in the category （C, C） are proved. Secondly, it is verified that （A, [, ] ） is a left （strict） Jacobi braided Lie algebra if and only if （A, [, ] ） is a braided Lie algebra, where A is an associative algebra in the category （C, C）. Finally, as an application, the structures of braided Lie algebras are given in the category of Yetter-Drinfel＇d modules and the category of Hopf bimodules.
基金Supported by the National Natural Science Foundation of China(No.11601486)Foundation of Zhejiang Educational Commitee(No.Y201738645)Project of Zhejiang College,Shanghai University of Finance and Economics(No.2018YJYB01).
文摘We investigate how the category of comodules of bimonads can be made into a monoidal category.It suffices that the monad and comonad in question are bimonads,with some extra compatibility relation.On a monoidal category of comodules of bimonads,we cons true t a braiding and get the necessary and sufficien t conditions making it a braided monoidal category.As an application,we consider the category of comodules of corings and the category of entwined modules.
基金The National Natural Science Foundation of China(No.11371088,11571173,11871144)。
文摘Firstly,the notion of the left-left Yetter-Drinfeld quasicomodule M=(M,·,ρ)over a Hopf coquasigroup H is given,which generalizes the left-left Yetter-Drinfeld module over Hopf algebras.Secondly,the braided monoidal category HHYDQCM is introduced and the specific structure maps are given.Thirdly,Sweedler's dual of infinite-dimensional Hopf algebras in HHYDQCM is discussed.It proves that if(B,mB,μB,ΔB,εB)is a Hopf algebra in HHYDQCM with antipode SB,then(B^0,(mB0)^op,εB^*,(ΔB0)^op,μB^*)is a Hopf algebra in HHYDQCM with antipode SB^*,which generalizes the corresponding results over Hopf algebras.
基金Supported by the National Natural Science Foundation of China(Grant Nos.110470301117105511071147)
文摘In this paper, we introduce the dual Hom-quasi-Hopf algebra and prove that the comodules category of a （braided） dual Hom-quasi-bialgebra is a monoidal category. Finally, we give a categorical realization of dual Hom-quasi-Hopf algebras.
基金Ministerio de Educacidn y Ciencia Projects MTM2006-14908-C02-01,MTM2007-62427FEDER
文摘In this paper, we give a necessary and sufficient condition for a comodule algebra over a weak Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the Hopf algebra setting. Also, from these results, we deduce a version of Maschke＇s Theorem for （H, B）-Hopf modules associated with a weak Hopf algebra H and a right H-comodule algebra B.
文摘In this paper, we give a sufficient condition for double crossproduct X A to be X A for some skew pairing T if X A is a 2-cocycle deformation of X A. Then we give a sufficient and necessary condition for X A to be X A by using natural isomorphism terminology.
基金supported by National Natural Science Foundation of China(Grant Nos.11571316 and 11001245)Natural Science Foundation of Zhejiang Province(Grant No.LY16A010003)
文摘We introduce the notions of differential graded(DG) Poisson algebra and DG Poisson module. Let A be any DG Poisson algebra. We construct the universal enveloping algebra of A explicitly, which is denoted by A^(ue). We show that A^(ue) has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over A is isomorphic to the category of DG modules over A^(ue). Furthermore, we prove that the notion of universal enveloping algebra A^(ue) is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.
基金Supported by Ministerio de Educació n, Xunta de Galicia and by FEDER (Grant Nos. MTM2010-15634,MTM2009-14464-C02-01, PGIDT07PXB322079PR)
文摘In this paper we obtain a criterion under which the bijectivity of the canonical morphism of a weak Galois extension associated to a weak invertible entwining structure is equivalent to the existence of a strong connection form. Also we obtain an explicit formula for a strong connection under equivariant projective conditions or under coseparability conditions.
基金Acknowledgements The authors would like to thank the referees for a number of helpful comments that greatly improved the presentation of this paper. The first author also thanks Prof. Ke Wu and Prof. Shikun Wang for stimulating discussion and help in preparation of this paper. This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11047030, 11171055, 11571145） and the Science and Technology Program of Henan Province （No. 152300410061）.
文摘We give a monoidal category approach to Hom-coassociative coalgebra by imposing the Hom-coassociative law up to some isomorphisms on the comultiplication map and requiring that these isomorphisms satisfy the copentagon axiom and obtain a Hom-coassociative 2-coalgebra, which is a 2- category. Second, we characterize Hom-bialgebras in terms of their categories of modules. Finally, we give a categorical realization of Hom-quasi-Hopf algebras using Hom-coassociative 2-coalgebra.
基金supported by CONICYT/FONDECYT/INICIACIOóN(Grant No.11170394)。
文摘While the Yoneda embedding and its generalizations have been studied extensively in the literature,the so-called tensor embedding has only received a little attention.In this paper,we study the tensor embedding for closed symmetric monoidal categories and show how it is connected to the notion of geometrically purity,which has recently been investigated in the works of Enochs et al.(2016)and Estrada et al.(2017).More precisely,for a Grothendieck cosmos,i.e.,a bicomplete Grothendieck category V with a closed symmetric monoidal structure,we prove that the geometrically pure exact category(V,ε■)has enough relative injectives;in fact,every object has a geometrically pure injective envelope.We also show that for some regular cardinalλ,the tensor embedding yields an exact equivalence between(V,ε■)and the category ofλ-cocontinuous V-functors from Presλ(V)to V,where the former is the full V-subcategory ofλ-presentable objects in V.In many cases of interest,λcan be chosen to be■0 and the tensor embedding identifies the geometrically pure injective objects in V with the(categorically)injective objects in the abelian category of V-functors from fp(V)to V.As we explain,the developed theory applies,e.g.,to the category Ch(R)of chain complexes of modules over a commutative ring R and to the category Qcoh(X)of quasi-coherent sheaves over a(suitably nice)scheme X.
基金supported by Xunta de Galicia (Grant No. PGIDT07PXB322079PR)Ministerio de Educación (Grant Nos. MTM2007-62427, MTM2006-14908-CO2-01)FEDER
文摘In this paper, we present the general theory and universal properties of weak crossed biproducts. We prove that every weak projection of weak bialgebras induces one of these weak crossed structures. Finally, we compute explicitly the weak crossed biproduct associated with a groupoid that admits an exact factorization.
基金supported by the project of Ministerio de Ciencia e Innovación(No.MTM2010-15634)Fondo Europeo de Desarrollo Regional
文摘The authors present the general theory of cleft extensions for a cocommutative weak Hopf algebra H. For a right H-comodule algebra, they obtain a bijective corre- spondence between the isomorphisms classes of H-cleft extensions AH → A, where AH is the subalgebra of coinvariants, and the equivalence classes of crossed systems for H over AH. Finally, they establish a bijection between the set of equivalence classes of crossed systems with a fixed weak H-module algebra structure and the second cohomology group H2φZ（AH） （H, Z（AH））, where Z（AH） is the center of AH.
基金supported by the National Natural Science Foundation of China(Grant No.11871144)the NNSF of Jiangsu Province(No.BK20171348)the Scientific Research Foundation of Nanjing Institute of Technology(No.YKJ202040).
文摘We introduce the notions of a four-angle Hopf quasimodule and an adjoint quasiaction over a Hopf quasigroup H in a,symmetric monoidal category C.li H possesses an adjoint quasiaction,we show that symmetric Yetter-Drinfeld categories are trivial,and hence we obtain a braided monoidal category equivalence between the category of right Yetter-Drinfeld modules over H and the category of four-angle Hopf modules over H under some suitable conditions.
文摘In this paper we define the notion of Brauer Clifford group for(S,■)-Azumaya algebras when S is a commutative algebra and■is a(k,S)-Lie algebra over a commutative ring k.This is the situation that arises in applications having connections to differential geometry.This Brauer-Clifford group turns out to be an example of a Brauer group of a.symmetric monoidal category.
